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Page 16-34

periodicity in the graph of the series.  This periodicity is easy to visualize by 
expanding the horizontal range of the plot to (-0.5,4):

Fourier series for a triangular wave

Consider the function 

which we assume to be periodic with period T = 2.  This function can be 
defined in the calculator, in ALG mode, by the expression

DEFINE(‘g(X) = IFTE(X<1,X,2-X)’)

If you started this example after finishing example 1 you already have a value 
of 2 stored in CAS variable PERIOD.  If you are not sure, check the value of this 
variable, and store a 2 in it if needed.   The coefficient c

0

 for the Fourier series 

is calculated as follows:

     

The calculator will request a change to Approx mode because of the integration 
of the function IFTE() included in the integrand.  Accepting, the change to 
Approx produces c

0

 = 0.5.  If we now want to obtain a generic expression for 

the coefficient c

n

 use:

⎩

⎨

⎧

<

<

−

<

<

=

2

1

,

2

1

0

,

)

(

x

if

x

x

if

x

x

g

Содержание 50G

Страница 1: ...HP g graphing calculator user s guide H Edition 1 HP part number F2229AA 90006 ...

Страница 2: ...AR PURPOSE HEWLETT PACKARD CO SHALL NOT BE LIABLE FOR ANY ERRORS OR FOR INCIDENTAL OR CONSEQUENTIAL DAMAGES IN CONNECTION WITH THE FURNISHING PERFORMANCE OR USE OF THIS MANUAL OR THE EXAMPLES CONTAINED HEREIN 2003 2006 Hewlett Packard Development Company L P Reproduction adaptation or translation of this manual is prohibited without prior written permission of Hewlett Packard Company except as all...

Страница 3: ...3 is entered to complete the operation The ALG mode on the other hand mimics the way you type arithmetic expressions in paper Thus the operation 2 3 in ALG mode will be entered in the calculator by pressing the keys 2 and 3 in that order To complete the operation we use the ENTER key Examples of applications of the different functions and operations in this calculator are illustrated in this user ...

Страница 4: ...d cable provided with your calculator you can connect your calculator with other calculators or computers This allows for fast and efficient exchange of programs and data with other calculators or computers The calculator provides a flash memory card port to facilitate storage and exchange of data with other users The programming capabilities of the calculator allow you or other users to develop e...

Страница 5: ...nd date 1 7 Introducing the calculator s keyboard 1 11 Selecting calculator modes 1 12 Operating Mode 1 13 Number Format and decimal dot or comma 1 17 Angle Measure 1 23 Coordinate System 1 24 Beep Key Click and Last Stack 1 25 Selecting CAS settings 1 26 Selecting Display modes 1 27 Selecting the display font 1 27 Selecting properties of the line editor 1 28 Selecting properties of the Stack 1 28...

Страница 6: ...editing summations derivatives and integrals 2 29 Organizing data in the calculator 2 33 Functions for manipulation of variables 2 34 The HOME directory 2 35 The CASDIR sub directory 2 35 Typing directory and variable names 2 37 Creating subdirectories 2 39 Moving among subdirectories 2 43 Deleting subdirectories 2 43 Variables 2 47 Creating variables 2 47 Checking variables contents 2 52 Replacin...

Страница 7: ...quare roots 3 5 Powers and roots 3 5 Base 10 logarithms and powers of 10 3 5 Using powers of 10 in entering data 3 6 Natural logarithms and exponential function 3 6 Trigonometric functions 3 6 Inverse trigonometric functions 3 6 Differences between functions and operators 3 7 Real number functions in the MTH menu 3 7 Hyperbolic functions and their inverses 3 9 Real number functions 3 11 Special fu...

Страница 8: ...culator to COMPLEX mode 4 1 Entering complex numbers 4 2 Polar representation of a complex number 4 3 Simple operations with complex numbers 4 4 Changing sign of a complex number 4 5 Entering the unit imaginary number 4 5 The CMPLX menus 4 5 CMPLX menu through the MTH menu 4 6 CMPLX menu in keyboard 4 7 Functions applied to complex numbers 4 8 Functions from the MTH menu 4 9 Function DROITE equati...

Страница 9: ...nu 5 9 DIVIS 5 9 FACTORS 5 9 LGCD 5 10 PROPFRAC 5 10 SIMP2 5 10 INTEGER menu 5 10 POLYNOMIAL menu 5 10 MODULO menu 5 11 Applications of the ARITHMETIC menu 5 12 Modular arithmetic 5 12 Finite arithmetic rings in the calculator 5 14 Polynomials 5 17 Modular arithmetic with polynomials 5 17 The CHINREM function 5 17 The EGCD function 5 18 The GCD function 5 18 The HERMITE function 5 18 The HORNER fu...

Страница 10: ...s 5 25 The CONVERT Menu and algebraic operations 5 26 UNITS convert menu Option 1 5 26 BASE convert menu Option 2 5 27 TRIGONOMETRIC convert menu Option 3 5 27 MATRICES convert menu Option 5 5 27 REWRITE convert menu Option 4 5 27 Chapter 6 Solution to single equations 6 1 Symbolic solution of algebraic equations 6 1 Function ISOL 6 1 Function SOLVE 6 2 Function SOLVEVX 6 3 Function ZEROS 6 4 Nume...

Страница 11: ...a lake into an open channel 7 5 Using the Multiple Equation Solver MES 7 9 Application 1 Solution of triangles 7 9 Application 2 Velocity and acceleration in polar coordinates 7 17 Chapter 8 Operations with lists 8 1 Definitions 8 1 Creating and storing lists 8 1 Composing and decomposing lists 8 2 Operations with lists of numbers 8 2 Changing sign 8 3 Addition subtraction multiplication division ...

Страница 12: ...efinitions 9 1 Entering vectors 9 2 Typing vectors in the stack 9 2 Storing vectors into variables 9 3 Using the Matrix Writer MTRW to enter vectors 9 3 Building a vector with ARRY 9 6 Identifying extracting and inserting vector elements 9 7 Simple operations with vectors 9 9 Changing sign 9 9 Addition subtraction 9 9 Multiplication by a scalar and division by a scalar 9 9 Absolute value function ...

Страница 13: ...w vector 9 21 Transforming a list into a vector 9 23 Transforming a vector or matrix into a list 9 24 Chapter 10 Creating and manipulating matrices 10 1 Definitions 10 1 Entering matrices in the stack 10 2 Using the Matrix Writer 10 2 Typing in the matrix directly into the stack 10 3 Creating matrices with calculator functions 10 3 Functions GET and PUT 10 6 Functions GETI and PUTI 10 6 Function S...

Страница 14: ... CSWP 10 20 Manipulating matrices by rows 10 21 Function ROW 10 22 Function ROW 10 23 Function ROW 10 23 Function ROW 10 24 Function RSWP 10 24 Function RCI 10 25 Function RCIJ 10 25 Chapter 11 Matrix Operations and Linear Algebra 11 1 Operations with matrices 11 1 Addition and subtraction 11 2 Multiplication 11 2 Characterizing a matrix The matrix NORM menu 11 7 Function ABS 11 8 Function SNRM 11...

Страница 15: ...ian and Gauss Jordan elimination 11 29 Step by step calculator procedure for solving linear systems 11 38 Solution to linear systems using calculator functions 11 41 Residual errors in linear system solutions Function RSD 11 44 Eigenvalues and eigenvectors 11 45 Function PCAR 11 45 Function EGVL 11 46 Function EGV 11 46 Function JORDAN 11 47 Function MAD 11 48 Matrix factorization 11 49 Function L...

Страница 16: ...12 8 Graph of ln X 12 8 Graph of the exponential function 12 10 The PPAR variable 12 11 Inverse functions and their graphs 12 11 Summary of FUNCTION plot operation 12 13 Plots of trigonometric and hyperbolic functions 12 16 Generating a table of values for a function 12 17 The TPAR variable 12 17 Plots in polar coordinates 12 18 Plotting conic curves 12 20 Parametric plots 12 22 Generating a table...

Страница 17: ...e drawing 12 43 DOT and DOT 12 44 MARK 12 44 LINE 12 44 TLINE 12 45 BOX 12 45 CIRCL 12 45 LABEL 12 45 DEL 12 46 ERASE 12 46 MENU 12 46 SUB 12 46 REPL 12 46 PICT 12 46 X Y 12 47 Zooming in and out in the graphics display 12 47 ZFACT ZIN ZOUT and ZLAST 12 47 BOXZ 12 48 ZDFLT ZAUTO 12 48 HZIN HZOUT VZIN and VZOUT 12 48 CNTR 12 48 ZDECI 12 48 ZINTG 12 48 ZSQR 12 49 ZTRIG 12 49 ...

Страница 18: ...of derivatives 13 7 Analyzing graphics of functions 13 8 Function DOMAIN 13 9 Function TABVAL 13 9 Function SIGNTAB 13 10 Function TABVAR 13 10 Using derivatives to calculate extreme points 13 12 Higher order derivatives 13 13 Anti derivatives and integrals 13 14 Functions INT INTVX RISCH SIGMA and SIGMAVX 13 14 Definite integrals 13 15 Step by step evaluation of derivatives and integrals 13 16 In...

Страница 19: ...s of two variables 14 5 Using function HESS to analyze extrema 14 6 Multiple integrals 14 8 Jacobian of coordinate transformation 14 9 Double integral in polar coordinates 14 9 Chapter 15 Vector Analysis Applications 15 1 Definitions 15 1 Gradient and directional derivative 15 1 A program to calculate the gradient 15 2 Using function HESS to obtain the gradient 15 2 Potential of a gradient 15 3 Di...

Страница 20: ...series 16 26 Function FOURIER 16 28 Fourier series for a quadratic function 16 28 Fourier series for a triangular wave 16 34 Fourier series for a square wave 16 38 Fourier series applications in differential equations 16 40 Fourier Transforms 16 42 Definition of Fourier transforms 16 45 Properties of the Fourier transform 16 47 Fast Fourier Transform FFT 16 47 Examples of FFT applications 16 48 So...

Страница 21: ...s 17 1 The MTH PROBABILITY sub menu part 1 17 1 Factorials combinations and permutations 17 1 Random numbers 17 2 Discrete probability distributions 17 3 Binomial distribution 17 4 Poisson distribution 17 5 Continuous probability distributions 17 6 The gamma distribution 17 6 The exponential distribution 17 6 The beta distribution 17 7 The Weibull distribution 17 7 Functions for continuous distrib...

Страница 22: ...ions 18 19 Confidence intervals 18 22 Estimation of Confidence Intervals 18 23 Definitions 18 23 Confidence intervals for the population mean when the population vari ance is known 18 24 Confidence intervals for the population mean when the population vari ance is unknown 18 24 Confidence interval for a proportion 18 25 Sampling distribution of differences and sums of statistics 18 25 Confidence i...

Страница 23: ... testing in linear regression 18 52 Procedure for inference statistics for linear regression using the calcula tor 18 54 Multiple linear fitting 18 57 Polynomial fitting 18 59 Selecting the best fitting 18 62 Chapter 19 Numbers in Different Bases 19 1 Definitions 19 1 The BASE menu 19 1 Functions HEX DEC OCT and BIN 19 2 Conversion between number systems 19 3 Wordsize 19 4 Operations with binary i...

Страница 24: ...pe 21 5 The PRG menu 21 5 Navigating through RPN sub menus 21 6 Functions listed by sub menu 21 7 Shortcuts in the PRG menu 21 9 Keystroke sequence for commonly used commands 21 10 Programs for generating lists of numbers 21 13 Examples of sequential programming 21 15 Programs generated by defining a function 21 15 Programs that simulate a sequence of stack operations 21 17 Interactive input in pr...

Страница 25: ...e FOR construct 21 59 The DO construct 21 61 The WHILE construct 21 63 Errors and error trapping 21 64 DOERR 21 64 ERRN 21 65 ERRM 21 65 ERR0 21 65 LASTARG 21 65 Sub menu IFERR 21 65 User RPL programming in algebraic mode 21 67 Chapter 22 Programs for graphics manipulation 22 1 The PLOT menu 22 1 User defined key for the PLOT menu 22 1 Description of the PLOT menu 22 2 Generating plots with progra...

Страница 26: ... 22 27 More information on the ANIMATE function 22 29 Graphic objects GROBs 22 29 The GROB menu 22 31 A program with plotting and drawing functions 22 33 Modular programming 22 35 Running the program 22 36 A program to calculate principal stresses 22 38 Ordering the variables in the sub directory 22 38 A second example of Mohr s circle calculations 22 39 An input form for the Mohr s circle program...

Страница 27: ...ing time and date 25 2 TIME Tools 25 2 Calculations with dates 25 3 Calculating with times 25 4 Alarm functions 25 4 Chapter 26 Managing memory 26 1 Memory Structure 26 1 The HOME directory 26 2 Port memory 26 2 Checking objects in memory 26 3 Backup objects 26 4 Backing up objects in port memory 26 4 Backing up and restoring HOME 26 5 Storing deleting and restoring backup objects 26 6 Using data ...

Страница 28: ...hapter 27 The Equation Library 27 1 Solving a Problem with the Equation Library 27 1 Using the Solver 27 2 Using the menu keys 27 3 Browsing in the Equation Library 27 4 Viewing equations 27 4 Viewing variables and selecting units 27 5 Viewing the picture 27 5 Using the Multiple Equation Solver 27 6 Defining a set of equations 27 8 Interpreting results from the Multiple Equation Solver 27 10 Check...

Страница 29: ...dix I Command catalog list I 1 Appendix J MATHS menu J 1 Appendix K MAIN menu K 1 Appendix L Line editor commands L 1 Appendix M Table of Built In Equations M 1 Appendix N Index N 1 Limited Warranty LW 1 Service LW 2 Regulatory information LW 4 Disposal of Waste Equipment by Users in Private Household in the European Union LW 6 ...

Страница 30: ... uses 4 AAA LR03 batteries as main power and a CR2032 lithium battery for memory backup Before using the calculator please install the batteries according to the following procedure To install the main batteries a Make sure the calculator is OFF Slide up the battery compartment cover as illustrated b Insert 4 new AAA LR03 batteries into the main compartment Make sure each battery is inserted in th...

Страница 31: ...ted at the lower left corner of the keyboard Press it once to turn your calculator on To turn the calculator off press the right shift key first key in the second row from the bottom of the keyboard followed by the key Notice that the key has a OFF label printed in the upper right corner as a reminder of the OFF command Adjusting the display contrast You can adjust the display contrast by holding ...

Страница 32: ...ive You can then navigate through the file directory tree to select any directory of interest As you navigate through the file directory the second line of the display will change to reflect the proper file directory and sub directory At the bottom of the display you will find a number of labels namely EDIT VIEW RCL STO PURGE CLEAR associated with the six soft menu keys F1 through F6 ABCDEF The si...

Страница 33: ...SE boxes Menus or SOFT menus associate labels in the lower part of the screen with the six soft menu keys Athrough F By pressing the appropriate soft menu key the function shown in the associated label gets activated For example with the TOOL menu active pressing the CLEAR key F activates function CLEAR which erases clears up the contents of the screen To see this function in action type a number ...

Страница 34: ... CHOOSE box use To move to the bottom of the current page use To move to the top of the entire menu use To move to the bottom of the entire menu use Selecting SOFT menus or CHOOSE boxes You can select the format in which your menus will be displayed by changing a setting in the calculator system flags A system flag is a calculator variable that controls a certain calculator operation or mode For m...

Страница 35: ...akes us back to the first menu page To revert to the CHOOSE boxes setting use H FLAGS CHK OK OK Note With the SOFT menu setting for system flag 117 the keystroke combination hold will show a list of the functions in the current soft menu For example for the two first pages in the BASE menu you will get Notes 1 The TOOL menu obtained by pressing I will always produce a SOFT menu 2 Most of the examp...

Страница 36: ...key is the third key from the left in the third row of keys in the keyboard In this case only the first two soft menu keys have commands associated with them These commands are CASCM A CASCMD CAS CoMmanD used to launch a command from the CAS by selecting from a list HELP B HELP facility describing the commands available Pressing the L key will show the original TOOL menu Another way to recover the...

Страница 37: ...he OK soft menu key The following input form see Appendix 1 A for adjusting time and date is shown Setting the time of the day Using the number keys 1234567890 start by adjusting the hour of the day Suppose that we change the hour to 11 by pressing 11 as the hour field in the SET TIME AND DATE input form is highlighted This results in the number 11 being entered in the lower line of the input form...

Страница 38: ... Θ If using the W key the setting in the time format field will change to either of the following options o AM indicates that displayed time is AM time o PM indicates that displayed time is PM time o 24 hr indicates that that the time displayed uses a 24 hour format where18 00 for example represents 6pm The last selected option will become the set option for the time format by using this procedure...

Страница 39: ...t set the date format The default format is M D Y month day year To modify this format press the down arrow key This will highlight the date format as shown below Use the CHOOS soft menu key to see the options for the date format Highlight your choice by using the up and down arrow keys and press the OK soft menu key to make the selection ...

Страница 40: ...eys combined with 3 5 or 6 columns Row 1 has 6 keys rows 2 and 3 have 3 keys each and rows 4 through 10 have 5 keys each There are 4 arrow keys located on the right hand side of the keyboard in the space occupied by rows 2 and 3 Each key has three four or five functions The main key function correspond to the most prominent label in the key Also the left shift key key 8 1 the right ...

Страница 41: ... functions associated with the key only the first four are shown in the keyboard itself This is the way that the key looks in the keyboard Notice that the color and the position of the labels in the key namely SYMB MTH CAT and P indicate which is the main function SYMB and which of the other three functions is associated with the left shift MTH right shift CAT and P keys For detailed information o...

Страница 42: ...he RPN mode To select an operating mode first open the CALCULATOR MODES input form by pressing the H button The Operating Mode field will be highlighted Select the Algebraic or RPN operating mode by either using the key second from left in the fifth row from the keyboard bottom or pressing the CHOOS soft menu key If using the latter approach use up and down arrow keys to select the mode and press ...

Страница 43: ...for writing the expression shown above use the following keystrokes OR3 Ü5 1 3 3 23Q3 2 5 After pressing the calculator displays the expression 3 5 1 3 3 23 3 EXP 2 5 Pressing again will provide the following value Accept Approx mode on if asked by pressing OK Note The integer values used above e g 3 5 1 represent exact values The EXP 2 5 however cannot be expressed as an exact value therefore a s...

Страница 44: ...ck levels Entering 3 puts the number 3 in stack level 1 Next entering 2pushes the 3 upwards to occupy stack level 2 Finally by pressing we are telling the calculator to apply the operator or program to the objects occupying levels 1 and 2 The result 5 is then placed in level 1 Let s try some other simple operations before trying the more complicated expression used earlier for the algebraic operat...

Страница 45: ... 5 3 4905156 into 1 Although RPN requires a little bit more thought than the algebraic ALG mode there are multiple advantages in using RPN For example in RPN mode you can see the equation unfolding step by step This is extremely useful to detect a possible input error Also as you become more efficient in this mode and learn more of the tricks you will be able to calculate expression faster and wil...

Страница 46: ...an use one of the following shortcuts Θ In ALG mode CF 95 selects RPN mode Θ In RPN mode 95 SF selects ALG mode For more information on calculator s system flags see Chapter 2 Number Format and decimal dot or comma Changing the number format allows you to customize the way real numbers are displayed by the calculator You will find this feature extremely useful in operations with powers of tens or ...

Страница 47: ... Press the key The number is rounded to the maximum 12 significant figures and is displayed as follows In the standard format of decimal display integer numbers are shown with no decimal zeros whatsoever Numbers with different decimal figures will be adjusted in the display so that only those decimal figures that are necessary will be shown More examples of numbers in standard format are shown nex...

Страница 48: ...ecision As we change the number of decimals to be displayed you will see the additional digits being shown again Θ Fixed format with decimals This mode is mainly used when working with limited precision For example if you are doing financial calculation using a FIX 2 mode is convenient as it can easily represent monetary units to a 1 100 precision Press the H button Next use the down arrow key to ...

Страница 49: ...the digit after 6 is 5 Θ Scientific format The scientific format is mainly used when solving problems in the physical sciences where numbers are usually represented as a number with limited precision multiplied by a power of ten To set this format start by pressing the H button Next use the down arrow key to select the option Number format Press the CHOOS soft menu key and select the option Scient...

Страница 50: ...int Scientific notation always includes one integer figure as shown above For this case therefore the number of significant figures is four Θ Engineering format The engineering format is very similar to the scientific format except that the powers of ten are multiples of three To set this format start by pressing the H button Next use the down arrow key to select the option Number format Press the...

Страница 51: ...as Θ Decimal comma vs decimal point Decimal points in floating point numbers can be replaced by commas if the user is more familiar with such notation To replace decimal points for commas change the FM option in the CALCULATOR MODES input form to commas as follows Notice that we have changed the Number Format to Std Θ Press the H button Next use the down arrow key once and the right arrow key high...

Страница 52: ...r in a complete circumference or π 2 radians π 2 r in a right angle This notation is mainly used when solving mathematics and physics problems This is the default mode of the calculator Θ Grades There are 400 grades 400 g in a complete circumference or 100 grades 100 g in a right angle This notation is similar to the degree mode and was introduced in order to simplify the degree notation but is se...

Страница 53: ... three linear coordinates x y z measured from the origin along each of three mutually perpendicular axes in 2 d mode z is assumed to be 0 In a Cylindrical or Polar coordinate system the coordinates of a point are given by r θ z where r is a radial distance measured from the origin on the xy plane θ is the angle that the radial distance r forms with the positive x axis measured as positive in a cou...

Страница 54: ...t the preferred mode and press the OK soft menu key to complete the operation For example in the following screen the Polar coordinate mode is selected Beep Key Click and Last Stack The last line of the CALCULATOR MODES input form include the options _Beep _Key Click _Last Stack By choosing the check mark next to each of these options the corresponding option is activated These options are describ...

Страница 55: ...he left arrow key š to select the _Key Click option Use the CHK soft menu key to change the selection Θ Press the left arrow key š to select the _Beep option Use the CHK soft menu key to change the selection Press the OK soft menu key to complete the operation Selecting CAS settings CAS stands for Computer Algebraic System This is the mathematical core of the calculator where the symbolic mathemat...

Страница 56: ...n of interest e g the _Small _Full page and _Indent options in the Edit line above Θ To select the Font for the display highlight the field in front of the Font option in the DISPLAY MODES input form and use the CHOOS soft menu key Θ After having selected and unselected all the options that you want in the DISPLAY MODES input form press the OK soft menu key This will take you back to the CALCULATO...

Страница 57: ...ay change to accommodate the different font Selecting properties of the line editor First press the H button to activate the CALCULATOR MODES input form Within the CALCULATOR MODES input form press the DISP soft menu key to display the DISPLAY MODES input form Press the down arrow key once to get to the Edit line This line shows three properties that can be modified When these properties are selec...

Страница 58: ...ings either in algebraic or RPN mode use the equation writer to type the following definite integral O Á0 è x x In Algebraic mode the following screen shows the result of these keystrokes with neither _Small nor _Textbook are selected With the _Small option selected only the display looks as shown below With the _Textbook option selected default value regardless of whether the _Small option is sel...

Страница 59: ...n the CALCULATOR MODES input form press the DISP soft menu key to display the DISPLAY MODES input form Press the down arrow key four times to get to the Header line The value 2 is assigned to the Header field by default This means that the top part of the display will contain two lines one showing the current settings of the calculator and a second one showing the current sub directory within the ...

Страница 60: ...ption is selected the time of the day and date will be shown in the upper right corner of the display If the _Analog option is also selected an analog clock rather than a digital clock will be shown in the upper right corner of the display If the _Clock option is not selected or the header is not present or too small the date and time of day will not be shown in the display ...

Страница 61: ... Note that real must be entered with a decimal point even if the number does not have a fractional part Otherwise the number is taken as an integer number which is a different calculator objects Reals behave as you would expect a number to when used in a mathematical operation Integers These objects represent integer numbers numbers without fractional part and do not have limits except the memory ...

Страница 62: ...tion that on reals or integers Vector and matrix operations utilize objects of type 3 real arrays and if needed type 4 complex arrays Objects type 2 strings are simply lines of text enclosed between quotes produced with the alphanumeric keyboard A list is just a collection of objects enclosed between curly brackets and separated by spaces in RPN mode the space key is labeled or by commas in algebr...

Страница 63: ...in a similar fashion as folders are used in a personal computer Libraries objects of type 16 are programs residing in memory ports that are accessible within any directory or sub directory in your calculator They resemble built in functions objects of type 18 and built in commands objects of type 19 in the way they are used Editing expressions on the screen In this section we present examples of e...

Страница 64: ...l mode with three decimal places see Chapter 1 In this case when the expression is entered directly into the stack As soon as you press the calculator will attempt to calculate a value for the expression If the expression is entered between quotes however the calculator will reproduce the expression as entered In the following example we enter the same expression as above but using quotes For this...

Страница 65: ... also set the CAS to Exact and the display to Textbook The keystrokes to enter the expression between quotes are the same used earlier i e 5 Ü1 1 7 5 ÜR3 2Q3 Resulting in the output Press once more to keep two copies of the expression available in the stack for evaluation We first evaluate the expression using the function EVAL and next using the function NUM Here are the steps explained in detail...

Страница 66: ... EXACT rather than the intended expression The incorrect expression was entered by using 5 Ü1 1 1 75 ÜR5 2Q3 To enter the line editor use The display now looks as follows Note Avoid mixing integer and real data to avoid conflicts in the calculations For many physical science and engineering applications including numerical solution of equation statistics applications etc the APPROX mode see Append...

Страница 67: ... key ƒ twice to erase the characters 1 Θ Press the right arrow key once to move the cursor to the right of the 7 Θ Type a decimal point with Θ Press the right arrow key until the cursor is immediately to the right of the 5 Θ Press the delete key ƒ once to erase the Character 5 Θ Type a 3 with 3 Θ Press to return to the stack The edited expression is now available in the stack Editing of a line of ...

Страница 68: ...similar to that of an arithmetic expression see exercise above Suppose that we want to modify the expression entered above to read To edit this algebraic expression using the line editor use This activates the line editor showing the expression to be edited as follows The editing cursor is shown as a blinking left arrow over the first character in the line to be edited As in an earlier exercise on...

Страница 69: ...ress the right arrow key once and the delete key ƒ once to delete the right parenthesis of the set inserted above Θ Press the right arrow key 4 times to move the cursor to the right of the b Θ Type Ü to enter a second set of parentheses Θ Press the delete key ƒ once to delete the left parenthesis of the set inserted above Θ Press to return to normal calculator display The result is shown next Noti...

Страница 70: ...ows you to modify and work apply functions on all or part of the equation The equation writer EQW therefore allows you to perform complex mathematical operations directly or in a step by step mode as you would do on paper when solving for example calculus problems The Equation Writer is launched by pressing the keystroke combination O the third key in the fourth row from the top in the keyboard Th...

Страница 71: ...shown below The six soft menu keys for the Equation Writer activate the following functions CMDS allows access to the collection of CAS commands listed in alphabetical order This is useful to insert CAS commands in an expression available in the Equation Writer HELP activates the calculator s CAS help facility to provide information and examples of CAS commands Some examples for the use of the Equ...

Страница 72: ...ession looks as follows Suppose that you want to replace the quantity between parentheses in the denominator i e 5 1 3 with 5 π2 2 First we use the delete key ƒ delete the current 1 3 expression and then we replace that fraction with π2 2 as follows ƒƒƒ ìQ2 When we hit this point the screen looks as follows In order to insert the denominator 2 in the expression we need to highlight the entire π2 e...

Страница 73: ...e right arrow or the upper arrow keys repeatedly until the entire expression is highlighted i e seven times producing Once the expression is highlighted as shown above type 1 3 to add the fraction 1 3 Resulting in NOTE Alternatively from the original position of the cursor to the right of the 2 in the denominator of π2 2 we can use the keystroke combination interpreted as 3 1 2 5 2 5 5 2 π ...

Страница 74: ...hin the Equation Writer highlight the part that you want to evaluate and press the EVAL soft menu key For example to evaluate the entire expression in this exercise first highlight the entire expression by pressing Then press the EVAL soft menu key If your calculator is set to Exact CAS mode i e the _Approx CAS mode is not checked then you will get the following symbolic result If you want to reco...

Страница 75: ...e arrow keys to select that particular sub expression Here is a way to do it Highlights only the first fraction Highlights the numerator of the first fraction Highlights denominator of the first fraction Highlights first term in denominator of first fraction Highlights second term in denominator of first fraction Highlights first factor in second term in denominator of first fraction Highlights ex...

Страница 76: ...ing in Then press the EVAL soft menu key to obtain Let s try a numerical evaluation of this term at this point Use ï to obtain Let s highlight the fraction to the right and obtain a numerical evaluation of that term too and show the sum of these two decimal values in small font format by using ï C we get To highlight and evaluate the expression in the Equation Writer we use D resulting in ...

Страница 77: ...eatures of the Equation Editor to transform it into the following expression In the previous exercises we used the arrow keys to highlight sub expressions for evaluation In this case we will use them to trigger a special editing cursor After you have finished entering the original expression the typing cursor a left pointing arrow will be located to the right of the 3 in the denominator of the sec...

Страница 78: ...ce more to delete the 2 and then 3 to enter a 3 At this point the screen looks as follows Next press the down arrow key to trigger the clear editing cursor highlighting the 3 in the denominator of π 2 3 Press the left arrow key š once to highlight the exponent 2 in the expression π 2 3 Next press the delete key ƒ once to change the cursor into the insertion cursor Press ƒ once more to delete the 2...

Страница 79: ...r editing cursor In this mode use the left or right arrow keys š to move from term to term in an expression When you reach a point that you need to edit use the delete key ƒ to trigger the insertion cursor and proceed with the edition of the expression Creating algebraic expressions An algebraic expression is very similar to an arithmetic expression except that English and Greek letters may be inc...

Страница 80: ...roke combinations was listed in an earlier section The expression tree The expression tree is a diagram showing how the Equation Writer interprets an expression See Appendix E for a detailed example The CURS function The CURS function CURS in the Equation Writer menu the B key converts the display into a graphical display and produces a graphical cursor that can be controlled with the arrow keys š...

Страница 81: ...use the delete key ƒ to trigger the insertion cursor and proceed with the edition of the expression To see the clear editing cursor in action let s start with the algebraic expression that we entered in the exercise above Press the down arrow key at its current location to trigger the clear editing cursor The 3 in the exponent of θ will be highlighted Use the left arrow key š to move from element ...

Страница 82: ...ed the exercise immediately above you should have the clear editing cursor on the number 2 in the first factor in the expression Follow these keystrokes to edit the expression 2 Enters the factorial for the 3 in the square root entering the factorial changes the cursor to the selection cursor Selects the μ in the exponential function 3 f Modifies exponential function argument Selects Δy R Places a...

Страница 83: ...ied any more according to the CAS rules Trying the keystrokes D again does not produce any changes on the expression Another sequence of D keystrokes however modifies the expression as follows One more application of the D keystrokes produces more changes This expression does not fit in the Equation Writer screen We can see the entire expression by using a smaller size font Press the BIG soft menu...

Страница 84: ... argument of the SIN function namely transformed into This may not seem like a simplification but it is in the sense that the cubic root function has been replaced by the inverse functions exp LN Factoring an expression In this exercise we will try factoring a polynomial expression To continue the previous exercise press the key Then launch the Equation Writer again by pressing the O key Type the ...

Страница 85: ...nal expression Using the CMDS menu key With the original polynomial expression used in the previous exercise still selected press the L key to show the CMDS and HELP soft menu keys These two commands belong to the second part of the soft menu available with the Equation Writer Let s try this example as an application of the CMDS soft menu key Press the CMDS soft menu key to get the list of CAS com...

Страница 86: ...y to get Next press the L key to recover the original Equation Writer menu and press the EVAL soft menu key to evaluate this derivative The result is Using the HELP menu Press the L key to show the CMDS and HELP soft menu keys Press the HELP soft menu key to get the list of CAS commands Then press d to select the command DERVX Press the OK soft menu key to get information on the command DERVX ...

Страница 87: ... of these editing functions are as follows BEGIN marks the beginning of a string of characters for editing END marks the ending of a string of characters for editing COPY copies the string of characters selected by BEGIN and END CUT cuts the string of characters selected by BEGIN and END PASTE pastes a string of characters previously copied or cut into the current cursor position To see and exampl...

Страница 88: ...quation Writer since we can select strings of characters by using the arrow keys Functions BEGIN and END are more useful when editing an expression with the line editor For example let s select the expression x 2 λ Δy from this expression but using the line editor within the Equation Writer as follows A The line editor screen will look like this quotes shown only if calculator in RPN mode To selec...

Страница 89: ...re commonly used for calculus probability and statistics applications In this section we show some examples of such operations created with the equation writer Use ALG mode Summations We will use the Equation Writer to enter the following summation Press O to activate the Equation Writer Then press to enter the summation sign Notice that the sign when entered into the Equation Writer screen provid...

Страница 90: ... use To evaluate the summation again you can use the D soft menu key This shows again that You can use the Equation Writer to prove that This summation representing an infinite series is said to diverge Double summations are also possible for example Derivatives We will use the Equation Writer to enter the following derivative Press O to activate the Equation Writer Then press to enter the partial...

Страница 91: ...that the general expression for a derivative in the line editor or in the stack is variable function of variables Press to return to the Equation Writer The resulting screen is not the derivative we entered however but its symbolic value namely To recover the derivative expression use To evaluate the derivative again you can use the D soft menu key This shows again that Second order derivatives ar...

Страница 92: ...line editor press and the A soft menu key to show This indicates that the general expression for a derivative in the line editor or in the stack is lower_limit upper_limit integrand variable_of_integration Press to return to the Equation Writer The resulting screen is not the definite integral we entered however but its symbolic value namely To recover the derivative expression use To evaluate the...

Страница 93: ...ager screen This screen gives a snapshot of the calculator s memory and of the directory tree The screen shows that the calculator has three memory ports or memory partitions port 0 IRAM port 1 ERAM and port 2 FLASH Memory ports are used to store third party application or libraries as well as for backups The size of the three different ports is also indicated The fourth and subsequent lines in th...

Страница 94: ...tents of a highlighted variable EVAL To evaluate a highlighted variable TREE To see the directory tree where the variable is contained If you press the L key the next set of functions is made available PURGE To purge or delete a variable RENAM To rename a variable NEW To create a new variable ORDER To order a set of variables in the directory SEND To send a variable to another calculator or comput...

Страница 95: ...e the HOME directory contains nothing but the CASDIR Pressing J will show the variables in the soft menu keys Subdirectories To store your data in a well organized directory tree you may want to create subdirectories under the HOME directory and more subdirectories within subdirectories in a hierarchy of directories similar to folders in modern computers The subdirectories will be given names that...

Страница 96: ...EPS Column number 3 shows another specification for the variable type e g ALG means an algebraic expression GROB stands for graphics object INTG means an integer numeric variable LIST means a list of data GNAME means a global name and REAL means a real or floating point numeric variable The fourth and last column represents the size in bytes of the variable truncated without decimals i e nibbles T...

Страница 97: ...lies to both the Algebraic and RPN calculator operating modes Variables in CASDIR The default variables contained in the CASDIR directory are the following PRIMIT Latest primitive anti derivative calculated not a default variable but one created by a previous exercise CASINFO a graph that provides CAS information MODULO Modulo for modular arithmetic default 13 REALASSUME List of variable names ass...

Страница 98: ... locked in this fashion pressing the before a letter key produces an upper case letter To unlock lower case press To unlock the upper case locked keyboard press Let s try some exercises typing directory variable names in the stack Assuming that the calculator is in the Algebraic mode of operation although the instructions work as well in RPN mode try the following keystroke sequences With these co...

Страница 99: ...ctions activated in the FILES menu Press to activate the FILES menu If the HOME directory is not already highlighted in the screen i e use the up and down arrow keys to highlight it Then press the OK soft menu key The screen may look like this showing that only one object exists currently in the HOME directory namely the CASDIR sub directory Let s create another sub directory called MANS for MANua...

Страница 100: ...K soft menu key to specify that you are creating a directory and press OK to exit the input form The variable listing for the HOME directory will be shown in the screen as follows The screen indicates that there is a new directory MANS within the HOME directory Next we will create a sub directory named INTRO for INTROduction within MANS to hold variables created as exercise in subsequent sections ...

Страница 101: ...ows Press the INTRO soft menu key to move into the INTRO sub directory This will show an empty sub directory Later on we will do some exercises in creating variables Using the command CRDIR The command CRDIR can be used to create directories This command is available through the command catalog key the N key second key in fourth row of keys from the top of the keyboard through the programming menu...

Страница 102: ... CRDIR in Algebraic mode Once you have selected the CRDIR through one of the means shown above the command will be available in your stack as follows At this point you need to type a directory name say chap1 chap1 The name of the new directory will be shown in the soft menu keys e g Command CRDIR in RPN mode To use the CRDIR in RPN mode you need to have the name of the directory already available ...

Страница 103: ...to and then press the CHDIR CHange DIRectory or A soft menu key This will show the contents of the sub directory you moved to in the soft menu key labels Deleting subdirectories To delete a sub directory use one of the following procedures Using the FILES menu Press the key to trigger the FILES menu Select the directory containing the sub directory you want to delete and press the CHDIR if needed ...

Страница 104: ...l be returned to the screen listing the contents of the sub directory The ABORT command however will show an error message and you will have to press OK before returning to the variable listing Using the command PGDIR The command PGDIR can be used to purge directories Like the command CRDIR the PGDIR command is available through the N or through the key or it can simply be typed in Through the cat...

Страница 105: ...row key to select the 6 PGDIR option and press OK Command PGDIR in Algebraic mode Once you have selected the PGDIR through one of the means shown above the command will be available in your stack as follows At this point you need to type the name of an existing directory say S4 s4 As a result sub directory S4 is deleted Instead of typing the name of the directory you can simply press the correspon...

Страница 106: ...mode To use the PGDIR in RPN mode you need to have the name of the directory between quotes already available in the stack before accessing the command For example s2 Then access the PGDIR command by either of the means shown above e g through the N key Press the OK soft menu key to activate the command and delete the sub directory ...

Страница 107: ...ting with a letter either English or Greek Some non alphabetic characters such as the arrow can be used in a variable name if combined with an alphabetical character Thus A is a valid variable name but is not Valid examples of variable names are A B a b α β A1 AB12 A12 Vel Z0 z1 etc A variable can not have the same name than a function of the calculator You can not have a SIN variable for example ...

Страница 108: ...rectory as shown in this screen Press OK to enter the directory You will get a files listing with no entries the INTRO sub directory is empty at this point Press the L key to move to the next set of soft menu keys and press the NEW soft menu key This will produce the NEW VARIABLE input form Name Contents Type A 12 5 real α 0 25 real A12 3 105 real Q r m r algebraic R 3 2 1 vector z1 3 5i complex p...

Страница 109: ...g variable listing The listing indicates a real variable R whose name is A and that occupies 10 5 bytes of memory To see the contents of the variable in this screen press L VIEW Press the GRAPH soft menu key to see the contents in a graphical format Press the TEXT soft menu key to see the contents in text format Press OK to return to the variable list Press once more to return to normal display Va...

Страница 110: ...5 into variable α 0 25 K a AT this point the screen will look as follows This expression means that the value 0 25 is being stored into α the symbol suggests the operation Press to create the variable The variable is now shown in the soft menu key labels when you press J The following are the keystrokes required to enter the remaining variables A12 3V5K a12 Q r Ü m r K q R Ô3 í2 í1 K r Name Conten...

Страница 111: ... 0 25 on the level 2 of the stack and α on the level 1 of the stack you can use the K key to create the variable The variable is now shown in the soft menu key labels when you press J To enter the value 3 105 into A12 we can use a shorter version of the procedure 3V5 a12 K Here is a way to enter the contents of Q Q r Ü m r q K To enter the value of R we can use an even shorter version of the proce...

Страница 112: ...contents of a variable in an earlier exercise when we created the variable A In this section we will show a simple way to look into the contents of a variable Pressing the soft menu key label for the variable This procedure will show the contents of a variable as long as the variable contains a numerical value or an algebraic value or an array For example for the variables listed above press the f...

Страница 113: ... an input to be referred to as r is to be provided to the program The action from the program is to take that value of r and evaluate the algebraic π r 2 In the example shown above r took the value of 5 and thus the value of πr2 π 25 is returned This program therefore calculates the area of a circle given its radius r RPN mode In RPN mode you only need to press the corresponding soft menu key labe...

Страница 114: ...ebraic mode you need to place parentheses to enter the argument Using the right shift key followed by soft menu key labels In Algebraic mode you can display the content of a variable by pressing J and then the corresponding soft menu key Try the following examples J p1 z1 R Q A12 Note In RPN mode you don t need to press just J and then the corresponding soft menu key This produces the following sc...

Страница 115: ...the replacement of a variable s content Using the STO command Using as illustration the six variables p1 z1 R Q A12 a and A created earlier we will proceed to change the contents of variable A12 currently a numerical variable with the algebraic expression β 2 using the STO command First using the Algebraic operating mode b 2 K A12 Check the new contents of variable A12 by using A12 Using the RPN o...

Страница 116: ...g the contents of z1 to a bi is the following î K z1 To check the new contents of z1 use z1 Copying variables The following exercises show different ways of copying variables from one sub directory to another Using the FILES menu To copy a variable from one directory to another you can use the FILES menu For example within the sub directory HOME MANS INTRO we have variables p1 z1 R Q A12 α and A S...

Страница 117: ... HOME directory and press OK If you now press twice the screen will show the contents of the HOME directory including a copy of variable R Using the history in Algebraic mode Here is a way to use the history stack to copy a variable from one directory to another with the calculator set to the Algebraic mode Suppose that we are within the sub directory HOME MANS INTRO and want to copy the contents ...

Страница 118: ...nd the name of the variable in the stack The calculator screen will look like this Now use to move to the HOME directory and press K to complete the operation Use z1 to verify the contents of the variable Copying two or more variables using the stack in Algebraic mode The following is an exercise to demonstrate how to copy two or more variables using the stack when the calculator is in Algebraic m...

Страница 119: ... in a directory In this section we illustrate the use of the ORDER command to reorder the variables in a directory We assume we start within the sub directory HOME MANS containing the variables A12 R Q z1 A and the sub directory INTRO as shown below Copy A12 from INTRO into MANS Algebraic mode In this case we have the calculator set to Algebraic mode Suppose that we want to change the order of the...

Страница 120: ... ORDER from the DIRECTORY menu The result is the following screen Moving variables using the FILES menu To move a variable from one directory to another you can use the FILES menu For example within the sub directory HOME MANS INTRO we have variables p1 z1 R Q A12 α and A Suppose that we want to move variable A12 to sub directory HOME MANS Here is how to do it Press OK to show a variable list Use ...

Страница 121: ...hin the sub directory HOME MANS INTRO we have variables p1 z1 R Q α and A left Suppose that we delete variable A Here is how to do it Press OK to produce the variable list Use the down arrow key to select variable A the last in the list then press L PURGE YES The screen will now show the contents of sub directory INTRO without variable A Using function PURGE in the stack in Algebraic mode We start...

Страница 122: ... now show the remaining variables Using function PURGE in the stack in RPN mode We start again at subdirectory HOME MANS INTRO containing variables p1 z1 Q R and α We will use command PURGE to delete variable p1 Press p1 I PURGE The screen will now show variable p1 removed To delete two variables simultaneously say variables R and Q first create a list in RPN mode the elements of the list need not...

Страница 123: ...of CMD let s enter the following entries in ALG mode Press after each entry Next use the CMD function to show the four most recent commands entered by the user i e You can use the up and down arrow keys to navigate through these commands and highlight any of them that you want to entry anew Once you have selected the command to enter press OK The CMD function operates in the same fashion when the ...

Страница 124: ...urrent system flag setting press the H button and then the FLAGS soft menu key i e F1 You will get a screen labeled SYSTEM FLAGS listing flag numbers and the corresponding setting Note In this screen as only system flags are present only the absolute value of the flag number Is displayed A flag is said to be set if you see a check mark in front of the flag number Otherwise the flag is not set or c...

Страница 125: ... 01 i e select Principal Value Press OK twice to return to normal calculator display We will try solving a quadratic equation solution say t2 5t 6 0 with command QUAD Algebraic mode Use the following keystroke sequence N q use the up and down arrow keys to select command QUAD press OK To enter the equation as the first argument of function QUAD use the following keystrokes O t Q2 5 t 6 Å0 í t The ...

Страница 126: ...e current flag setting by pressing the H button and then the FLAGS soft menu key Make sure to clear system flag 01 which was left set from the previous exercise Use the up and down arrow keys to move about the system flag list Some flags of interest and their preferred value for the purpose of the exercises that follow in this manual are 02 Constant symb Constant values e g π are kept as symbols 0...

Страница 127: ... For example to use the ORDER command to reorder variables in a directory we use in algebraic mode Show PROG menu list and select MEMORY OK Show the MEMORY menu list and select DIRECTORY OK Show the DIRECTORY menu list and select ORDER OK activate the ORDER command There is an alternative way to access these menus as soft MENU keys by setting flag 117 To set this flag try the following H FLAGS ...

Страница 128: ...tart with Notice that instead of a menu list we get soft menu labels with the different options in the PROG menu i e Press B to select the MEMORY soft menu MEM The display now shows Press E to select the DIRECTORY soft menu DIR The ORDER command is not shown in this screen To find it we use the L key to find it To activate the ORDER command we press the C ORDER soft menu key Although not applied t...

Страница 129: ...amples in this guide you should clear the flag before continuing Selected CHOOSE boxes Some menus will only produce CHOOSE boxes e g The APPS APPlicationS menu activated with the G key first key in the second row of keys from the top of the keyboard The CAT CATalog menu activated with the N key second key in the fourth row of keys from the top of the keyboard The HELP menu activated with I L HELP ...

Страница 130: ...Page 2 70 The CMDS CoMmanDS menu activated within the Equation Writer i e O L CMDS ...

Страница 131: ...ple you may see the following setting RAD XYZ DEC R X This stands for RADians for angular measurements XYZ for Rectangular Cartesian coordinates DECimal number base Real numbers preferred means exact results and X is the value of the default independent variable Another possible listing of options could be DEG R Z HEX C t This stands for DEGrees as angular measurements R Z for Polar coordinates HE...

Страница 132: ...ack are listed in the left hand side of the screen When the ALGEBRAIC mode is selected there are no numbered stack levels and the word ALG is listed in the top line of the display towards the right hand side The difference between these operating modes was described in detail in Chapter 1 Real number calculations To perform real number calculations it is preferred to have the CAS set to Real as op...

Страница 133: ...to the object on the first level of the stack The inverse function Use the Ykey In ALG mode press Y first followed by a number or algebraic expression e g Y2 Result or 0 5 In RPN mode enter the number first then use the key e g 4 Y Result or 0 25 Addition subtraction multiplication division Use the proper operation key namely In ALG mode press an operand then an operator then an operand followed b...

Страница 134: ...n RPN mode you do not need the parenthesis calculation is done directly on the stack 5 3 2 7 2 2 In RPN mode typing the expression between quotes will allow you to enter the expression like in algebraic mode Ü5 3 2 Ü7 2 2 μ For both ALG and RPN modes using the Equation Writer O5 3 2 7 2 2 The expression can be evaluated within the Equation writer by using EVAL or EVAL Absolute value function The a...

Страница 135: ...in ALG mode enter the base y followed by the Q key and then the exponent x e g 5 2Q1 25 In RPN mode enter the number first then the function e g 5 2 1 25 Q The root function XROOT y x is available through the keystroke combination When calculating in the stack in ALG mode enter the function XROOT followed by the arguments y x separated by commas e g 3 í 27 In RPN mode enter the argument y first th...

Страница 136: ... 3 In RPN mode the argument is entered before the function 2 45 2 3 Trigonometric functions Three trigonometric functions are readily available in the keyboard sine S cosine T and tangent U The arguments of these functions are angles therefore they can be entered in any system of angular measure degrees radians grades For example with the DEG option selected we can calculate the following trigonom...

Страница 137: ...ns and operators Functions like ABS SQ LOG ALOG LN EXP SIN COS TAN ASIN ACOS ATAN require a single argument Thus their application is ALG mode is straightforward e g ABS x Some functions like XROOT require two arguments e g XROOT x y This function has the equivalent keystroke sequence Operators on the other hand are placed after a single argument or between two arguments The factorial operator for...

Страница 138: ...plex numbers which will be discussed in the next chapter Option 10 CONSTANTS provides access to the constants in the calculator This option will be presented later in this section Finally option 11 SPECIAL FUNCTIONS includes functions of advanced mathematics that will be discussed in this section also In general to apply any of these functions you need to be aware of the number and order of the ar...

Страница 139: ...is menu contains also the functions EXPM x exp x 1 LNP1 x ln x 1 Finally option 9 MATH returns the user to the MTH menu For example in ALG mode the keystroke sequence to calculate say tanh 2 5 is the following Select MTH menu 4 OK Select the 4 HYPERBOLIC menu 5 OK Select the 5 TANH function 2 5 Evaluate tanh 2 5 The screen shows the following output In the RPN mode the keystrokes to perform this c...

Страница 140: ...oft menu keys as follows left hand side in ALG mode right hand side in RPN mode Pressing L shows the remaining options Thus to select for example the hyperbolic functions menu with this menu format press HYP to produce Finally in order to select for example the hyperbolic tangent tanh function simply press TANH Note Pressing will return to the first set of MTH options Also using the combination wi...

Страница 141: ...unction As an exercise of applications of hyperbolic functions verify the following values SINH 2 5 6 05020 ASINH 2 0 1 4436 COSH 2 5 6 13228 ACOSH 2 0 1 3169 TANH 2 5 0 98661 ATANH 0 2 0 2027 EXPM 2 0 6 38905 LNP1 1 0 0 69314 Once again the general procedure shown in this section can be applied for selecting options in any calculator menu Real number functions Selecting option 5 REAL in the MTH m...

Страница 142: ...ollows y x calculates the x percentage of y CH y x calculates 100 y x x i e the percentage change the difference between two numbers T y x calculates 100 x y i e the percentage total the portion that one number x is of another y These functions require two arguments we illustrate the calculation of T 15 45 i e calculation 15 of 45 next We assume that the calculator is set to ALG mode and that syst...

Страница 143: ... T function As an exercise for percentage related functions verify the following values 5 20 1 CH 22 25 13 6363 T 500 20 4 Minimum and maximum Use these functions to determine the minimum or maximum value of two arguments MIN x y minimum value of x and y MAX x y maximum value of x and y As an exercise verify that MIN 2 2 2 MAX 2 2 2 Modulo MOD y mod x residual of y x i e if x and y are integer num...

Страница 144: ...e fractional part of a real number As an exercise verify that ABS 3 3 3 SIGN 5 1 MANT 2540 2 540 XPON 2540 3 IP 2 35 2 FP 2 35 0 35 Rounding truncating floor and ceiling functions RND x y rounds up y to x decimal places TRNC x y truncate y to x decimal places FLOOR x closest integer that is less than or equal to x CEIL x closest integer that is greater than or equal to x As an exercise verify that...

Страница 145: ...mma function is defined as Factorial of a number The factorial of a positive integer number n is defined as n n n 1 n 2 3 2 1 with 0 1 The factorial function is available in the calculator by using 2 In both ALG and RPN modes enter the number first followed by the sequence 2 Example 5 2 The Gamma function defined above has the property that Γ α α 1 Γ α 1 for α 1 Therefore it can be related to the ...

Страница 146: ... Calculator constants The following are the mathematical constants used by your calculator Θ e the base of natural logarithms Θ i the imaginary unit ii 2 1 Θ π the ratio of the length of the circle to its diameter Θ MINR the minimum real number available to the calculator Θ MAXR the maximum real number available to the calculator To have access to these constants select option 11 CONSTANTS in the ...

Страница 147: ...ave units associated with them Thus it is possible to calculate results involving a consistent system of units and produce a result with the appropriate combination of units The UNITS menu The units menu is launched by the keystroke combination Û associated with the 6 key With system flag 117 set to CHOOSE boxes the result is the following menu Option 1 Tools contains functions used to operate on ...

Страница 148: ... 5_N For extensive operations with units SOFT menus provide a more convenient way of attaching units Change system flag 117 to SOFT menus see Chapter 1 and use the keystroke combination Û to get the following menus Press L to move to the next menu page Pressing on the appropriate soft menu key will open the sub menu of units for that particular selection For example for the SPEED sub menu the foll...

Страница 149: ...i fermi AREA m 2 square meter cm 2 square centimeter b barn yd 2 square yard ft 2 square feet in 2 square inch km 2 square kilometer ha hectare a are mi 2 square mile miUS 2 square statute mile acre acre VOLUME m 3 cubic meter st stere cm 3 cubic centimeter yd 3 cubic yard ft 3 cubic feet in 3 cubic inch l liter galUK UK gallon galC Canadian gallon gal US gallon qt quart pt pint ml mililiter cu US...

Страница 150: ... gram force kip kilopound force lbf pound force pdl poundal ENERGY J joule erg erg Kcal kilocalorie Cal calorie Btu International table btu ft lbf foot pound therm EEC therm MeV mega electron volt eV electron volt POWER W watt hp horse power PRESSURE Pa pascal atm atmosphere bar bar psi pounds per square inch torr torr mmHg millimeters of mercury inHg inches of mercury inH20 inches of water TEMPER...

Страница 151: ...Bq becquerel Ci curie R roentgen VISCOSITY P poise St stokes Units not listed Units not listed in the Units menu but available in the calculator include gmol gram mole lbmol pound mole rpm revolutions per minute dB decibels These units are accessible through menu 117 02 triggered by using MENU 117 02 in ALG mode or 117 02 MENU in RPN mode The menu will show in the screen as follows use to show lab...

Страница 152: ...nu OK Select the VISCOSITY option OK Select the UNITS menu Convert the units This results in the following screen i e 1 poise 0 1 kg m s In RPN mode system flag 117 set to CHOOSE boxes 1 Enter 1 no underline Û Select the UNITS menu OK Select the VISCOSITY option OK Select the unit P poise Û Select the UNITS menu OK Select the TOOLS menu OK Select the UBASE function In ALG mode system flag 117 set ...

Страница 153: ...Here is the sequence of steps to enter this number in ALG mode system flag 117 set to CHOOSE boxes 5 Ý Enter number and underscore Û Access the UNITS menu 8 OK Select units of force 8 Force OK Select Newtons N Enter quantity with units in the stack The screen will look like the following To enter this same quantity with the calculator in RPN mode use the following keystrokes 5 Enter number do not ...

Страница 154: ...FORCE Select units of force N Select Newtons N Enter quantity with units in the stack The same quantity entered in RPN mode uses the following keystrokes 5 Enter number no underscore Û Access the UNITS menu L FORCE Select units of force N Select Newtons N Unit prefixes You can enter prefixes for units according to the following table of prefixes from the SI system The prefix abbreviation is shown ...

Страница 155: ...123 Ý p m Using UBASE to convert to the default unit 1 m results in Operations with units Once a quantity accompanied with units is entered into the stack it can be used in operations similar to plain numbers except that quantities with units cannot be used as arguments of functions say SQ or SIN Thus attempting to calculate LN 10_m will produce an error message Error Bad Argument Type Here are so...

Страница 156: ...formed to SI units with function UBASE produces Addition and subtraction can be performed in ALG mode without using parentheses e g 5 m 3200 mm can be entered simply as 5_m 3200_mm More complicated expression require the use of parentheses e g 12_mm 1_cm 2 2_s Note Recall that the ANS 1 variable is available through the keystroke combination î associated with the key ...

Страница 157: ...erations 5_m 3200_mm 12_mm 1_cm 2 2_s These last two operations produce the following output Units manipulation tools The UNITS menu contains a TOOLS sub menu which provides the following functions CONVERT x y convert unit object x to units of object y UBASE x convert unit object x to SI units UVAL x extract the value from unit object x Note Units are not allowed in expressions entered in the equa...

Страница 158: ...unction UVAL requires only one argument functions CONVERT UFACT and UNIT require two arguments Try the following exercises The output shown below was developed in ALG mode with system flat 117 set to SOFT menu Examples of CONVERT These examples produce the same result i e to convert 33 watts to btus CONVERT 33_W 1_hp CONVERT 33_W 11_hp These operations are shown in the screen as Examples of UVAL U...

Страница 159: ...d in a constants library activated with the command CONLIB To launch this command you could simply type it in the stack conlib or you can select the command CONLIB from the command catalog as follows First launch the catalog by using N c Next use the up and down arrow keys to select CONLIB Finally press the F OK soft menu key Press if needed The constants library screen will look like the followin...

Страница 160: ...ts values are shown in English units UNIT when selected constants are shown with units attached VALUE when selected constants are shown without units STK copies value with or without units to the stack QUIT exit constants library Active only if the function VALUE is active This is the way the top of the CONSTANTS LIBRARY screen looks when the option VALUE is selected units in the SI system ...

Страница 161: ...k select the variable name and press then press QUIT For the calculator set to the ALG the screen will look like this The display shows what is called a tagged value Vm 359 0394 In here Vm is the tag of this result Any arithmetic operation with this number will ignore the tag Try for example 2 î which produces The same operation in RPN mode will require the following keystrokes after the value of ...

Страница 162: ...ge of this menu press L we find the following items In this menu page there is one function TINC and a number of units described in an earlier section on units see above The function of interest is TINC temperature increment command Out of all the functions available in this MENU UTILITY menu namely ZFACTOR FANNING DARCY F0λ SIDENS TDELTA and TINC functions FANNING and DARCY are described in Chapt...

Страница 163: ...n wavelengths 0 and λ If no units are attached to T and λ it is implied that T is in K and λ in m Example in ALG mode Function SIDENS Function SIDENS T calculates the intrinsic density of silicon in units of 1 cm3 as a function of temperature T T in K for T between 0 and 1685 K For example Function TDELTA Function TDELTA T0 Tf yields the temperature increment Tf T0 The result is returned with the ...

Страница 164: ...ey The function must be entered in the following format Function_name arguments expression_containing_arguments For example we could define a simple functionH x ln x 1 exp x Suppose that you have a need to evaluate this function for a number of discrete values and therefore you want to be able to press a single button and get the result you want without having to type the expression in the right h...

Страница 165: ...r a value that is temporarily assigned to the name x referred to as a local variable evaluate the expression between quotes that contain that local variable and show the evaluated expression To activate the function in ALG mode type the name of the function followed by the argument between parentheses e g H Ü2 Some examples are shown below In the RPN mode to activate the function enter the argumen...

Страница 166: ...be the function listed above Function IFTE is accessible from the function catalog N The symbol greater than is available as associated with the Y key To define this function in ALG mode use the command DEF f x IFTE x 0 x 2 1 2 x 1 then press In RPN mode type the function definition between apostrophes f x IFTE x 0 x 2 1 2 x 1 then press à Press J to recover your variable menu The function f shoul...

Страница 167: ...icated function such as you can combine several levels of the IFTE function i e g x IFTE x 2 x IFTE x 0 x 1 IFTE x 2 x 1 x 2 Define this function by any of the means presented above and check that g 3 3 g 1 0 g 1 0 g 3 9 2 2 0 1 0 2 1 2 2 x x x x x x x x x g ...

Страница 168: ...omplex number can also be represented in polar coordinates polar representation as z re iθ r cosθ i r sinθ where r z is the magnitude of the complex number z and θ Arg z arctan y x is the argument of the complex number z The relationship between the Cartesian and polar representation of complex numbers is given by the Euler formula e iθ cos θ i sin θ The complex conjugate of a complex number z x i...

Страница 169: ...lator in ALG mode the complex number 3 5 1 2 is entered as Ü3 5 í 1 2 A complex number can also be entered in the form x iy For example in ALG mode 3 5 1 2i is entered as 3 5 1 2 The following screen results after entering these complex numbers In RPN mode these numbers will be entered using the following keystrokes Ü3 5 í1 2 Notice that the change sign keystroke is entered after the number 1 2 ha...

Страница 170: ...is in standard notation and the angular measure is set to radians you can always change to radians by using function RAD The result shown above represents a magnitude 3 7 and an angle 0 33029 The angle symbol is shown in front of the angle measure Return to Cartesian or rectangular coordinates by using function RECT available in the catalog N A complex number in polar representation is written as ...

Страница 171: ...eat that i2 1 Operations with complex numbers are similar to those with real numbers For example with the calculator in ALG mode and the CAS set to Complex we ll attempt the following sum 3 5i 6 3i Notice that the real parts 3 6 and imaginary parts 5 3 are combined together and the result given as an ordered pair with real part 9 and imaginary part 2 Try the following operations on your own 5 2i 3...

Страница 172: ...ons such as magnitude argument real and imaginary parts and complex conjugate are available through the CMPLX menus detailed later The CMPLX menus There are two CMPLX CoMPLeX numbers menus available in the calculator One is available through the MTH menu introduced in Chapter 3 and one directly into the keyboard ß The two CMPLX menus are presented next Thus the inverse function INV activated with ...

Страница 173: ...x number x y out of real numbers x and y ABS z Calculates the magnitude of a complex number or the absolute value of a real number ARG z Calculates the argument of a complex number The remaining options options 7 through 10 are the following SIGN z Calculates a complex number of unit magnitude as z z NEG Changes the sign of z CONJ z Produces the complex conjugate of z Examples of applications of t...

Страница 174: ...ute value Also the result of function ARG which represents an angle will be given in the units of angle measure currently selected In this example ARG 3 5 i 1 0303 is given in radians In the next screen we present examples of functions SIGN NEG which shows up as the negative sign and CONJ CMPLX menu in keyboard A second CMPLX menu is accessible by using the right shift option associated with the 1...

Страница 175: ...ernative to the MTH based CMPLX menu containing the basic complex number functions Try the examples shown earlier using the keyboard based CMPLX menu for practice Functions applied to complex numbers Many of the keyboard based functions defined in Chapter 3 for real numbers e g SQ LN ex LOG 10X SIN COS TAN ASIN ACOS ATAN can be applied to complex numbers The result is another complex number as ill...

Страница 176: ...ITE equation of a straight line Function DROITE takes as argument two complex numbers say x1 iy1 and x2 iy2 and returns the equation of the straight line say y a bx that contains the points x1 y1 and x2 y2 For example the line between points A 5 3 and B 6 2 can be found as follows example in Algebraic mode Note When using trigonometric functions and their inverses with complex numbers the argument...

Страница 177: ...Page 4 10 Function DROITE is found in the command catalog N Using EVAL ANS 1 simplifies the result to ...

Страница 178: ...evel 1 or by using the equation writer O For example to enter the algebraic object π D 2 4 directly into stack level 1 use ì dQ2 4 The resulting screen is shown next for both the ALG mode left hand side and the RPN mode right hand side An algebraic object can also be built in the Equation Writer and then sent to the stack The operation of the Equation Writer was described in Chapter 2 As an exerci...

Страница 179: ...rn how to create variables and store values in them Here are the keystrokes for storing variables A1 in ALG mode ì rQ2 K a1 resulting in The keystrokes corresponding to RPN mode are ì r Q2 a1 K After storing the variable A2 and pressing the key the screen will show the variables as follows In ALG mode the following keystrokes will show a number of operations with the algebraics contained in variab...

Страница 180: ...CHOOSE boxes the ALG menu shows the following functions Rather than listing the description of each function in this manual the user is invited to look up the description using the calculator s help facility I L HELP To locate a particular function type the first letter of the function For example for function COLLECT we type c then use the up and down arrow keys to locate COLLECT within the help ...

Страница 181: ...NEXT CASCMD allows you to browse through all the CAS commands It provides not only information on each command but also provides an example of its application This example can be copied onto your stack by pressing the ECHO soft menu key For example for the EXPAND entry shown above press the ECHO soft menu key to get the following example copied to the stack press to execute the command We leave fo...

Страница 182: ...se these or any other functions in the RPN mode you must enter the argument first and then the function For example the example for TEXPAND in RPN mode will be set up as x y At this point select function TEXPAND from menu ALG or directly from the catalog N to complete the operation ...

Страница 183: ...shown in the right hand figure In RPN mode this can be accomplished by entering first the expression where the substitution will be performed x x2 followed by a list see Chapter 8 containing the substitution variable an space and the value to be substituted i e x 2 The final step is to press the keystroke combination The required keystrokes are the following x xQ2 ä x 2 In ALG mode substitution of...

Страница 184: ...ial trigonometric and hyperbolic functions in terms of trigonometric identities or in terms of exponential functions The menus containing functions to replace trigonometric functions can be obtained directly from the keyboard by pressing the right shift key followed by the 8 key i e Ñ The combination of this key with the left shift key i e Ð produces a menu that lets you replace expressions in ter...

Страница 185: ... shows the following functions These functions allow to simplify expressions by replacing some category of trigonometric functions for another one For example the function ACOS2S allows to replace the function arccosine acos x with its expression in terms of arcsine asin x Description of these commands and examples of their applications are available in the calculator s help facility IL HELP The u...

Страница 186: ...s 5 through 9 DIVIS FACTORS LGCD PROPFRAC SIMP2 correspond to common functions that apply to integer numbers or to polynomials The remaining options 1 INTEGER 2 POLYNOMIAL 3 MODULO and 4 PERMUTATION are actually sub menus of functions that apply to specific mathematical objects This distinction between sub menus options 1 through 4 and plain functions options 5 through 9 is made clear when system ...

Страница 187: ...ean division of two integers IEGCD Returns u v such that au bv gcd a b IQUOT Euclidean quotient of two integers IREMAINDER Euclidean remainder of two integers ISPRIME Test if an integer number is prime NEXTPRIME Next prime for a given integer number PA2B2 Prime number as square norm of a complex number PREVPRIME Previous prime for a given integer number POLYNOMIAL menu ABCUV Bézout polynomial equa...

Страница 188: ...e Sylvester matrix of 2 polynomials REMAINDER Euclidean reminder of two polynomials STURM Sturm sequences for a polynomial STURMAB Sign at low bound and number of zeros between bounds MODULO menu ADDTMOD Add two expressions modulo current modulus DIVMOD Divides 2 polynomials modulo current modulus DIV2MOD Euclidean division of 2 polynomials with modular coefficients EXPANDMOD Expands simplify poly...

Страница 189: ...modulus n which is a positive integer follow the rules that if j and k are any two nonnegative integer numbers both smaller than n if j k n then j k is defined as j k n For example in the case of the clock i e for n 12 6 9 3 To distinguish this equality from infinite arithmetic equalities the symbol is used in place of the equal sign and the relationship between the numbers is referred to as a con...

Страница 190: ...arithmetic you cannot define 5 6 mod 12 because the multiplication table of 6 does not show the result 5 in modulus 12 arithmetic This multiplication table is shown below Formal definition of a finite arithmetic ring The expression a b mod n is interpreted as a is congruent to b modulo n and holds if b a is a multiple of n With this definition the rules of arithmetic simplify to the following If a...

Страница 191: ...d the corresponding calculator s finite arithmetic ring is given by 3 2 1 0 1 2 3 Modular arithmetic in the calculator To launch the modular arithmetic menu in the calculator select the MODULO sub menu within the ARITHMETIC menu Þ The available menu includes functions ADDTMOD DIVMOD DIV2MOD EXPANDMOD FACTORMOD GCDMOD INVMOD MOD MODSTO MULTMOD POWMOD and SUBTMOD Brief descriptions of these function...

Страница 192: ...13 4 mod 12 66 6 1 mod 12 DIV2MOD examples 2 3 mod 12 does not exist 26 12 mod 12 does not exist 125 17 mod 12 1 with remainder 0 68 7 4 mod 12 with remainder 0 7 5 1 mod 12 with remainder 0 POWMOD examples 23 4 mod 12 35 3 mod 12 510 1 mod 12 118 1 mod 12 62 0 mod 12 99 3 mod 12 In the examples of modular arithmetic operations shown above we have used numbers that not necessarily belong to the ri...

Страница 193: ...1 7 5 mod 12 1 3 mod 12 does not exist 1 11 1 mod 12 The MOD operator The MOD operator is used to obtain the ring number of a given modulus corresponding to a given integer number On paper this operation is written as m mod n p and is read as m modulo n is equal to p For example to calculate 15 mod 8 enter Θ ALG mode 15 MOD 8 Θ RPN mode 15 8 MOD The result is 7 i e 15 mod 8 7 Try the following exe...

Страница 194: ...ion A X U X B X V X C X Specific examples of polynomial applications are provided next Modular arithmetic with polynomials The same way that we defined a finite arithmetic ring for numbers in a previous section we can define a finite arithmetic ring for polynomials with a given polynomial as modulo For example we can write a certain polynomial P X as P X X mod X2 or another polynomial Q X X 1 mod ...

Страница 195: ... the polynomials C X U X and V X so that C X U X A X V X B X For example for A X X 2 1 B X X 2 1 EGCD A X B X 2 1 1 i e 2 1 X 2 1 1 X 2 1 Also EGCD X 3 2 X 5 X 5 1 X 2 2 i e 5 X 2 2 X 1 X 3 2 X 5 The GCD function The function GCD Greatest Common Denominator can be used to obtain the greatest common denominator of two polynomials or of two lists of polynomials of the same length The two polynomials...

Страница 196: ...fore write X3 2X2 3X 1 X2 4X 5 X 2 11 A second example HORNER X 6 1 5 X 5 5 X 4 25 X 3 125 X 2 625 X 3125 5 15624 i e X6 1 X5 5 X4 25X3 125X2 625X 3125 X 5 15624 The variable VX A variable called VX exists in the calculator s HOME CASDIR directory that takes by default the value of X This is the name of the preferred independent variable for algebraic and calculus applications Avoid using the vari...

Страница 197: ... of two polynomials or of lists of polynomials of the same length Examples LCM 2 X 2 4 X 2 X 2 1 2 X 2 4 X 2 X 1 LCM X 3 1 X 2 2 X X 3 1 X 2 2 X The LEGENDRE function A Legendre polynomial of order n is a polynomial function that solves the differential equation To obtain the n th order Legendre polynomial use LEGENDRE n e g LEGENDRE 3 5 X 3 3 X 2 LEGENDRE 5 63 X 5 70 X 3 15 X 8 Note Matrices are ...

Страница 198: ...YL is used to obtain an expression Q X a P X i e to develop a polynomial in powers of X a This is also known as a Taylor polynomial from which the name of the function Polynomial TAYLor follow For example PTAYL X 3 2 X 2 2 X 3 6 X 2 10 X 6 In actuality you should interpret this result to mean X 2 3 6 X 2 2 10 X 2 6 Let s check by using the substitution X x 2 We recover the original polynomial but ...

Страница 199: ...μ X 3 0000012 X The PEVAL function The functions PEVAL Polynomial EVALuation can be used to evaluate a polynomial p x an xn an 1 x n 1 a2 x2 a1 x a0 given an array of coefficients an an 1 a2 a1 a0 and a value of x0 The result is the evaluation p x0 Function PEVAL is not available in the ARITHMETIC menu it must be accessed from the function catalog N Example PEVAL 1 5 6 1 5 281 The TCHEBYCHEFF func...

Страница 200: ...s arguments two numbers or polynomials representing the numerator and denominator of a rational fraction and returns the simplified numerator and denominator For example SIMP2 X 3 1 X 2 4 X 3 X 2 X 1 X 3 The PROPFRAC function The function PROPFRAC converts a rational fraction into a proper fraction i e an integer part added to a fractional part if such decomposition is possible For example PROPFRA...

Страница 201: ...2 If you press μ î or simply μ in RPN mode you will get X 6 8 X 5 5 X 4 50 X 3 X 7 13 X 6 61 X 5 105 X 4 45 X 3 297 X 2 81 X 243 The FROOTS function The function FROOTS obtains the roots and poles of a fraction As an example applying function FROOTS to the result produced above will result in 1 2 3 5 0 3 2 1 5 2 The result shows poles followed by their multiplicity as a negative number and roots f...

Страница 202: ... the calculator will show simplifications of fractions or operations with polynomials in a step by step fashion This is very useful to see the steps of a synthetic division The example of dividing is shown in detail in Appendix C The following example shows a lengthier synthetic division Note that DIV2 is available from the ARITH POLYNOMIAL menu 2 2 3 5 2 3 X X X X 1 1 2 9 X X ...

Страница 203: ... menu summarizes all conversion menus in the calculator The list of these menus is shown next The functions available in each of the sub menus are shown next UNITS convert menu Option 1 This menu is the same as the UNITS menu obtained by using Û The applications of this menu are discussed in detail in Chapter 3 ...

Страница 204: ...f this menu are discussed in detail in this Chapter MATRICES convert menu Option 5 This menu contains the following functions These functions are discussed in detail in Chapter 10 REWRITE convert menu Option 4 This menu contains the following functions Functions I R and R I are used to convert a number from integer I to real R or vice versa Integer numbers are shown without trailing decimal points...

Страница 205: ... of π if a fraction of π can be found for the number otherwise it converts the number to a fraction Examples are of these three functions are shown next Out of the functions in the REWRITE menu functions DISTRIB EXPLN EXP2POW FDISTRIB LIN LNCOLLECT POWEREXPAND and SIMPLIFY apply to algebraic expressions Many of these functions are presented in this Chapter However for the sake of completeness we p...

Страница 206: ...Page 5 29 LIN LNCOLLECT POWEREXPAND SIMPLIFY ...

Страница 207: ...ns DESOLVE and LDEC are used for the solution of differential equations the subject of a different chapter and therefore will not be presented here Similarly function LINSOLVE relates to the solution of multiple linear equations and it will be presented in a different chapter Functions ISOL and SOLVE can be used to solve for any unknown in a polynomial equation Function SOLVEVX solves a polynomial...

Страница 208: ...e or an equation For example in ALG mode try The same problem can be solved in RPN mode as illustrated below figures show the RPN stack before and after the application of function ISOL Function SOLVE Function SOLVE has the same syntax as function ISOL except that SOLVE can also be used to solve a set of polynomial equations The help facility entry for function SOLVE with the solution to equation ...

Страница 209: ...ions by using the down arrow key which triggers the line editor this operation can be used to access any output line that is wider than the calculator s screen The corresponding RPN screens for these two examples before and after the application of function SOLVE are shown next Use of the down arrow key in this mode will launch the line editor Function SOLVEVX The function SOLVEVX solves an equati...

Страница 210: ...ollowing equation will not processed by SOLVEVX Function ZEROS The function ZEROS finds the solutions of a polynomial equation without showing their multiplicity The function requires having as input the expression for the equation and the name of the variable to solve for Examples in ALG mode are shown next To use function ZEROS in RPN mode enter first the polynomial expression then the variable ...

Страница 211: ... 2 Solve diff eq is to be discussed in a later chapter on differential equations Item 4 Solve lin sys will be discussed in a later Chapter on matrices Item 6 MSLV Multiple equation SoLVer will be presented in the next chapter Following we present applications of items 3 Solve poly 5 Solve finance and 1 Solve equation in that order Appendix 1 A at the end of Chapter 1 contains instructions on how t...

Страница 212: ...ns to any polynomial equation of order n Some of the solutions could be complex numbers nevertheless As an example solve the equation 3s4 2s3 s 1 0 We want to place the coefficients of the equation in a vector an an 1 a1 a0 For this example let s use the vector 3 2 0 1 1 To solve for this polynomial equation using the calculator try the following Ï OK Select solve poly Ô3 í2 í 0 í 1 í1 OK Enter ve...

Страница 213: ...h the first number in the pair being the real part and the second number the imaginary part For example the number 0 432 0 389 a complex number will be written normally as 0 432 0 389i where i is the imaginary unit i e i2 1 Note The fundamental theorem of algebra indicates that there are n solutions for any polynomial equation of order n There is another theorem of algebra that indicates that if o...

Страница 214: ...MB Generate symbolic expression Return to stack The expression thus generated is shown in the stack as X 3 5 X 2 2 X 4 To generate the algebraic expression using the roots try the following example Assume that the polynomial roots are 1 3 2 1 Use the following keystrokes Ï OK Select Solve poly Ô1 í3 Enter vector of roots í2 í 1 OK SYMB Generate symbolic expression Return to stack The expression th...

Страница 215: ...peration of this solving environment we present some definitions needed to understand financial operations in the calculator Definitions Often to develop projects it is necessary to borrow money from a financial institution or from public funds The amount of money borrowed is referred to as the Present Value PV This money is to be repaid through n periods typically multiples or sub multiples of a ...

Страница 216: ...lue of PMT as 39 132 30 i e the borrower must pay the lender US 39 132 30 at the end of each month for the next 60 months to repay the entire amount The reason why the value of PMT turned out to be negative is because the calculator is looking at the money amounts from the point of view of the borrower The borrower has US 2 000 000 00 at time period t 0 then he starts paying i e adding US 39132 30...

Страница 217: ...screen now looks like this This means that at the end of 60 months the US 2 000 000 00 principal amount has been paid together with US 347 937 79 of interest with the balance being that the lender owes the borrower US 0 000316 Of course the balance should be zero The value shown in the screen above is simply round off error resulting from the numerical solution Press or twice to return to normal c...

Страница 218: ... any sub directory it will generate the variables N I YR PV PMT PYR FV to store the corresponding terms in the calculations You can see the contents of these variables by using n I YR PV PMT PYR FV You can either keep these variables for future use or use the PURGE function to erase them from your directory To erase all of the variables at once if using ALG mode try the following I PURGE J ä Enter...

Страница 219: ...e PV PMT Enter name of variable PMT PYR Enter name of variable PYR FV Enter name of variable FV Enter list of variables in stack I PURGE Purge variables in list Before the command PURGE is entered the RPN stack will look like this Solving equations with one unknown through NUM SLV The calculator s NUM SLV menu provides item 1 Solve equation solve different types of equations in a single variable i...

Страница 220: ...QUATION input form Also a field labeled x is provided To solve the equation all you need to do is highlight the field in front of X by using and press SOLVE The solution shown is X 4 5006E 2 Function STEQ Function STEQ available through the command catalog N will store its argument into variable EQ e g in ALG mode In RPN mode enter the equation between apostrophes and activate command STEQ Thus fu...

Страница 221: ...force a solution by providing an initial guess for the solution in the appropriate input field before solving the equation The calculator uses a search algorithm to pinpoint an interval for which the function changes sign which indicates the existence of a root or solution It then utilizes a numerical method to converge into the solution The solution the calculator seeks is determined by the initi...

Страница 222: ...owing Ï OK Access numerical solver to solve equations O Access the equation writer to enter equation At this point follow the instructions from Chapter 2 on how to use the Equation Writer to build an equation The equation to enter in the Eq field should look like this notice that we use only one sub index to refer to the variables i e exx is translated as ex etc this is done to save typing time Us...

Страница 223: ...should enter a value of 0 005 in the ex field and a zero in the ΔT field with ΔT 0 no thermal effects are included To solve for E highlight the E field and press SOLVE The result seeing with the EDIT feature is E 449000 psi Press SOLVE to return to normal display Notice that the results of the calculations performed within the numerical solver environment have been copied to the stack Also you wil...

Страница 224: ...m y y where b bottom width and m side slope of cross section We can type in the equation for E as shown above and use auxiliary variables for A and V so that the resulting input form will have fields for the fundamental variables y Q g m and b as follows Θ First create a sub directory called SPEN SPecific ENergy and work within that sub directory Θ Next define the following variables Θ Launch the ...

Страница 225: ...arger value of y say 15 highlight the y input field and solve for y once more The result is now 9 99990 i e y 9 99990 ft This example illustrates the use of auxiliary variables to write complicated equations When NUM SLV is activated the substitutions implied by the auxiliary variables are implemented and the input screen for the equation provides input field for the primitive or fundamental varia...

Страница 226: ... ν μ ρ is the kinematic viscosity of the fluid The calculator provides a function called DARCY that uses as input the relative roughness ε D and the Reynolds number in that order to calculate the friction factor f The function DARCY can be found through the command catalog For example for ε D 0 0001 Re 1000000 you can find the friction factor by using DARCY 0 0001 1000000 In the following screen t...

Страница 227: ...he following variables f A V Re In this case we stored the main equation Darcy Weisbach equation into EQ and then replaced several of its variables by other expressions through the definition of variables f A V and Re To see the combined equation use EVAL EQ In this example we changed the display setting so that we can see the entire equation in the screen Thus the equation we are solving after co...

Страница 228: ...e input values and solve for D The solution is 0 12 i e D 0 12 m If the equation is dimensionally consistent you can add units to the input values as shown in the figure below However you must add those units to the initial guess in the solution Thus in the example below we place 0_m in the D field before solving the problem The solution is shown in the screen to the right Press to return to norma...

Страница 229: ...the function CONST in the calculator by using We can solve for any term in the equation except G by entering the equation as This equation is then stored in EQ Launching the numerical solver for this equation results in an input form containing input fields for F G m1 m2 and r Let s solve this problem using units with the following values for the known variables m1 1 0 106 kg m2 1 0 1012 kg r 1 0 ...

Страница 230: ...he solver after activating it by editing the contents of the EQ field in the numerical solver input form If variable EQ has not been defined previously when you launch the numerical solver Ï OK the EQ field will be highlighted At this point you can either type a new equation by pressing EDIT You will be provided with a set of apostrophes so that you can type the expression between them Note When u...

Страница 231: ...lready existing in your directory to be entered into EQ This means that your equation would have to have been stored in a variable name previously to activating the numerical solver For example suppose that we have entered the following equations into variables EQ1 and EQ2 Now launch the numerical solver Ï OK and highlight the EQ field At this point press the CHOOS soft menu key Use the up and dow...

Страница 232: ...ve an equation for a given variable with a starting guess value In RPN mode the equation will be in stack level 3 while the variable name will be located in level 2 and the initial guess in level 1 The following figure shows the RPN stack before and after activating function ROOT In ALG mode you would use ROOT TAN θ θ θ 5 to activate function ROOT Variable EQ The soft menu key EQ in this sub menu ...

Страница 233: ... environment press J The access to the SOLVE menu is lost at this point so you have to activate it once more as indicated earlier to continue with the exercises below Example 2 Solving the equation Q at2 bt It is possible to store in EQ an equation involving more than one variable say Q at 2 bt In this case after activating the SOLVE soft menu and pressing ROOT SOLVR you will get the following scr...

Страница 234: ... by using X This gives the value X 9 4999 To check the value of the equation at this point press EXPR The results are Left 19 Right 19 To solve the next equation press L NEXQ The screen shows the soft menu keys as Say we enter the values k 2 s 12 Then solve for Y and press EXPR The results are now Y We then continue moving from the first to the second equation back and forth solving the first equa...

Страница 235: ...Θ The expression used in the solution must have consistent units or an error will result when trying to solve for a value The DIFFE sub menu The DIFFE sub menu provides a number of functions for the numerical solution of differential equations The functions provided are the following These functions are presented in detail in Chapter 16 The POLY sub menu The POLY sub menu performs operations on po...

Страница 236: ...urns the value 28 The SYS sub menu The SYS sub menu contains a listing of functions used to solve linear systems The functions listed in this sub menu are These functions are presented in detail in Chapter 11 The TVM sub menu The TVM sub menu contains functions for calculating Time Value of Money This is an alternative way to solve FINANCE problems see Chapter 6 The functions available are shown n...

Страница 237: ... previously For example having solved a TVM problem above we can solve for say N as follows n TVMRO The result is 10 Function AMORT This function takes a value representing a period of payment between 0 and n and returns the principal interest and balance for the values currently stored in the TVM variables For example with the data used earlier if we activate function AMORT for a value of 10 we g...

Страница 238: ... that the CAS is set to mode Exact before attempting a solution using this procedure Also the more complicated the expressions the longer the CAS takes in solving a particular system of equations Examples of this application follow Example 1 Projectile motion Use function SOLVE with the following vector arguments the first being the list of equations x x0 v0 COS θ0 t y y0 v0 SIN θ0 t g t 2 2 and t...

Страница 239: ...ce r from the cylinder s axis the normal stresses in the radial and transverse directions σrr and σθθ respectively are given by Notice that the right hand sides of the two equations differ only in the sign between the two terms Therefore to write these equations in the calculator I suggest you type the first term and store in a variable T1 then the second term and store it in T2 Writing the equati...

Страница 240: ...ate the equation for σθθ J T1 T2 s t Å Create the equation for σrr J T1 T2 s r Å Put together a vector with the two equations using function ARRY find it using the command catalog N after typing a 2 Now suppose that we want to solve for Pi and Po given a b r σrr and σθθ We enter a vector with the unknowns To solve for Pi and Po use the command SOLVE from the S SLV menu Î it may take the calculator...

Страница 241: ...ndled equally well with function LINSOLVE see Chapter 11 The following example shows function SOLVE applied to a system of polynomial equations Example 3 System of polynomial equations The following screen shot shows the solution of the system X2 XY 10 X2 Y2 5 using function SOLVE Solution to simultaneous equations with MSLV Function MSLV is available as the last option in the Ï menu The help faci...

Страница 242: ...e solution for this example is produced by using Activating function MSLV results in the following screen You may have noticed that while producing the solution the screen shows intermediate information on the upper left corner Since the solution provided by MSLV is numerical the information in the upper left corner shows the results of the iterative process used to obtain a solution The final sol...

Страница 243: ...mH of the cross section Typically one has to solve the equations of energy and Manning s simultaneously for y and Q Once these equations are written in terms of the primitive variables b m y g So n Cu Q and Ho we are left with a system of equations of the form f1 y Q 0 f2 y Q 0 We can build these two equations as follows We assume that we will be using the ALG and Exact modes in the calculator alt...

Страница 244: ...SLV for the solution we need to enter these values into the corresponding variable names This can be accomplished as follows Now we are ready to solve the equation First we need to put the two equations together into a vector We can do this by actually storing the vector into a variable that we will call EQS EQuationS As initial values for the variables y and Q we will use y 5 equal to the value o...

Страница 245: ...K and allow the solution to proceed An intermediate solution step may look like this The vector at the top representing the current value of y Q as the solution progresses and the value 358822986286 representing the criteria for convergence of the numerical method used in the solution If the system is well posed this value will diminish until reaching a value close to zero At that point a numerica...

Страница 246: ...ns by solving for one unknown from one equation at a time It is not really a solver to simultaneous solutions rather it is a one by one solver of a number of related equations To illustrate the use of the MES for solving multiple equations we present an application related to trigonometry in the next section The examples shown here are developed in the RPN mode Application 1 Solution of triangles ...

Страница 247: ...ther it takes the known variables and then searches in a list of equations until it finds one that can be solved for one of the unknown variables Then it searches for another equation that can be solved for the next unknowns and so on until all unknowns have been solved for Creating a working directory We will use the MES to solve for triangles by creating a list of equations corresponding to the ...

Страница 248: ...riangle Solution as follows Õ Open double quotes in stack Locks keyboard into lower case alpha triangle Enter text Triangle_ solution Enter text Solution Enter string Triangle Solution in stack Open single quotes in stack title Enter variable name TITLE K Store string into TITLE Creating a list of variables Next create a list of variable names in the stack that will look like this a b c α β γ A s ...

Страница 249: ...tack level 1 and places the title atop of the MES window and the list of variables as soft menu keys in the order indicated by the list In the present exercise we already have a title Triangle Solution and a list of variables a b c α β γ A s in stack levels 2 and 1 respectively ready to activate MITM Θ MSOLVR MES SOLVER activates the Multiple Equation Solver MES and waits for input by the user Run...

Страница 250: ...2 5423968763 You should have the values of the three angles listed in stack levels 3 through 1 Press twice to check that they add indeed to 180o Press L to move to the next variables menu To calculate the area use A The calculator first solves for all the other variables and then finds the area as A 7 15454401063 Note If you get a value that is larger than 180 try the following 10 α Re initialize ...

Страница 251: ...he setting up of the MES for this particular set of equations If you use Mpar to see the contents of the variable Mpar You will get the cryptic message Library Data The meaning of this is that the MES parameters are coded in a binary file which cannot be accessed by the editor Next we want to place them in the menu labels in a different order than the one listed above by following these steps 1 Cr...

Страница 252: ... Type in MSOLVR Enter program in stack Store the program in a variable called TRISOL for TRIangle SOLution by using trisol K Press J if needed to recover your list of variables A soft key label TRISO should be available in your menu Running the program solution examples To run the program press the TRISO soft menu key You will now have the MES menu corresponding to the triangle solution Let s try ...

Страница 253: ... the display To see the equations used in the solution of each variable press the EQNS soft menu key The display will now look like this The soft menu key PRINT is used to print the screen in a printer if available And EXIT returns you to the MES environment for a new solution if needed To return to normal calculator display press J The following table of triangle solutions shows the data input in...

Страница 254: ...se components of the velocity and acceleration of the particle given r r dr dt r d2 r dt2 θ θ d θ dt and θ d2 θ dt2 The following equations are used Create a subdirectory called POLC POLar Coordinates which we will use to calculate velocities and accelerations in polar coordinates Within that subdirectory enter the following variables a b c α ο β ο γ ο A 2 5 6 9837 7 2 20 229 75 84 771 8 6933 7 2 ...

Страница 255: ...ate the magnitude of the velocity v and the acceleration a when the polar components are known r rD rDD r radial coordinate r dot first derivative of r r double dot second derivative of r θD θDD θ dot first derivative of θ θ double dot second derivative of θ ________________________________________________________________ Suppose you are given the following information r 2 5 rD 0 5 rDD 1 5 θD 2 3 ...

Страница 256: ... are v 5 77169819031 ar 14 725 aθ 13 95 and a 20 2836911089 or b Solve for all variables at once by pressing ALL The calculator will flash the solutions as it finds them When the calculator stops you can press ALL to list all results For this case we have Pressing the soft menu key EQNS will let you know the equations used to solve for each of the values in the screen To use a new set of values pr...

Страница 257: ...Page 7 20 ...

Страница 258: ...ET h2 4 1 1 5 2 0 a a a a 1 2 3 3 2 1 1 2 3 In the examples shown below we will limit ourselves to numerical lists Creating and storing lists To create a list in ALG mode first enter the braces key ä associated with the key then type or enter the elements of the list separating them with commas í The following keystrokes will enter the list 1 2 3 4 and store it into variable L1 ä 1 í 2 í 3 í 4 K l...

Страница 259: ...2 while level 1 shows the number of elements in the list To compose a list in RPN mode place the elements of the list in the stack enter the list size and apply function LIST select it from the function catalog as follows N é then use the up and down arrow keys to locate function LIST The following screen shots show the elements of a list of size 4 before and after application of function LIST Not...

Страница 260: ... all elements in the list For example Addition subtraction multiplication division Multiplication and division of a list by a single number is distributed across the list for example Subtraction of a single number from a list will subtract the same number from each element in the list for example Addition of a single number to a list produces a list augmented by the number and not an addition of t...

Страница 261: ...on The plus sign when applied to lists acts a concatenation operator putting together the two lists rather than adding them term by term For example In order to produce term by term addition of two lists of the same length we need to use operator ADD This operator can be loaded by using the function catalog N The screen below shows an application of ADD to add lists L1 and L2 term by term Real num...

Страница 262: ...MTH menu Functions of interest from the MTH menu include from the HYPERBOLIC menu SINH ASINH COSH ACOSH TANH ATANH and from the REAL menu CH T MIN MAX MOD SIGN MANT XPON IP FP RND TRNC FLOOR CEIL D R R D Some of the functions that take a single argument are illustrated below applied to lists of real numbers SINH ASINH COSH ACOSH ...

Страница 263: ...o arguments The first two examples show cases in which only one of the two arguments is a list The results are lists with the function distributed according to the list argument For example 10 20 30 1 10 1 20 1 30 1 while 5 10 20 30 5 10 5 20 5 30 In the following example both arguments of function are lists of the same size In this case a term by term distribution of the arguments is performed i ...

Страница 264: ...ments are lists Examples of applications of function RND are shown next Lists of complex numbers The following exercise shows how to create a list of complex numbers given two lists of the same length one representing the real parts and one the imaginary parts of the complex numbers Use L1 ADD i L2 Functions such as LN EXP SQ etc can also be applied to a list of complex numbers e g ...

Страница 265: ...lex numbers The results are lists of real numbers Lists of algebraic objects The following are examples of lists of algebraic objects with the function SIN applied to them The MTH LIST menu The MTH menu provides a number of functions that exclusively to lists With flag 117 set to CHOOSE boxes Next with system flag 117 set to SOFT menus ...

Страница 266: ...y term addition of two lists of the same length examples of this operator were shown above Examples of application of these functions in ALG mode are shown next SORT and REVLIST can be combined to sort a list in decreasing order If you are working in RPN mode enter the list onto the stack and then select the operation you want For example to calculate the increment between consecutive elements in ...

Страница 267: ...ain the size also known as length of the list e g Extracting and inserting elements in a list To extract elements of a list we use function GET available in the PRG LIST ELEMENTS sub menu The arguments of function GET are the list and the number of the element you want to extract To insert an element into a list use function PUT also available in the PRG LST ELEMENTS sub menu The arguments of func...

Страница 268: ...mine the position of an element in a list use function POS having the list and the element of interest as arguments For example HEAD and TAIL functions The HEAD function extracts the first element in the list The TAIL function removes the first element of a list returning the remaining list Some examples are shown next The SEQ function Item 2 PROCEDURES in the PRG LIST menu contains the following ...

Страница 269: ... ALG mode we identify expression n2 index n start 1 end 4 and increment 1 The list produced corresponds to the values 12 22 32 42 In RPN mode you can list the different arguments of the function as follows before applying function SEQ The MAP function The MAP function available through the command catalog N takes as arguments a list of numbers and a function f X or a program of the form a and prod...

Страница 270: ...guments except that any function incorporating an addition must use the ADD operator rather than the plus sign For example if we define the function F X Y X 5 Y 2 shown here in ALG mode we can use lists e g variables L1 and L2 defined earlier in this Chapter to evaluate the function resulting in Since the function statement includes no additions the application of the function to list arguments is...

Страница 271: ...e edited expression into variable G Evaluating G L1 L2 now produces the following result As an alternative you can define the function with ADD rather than the plus sign from the start i e use DEFINE G X Y X DD 3 Y You can also define the function as G X Y X 3 Y ...

Страница 272: ...PN mode is very similar Just keep in mind that in RPN mode you place the arguments of functions in the stack before activating the function Harmonic mean of a list This is a small enough sample that we can count on the screen the number of elements n 10 For a larger list we can use function SIZE to obtain that number e g Suppose that we want to calculate the harmonic mean of the sample defined as ...

Страница 273: ...mean of list S is sh 1 6348 Geometric mean of a list The geometric mean of a sample is defined as To find the geometric mean of the list stored in S we can use the following procedure 1 Apply function ΠLIST to list S 2 Apply function XROOT x y i e keystrokes to the result in 1 n n n n k k g x x x x x L 2 1 1 ...

Страница 274: ... list W above can be defined by wk k Thus we can use function SEQ to generate this list and then store it into variable W as follows Given the data list s1 s2 sn and the weight list w1 w2 wn the weighted average of the data in S is defined as To calculate the weighted average of the data in list S with the weights in list W we can use th e following steps 1 Multiply lists S and W 2 Use function ΣL...

Страница 275: ...uped data Grouped data is typically given by a table showing the frequency w of data in data classes or bins Each class or bin is represented by a class mark s typically the midpoint of the class An example of grouped data is shown next Note ANS 1 refers to the most recent result 55 while ANS 2 refers to the previous to last result 121 Class Frequency Class mark count boundaries sk wk 0 2 1 5 2 4 ...

Страница 276: ...e data in S with weights W represents the mean value of the grouped data that we call s in this context where represents the total frequency count The mean value for the data in lists S and W therefore can be calculated using the procedure outlined above for the weighted average i e We ll store this value into a variable called XBAR The variance of this grouped data is defined as N s w w s w s n k...

Страница 277: ...Page 8 20 To calculate this last result we can use the following The standard deviation of the grouped data is the square root of the variance N s s w w s s w V n k k k n k k n k k k 1 2 1 1 2 ...

Страница 278: ...tor are A Ax Ay Az A Ax Ay Az or A Ax Ay Az A two dimensional version of this vector will be written as A Axi Ayj A Ax Ay A Ax Ay or A Ax Ay Since in the calculator vectors are written between brackets we will choose the notation A Ax Ay Az or A Ax Ay Az to refer to two and three dimensional vectors from now on The magnitude of a vector A is defined as A A unit vector in the direction of vector A ...

Страница 279: ... the keystroke combination Ô associated with the key The following are examples of vectors in the calculator 3 5 2 2 1 3 5 6 2 3 A general row vector 1 5 2 2 A 2 D vector 3 1 2 A 3 D vector t t 2 SIN t A vector of algebraics Typing vectors in the stack With the calculator in ALG mode a vector is typed into the stack by opening a set of brackets Ô and typing the components or elements of the vector...

Страница 280: ... the keyboard This command generates a species of spreadsheet corresponding to rows and columns of a matrix Details on using the Matrix Writer to enter matrices will be presented in a subsequent chapter For a vector we are interested in filling only elements in the top row By default the cell in the top row and first column is selected At the bottom of the spreadsheet you will find the following s...

Страница 281: ...aunch the Matrix Writer With VEC and GO selected enter 3 5 2 This produces 3 5 2 In RPN mode you can use the following keystroke sequence to produce the same result 3 5 2 2 With VEC deselected and GO selected enter 3 5 2 This produces 3 5 2 Although these two results differ only in the number of brackets used for the calculator they represent different mathematical objects The first one is a vecto...

Страница 282: ...h contains only one function DEL delete The function DEL will delete the contents of the selected cell and replace it with a zero To see these keys in action try the following exercise 1 Activate the Matrix Writer by using Make sure the VEC and GO keys are selected 2 Enter the following 1 2 3 L GOTO 2 OK 1 OK OK 2 1 5 4 5 6 7 8 9 3 Move the cursor up two positions by using Then press ROW The secon...

Страница 283: ...r you want them to appear in the array when read from left to right into the RPN stack 2 Enter n as the last entry 3 Use function ARRY The following screen shots show the RPN stack before and after applying function ARRY Summary of Matrix Writer use for entering vectors In summary to enter a vector using the Matrix Writer simply activate the writer and place the elements of the vector pressing aft...

Страница 284: ...he vector by using A i where i is an integer number less than or equal to the vector size For example create the following array and store it in variable A 1 2 3 4 5 To recall the third element of A for example you could type in A 3 into the calculator In ALG mode simply type A 3 In RPN mode type A 3 μ You can operate with elements of the array by writing and evaluating algebraic expressions such ...

Страница 285: ...r example if we want to change the contents of A 3 to read 4 5 instead of its current value of 3 use 4 5 a Ü3 K To verify that the change took place use A The result now shown is 1 2 4 5 4 5 To find the length of a vector you can use the function SIZE available through the command catalog N or through the PRG LIST ELEMENTS sub menu Some examples based on the arrays or vectors stored previously are...

Страница 286: ...a vector use the key e g Addition subtraction Addition and subtraction of vectors require that the two vector operands have the same length Attempting to add or subtract vectors of different length produces an error message Invalid Dimension e g v2 v3 u2 u3 A v3 etc Multiplication by a scalar and division by a scalar Multiplication by a scalar or division by a scalar is straightforward ...

Страница 287: ...ample BS 1 2 6 BS BS u3 will show in the screen as follows The MTH VECTOR menu The MTH menu contains a menu of functions that specifically to vector objects The VECTOR menu contains the following functions system flag 117 set to CHOOSE boxes Magnitude The magnitude of a vector as discussed earlier can be found with function ABS This function is also available from the keyboard Ê Examples of applic...

Страница 288: ...ting a cross product a 2 D vector of the form Ax Ay is treated as the 3 D vector Ax Ay 0 Examples in ALG mode are shown next for two 2 D and two 3 D vectors Notice that the cross product of two 2 D vectors will produce a vector in the z direction only i e a vector of the form 0 0 Cz Examples of cross products of one 3 D vector with one 2 D vector or vice versa are presented next Attempts to calcul...

Страница 289: ...sional vector Function V3 is used in the RPN mode to build a vector with the values in stack levels 1 2 and 3 The following screen shots show the stack before and after applying function V2 Changing coordinate system Functions RECT CYLIN and SPHERE are used to change the current coordinate system to rectangular Cartesian cylindrical polar or spherical coordinates The current system is shown highli...

Страница 290: ...G should be selected as the angular measure and z 2 3 we can enter this vector in the following way Ô5 í 6 25 í 2 3 Before pressing the screen will look as in the left hand side of the following figure After pressing the screen will look as in the right hand side of the figure For this example the numerical format was changed to Fix with three decimals Notice that the vector is displayed in Cartes...

Страница 291: ...ular nature The conversion from Cartesian to cylindrical coordinates is such that r x2 y2 1 2 θ tan 1 y x and z z For the case shown above the transformation was such that x y z 3 204 2 112 2 300 produced r θ z 3 536 25o 3 536 At this point change the angular measure to Radians If we now enter a vector of integers in Cartesian form even if the CYLINdrical coordinate system is active it will be sho...

Страница 292: ...ritten in cylindrical polar coordinates have now been changed to the spherical coordinate system The transformation is such that ρ r2 z2 1 2 θ θ and φ tan 1 r z However the vector that originally was set to Cartesian coordinates remains in that form Application of vector operations This section contains some examples of vector operations that you may encounter in Physics or Mechanics applications ...

Страница 293: ...1 calculates θ The steps are shown in the following screens ALG mode of course Thus the result is θ 122 891o In RPN mode use the following 3 5 6 2 1 3 DOT 3 5 6 BS 2 1 3 BS COS NUM Moment of a force The moment exerted by a force F about a point O is defined as the cross product M r F where r also known as the arm of the force is the position vector based at O and pointing towards the point of appl...

Страница 294: ...pace Given a point in space P0 x0 y0 z0 and a vector N Nxi Nyj Nzk normal to a plane containing point P0 the problem is to find the equation of the plane We can form a vector starting at point P0 and ending at point P x y z a generic point in the plane Thus this vector r P0P x x0 i y y0 j z z0 k is perpendicular to the normal vector N since r is contained entirely in the plane We learned that for ...

Страница 295: ...4i 6j 2k is 4x 6y 2z 24 0 In RPN mode use 2 3 1 x y z 4 6 2 DOT EXP ND Row vectors column vectors and lists The vectors presented in this chapter are all row vectors In some instances it is necessary to create a column vector e g to use the pre defined statistical functions in the calculator The simplest way to enter a column vector is by enclosing each vector element within brackets all contained...

Страница 296: ...ble by using TYPE Functions OBJ ARRY and LIST will be available in soft menu keys A B and C Function DROP is available by using STACK DROP Following we introduce the operation of functions OBJ LIST ARRY and DROP with some examples Function OBJ This function decomposes an object into its components If the argument is a list function OBJ will list the list elements in the stack with the number of el...

Страница 297: ...ws and 1 column To build a regular vector we enter the elements of the vector in the stack and in stack level 1 we enter the vector size as a list e g 1 2 3 ä 3 TYPE ARRY To build a column vector of n elements enter the elements of the vector in the stack and in stack level 1 enter the list n 1 For example 1 2 3 ä 1 í3 TYPE ARRY Function DROP This function has the same effect as the delete key ƒ T...

Страница 298: ...orm a row vector to a column vector In RPN mode enter the row vector and then press RXC Try for example 1 2 3 RXC After having defined this variable we can use it in ALG mode to transform a row vector into a column vector Thus change your calculator s mode to ALG and try the following procedure 1 2 3 J RXC Ü î resulting in Transforming a column vector into a row vector To illustrate this transform...

Страница 299: ...nction LIST to create a list 5 Use function ARRY to create the row vector These five steps can be put together into a UserRPL program entered as follows in RPN mode still å TYPE OBJ OBJ STACK DROP TYPE LIST ARRY cxr K A new variable CXR will be available in the soft menu labels after pressing J Press CXR to see the program contained in the variable CXR OBJ OBJ DROP RRY ...

Страница 300: ...de to ALG and try the following procedure 1 2 3 J CXR Ü î resulting in Transforming a list into a vector To illustrate this transformation we ll enter the list 1 2 3 in RPN mode Then follow the next exercise to transform a list into a vector 1 Use function OBJ to decompose the column vector 2 Type a 1 and use function LIST to create a list in stack level 1 3 Use function ARRY to create the vector ...

Страница 301: ...ble LXV we can use it in ALG mode to transform a list into a vector Thus change your calculator s mode to ALG and try the following procedure 1 2 3 J LXV Ü î resulting in Transforming a vector or matrix into a list To transform a vector into a list the calculator provides function AXL You can find this function through the command catalog as follows N axl OK As an example apply function AXL to the...

Страница 302: ...tion we can write matrix A as A aij n m The full matrix is shown next A matrix is square if m n The transpose of a matrix is constructed by swapping rows for columns and vice versa Thus the transpose of matrix A is AT aT ij m n aji m n The main diagonal of a square matrix is the collection of elements aii An identity matrix In n is a square matrix whose main diagonal elements are all equal to 1 an...

Страница 303: ...ed in Chapter 9 matrices can be entered into the stack by using the Matrix Writer For example to enter the matrix first start the matrix writer by using Make sure that the option GO is selected Then use the following keystrokes 2 5 4 2 2 ššš 3 1 9 2 8 2 1 5 At this point the Matrix Writer screen may look like this Press once more to place the matrix on the stack The ALG mode stack is shown next be...

Страница 304: ...brackets Ô and enclose each row of the matrix with an additional set of brackets Ô Commas í should separate the elements of each row as well as the brackets between rows Note In RPN mode you can omit the inner brackets after the first set has been entered thus instead of typing for example 1 2 3 4 5 6 7 8 9 type 1 2 3 4 5 6 7 8 9 For future exercises let s save this matrix under the name A In ALG ...

Страница 305: ...ATRICES CREATE menu available through Ø The MTH MATRIX MAKE sub menu let s call it the MAKE menu contains the following functions while the MATRICES CREATE sub menu let s call it the CREATE menu has the following functions ...

Страница 306: ...ll show how to access functions through use of the matrix MAKE menu At the end of this section we present a table with the keystrokes required to obtain the same functions with the CREATE menu when system flag 117 is set to SOFT menus If you have set that system flag flag 117 to SOFT menu the MAKE menu will be available through the keystroke sequence MATRX MAKE The functions available will be show...

Страница 307: ...ows Notice that we achieve the same result by simply typing 2 3 and pressing In RPN mode this exercise is performed by entering A 3 GET or by using 2 3 Suppose that we want to place the value π into element a31 of the matrix We can use function PUT for that purpose e g In RPN mode you can use J A 3 1 ì PUT Alternatively in RPN mode you can use ì 2 3 K To see the contents of variable A after this o...

Страница 308: ...n 3 1 i e now A 3 1 2 and the index list was increased by 1 by column first i e from 3 1 to 3 2 The matrix is in level 2 and the incremented index list is in level 1 Function SIZE Function SIZE provides a list showing the number of rows and columns of the matrix in stack level 1 The following screen shows a couple of applications of function SIZE in ALG mode In RPN mode these exercises are perform...

Страница 309: ...on CON The function takes as argument a list of two elements corresponding to the number of row and columns of the matrix to be generated and a constant value Function CON generates a matrix with constant elements For example in ALG mode the following command creates a 4 3 matrix whose elements are all equal to 1 5 Note The calculator also includes Function TRAN in the MATRICES OPERATIONS sub menu...

Страница 310: ... e g The resulting identity matrix will have the same dimensions as the argument matrix Be aware that an attempt to use a rectangular i e non square matrix as the argument of IDN will produce an error In RPN mode the two exercises shown above are created by using 4 IDN and A IDN Function RDM Function RDM Re DiMensioning is used to re write vectors and matrices as matrices and vectors The input to ...

Страница 311: ...can use 1 2 3 4 5 6 2 3 RDM to produce the matrix shown above Re dimensioning a matrix into another matrix In ALG mode we now use the matrix created above and re dimension it into a matrix of 3 rows and 2 columns In RPN mode we simply use 3 2 RDM Re dimensioning a matrix into a vector To re dimension a matrix into a vector we use as arguments the matrix followed by a list containing the number of ...

Страница 312: ...own above The random numbers generated are integer numbers uniformly distributed in the range 10 10 i e each one of those 21 numbers has the same probability of being selected Function RANM is useful for generating matrices of any size to illustrate matrix operations or the application of matrix functions Function SUB Function SUB extracts a sub matrix from an existing matrix provided you indicate...

Страница 313: ... matrix before pressing The screen shot to the right shows the application of function RPL to replace the matrix in NS 2 the 2 2 matrix into the 3 3 matrix currently located in NS 1 starting at position 2 2 If working in the RPN mode assuming that the 2 2 matrix was originally in the stack we proceed as follows 1 2 3 4 5 6 7 8 9 this last key swaps the contents of stack levels 1 and 2 1 2 another ...

Страница 314: ...a 3 2 matrix was to be created using as main diagonal elements as many elements as possible form the vector 1 2 3 4 5 The main diagonal for a rectangular matrix starts at position 1 1 and moves on to position 2 2 3 3 etc until either the number of rows or columns is exhausted In this case the number of columns 2 was exhausted before the number of rows 3 so the main diagonal included only the eleme...

Страница 315: ...rograms to build a matrix out of a number of lists of objects The lists may represent columns of the matrix program CRMC or rows of the matrix program CRMR The programs are entered with the calculator set to RPN mode and the instructions for the keystrokes are given for system flag 117 set to SOFT menus This section is intended for you to practice accessing programming functions in the calculator ...

Страница 316: ...ut of n lists of p elements each To create the program enter the following keystrokes Keystroke sequence Produces å STACK DUP DUP é n n å 1 STACK SWAP 1 SWAP BRCH FOR FOR FOR j j TYPE OBJ OBJ ARRY ARRY BRCH IF IF IF j j n n TEST BRCH IF THEN THEN j 1 j 1 STACK L ROLL ROLL BRCH IF END END BRCH FOR NEXT NEXT BRCH IF IF IF n 1 n 1 TEST BRCH IF THEN THEN 1 1 n 1 n 1 BRCH FOR FOR FOR j j j 1 j 1 STACK ...

Страница 317: ...n example try the following exercise 1 2 3 4 1 4 9 16 1 8 27 64 3 CRMC The following screen shots show the RPN stack before and after running program CRMC To use the program in ALG mode press CRMC followed by a set of parentheses Ü Within the parentheses type the lists of data representing the columns of the matrix separated by commas and finally a comma and the number of columns The command shoul...

Страница 318: ...e crmr K 1 2 3 4 1 4 9 16 1 8 27 64 3 CRMR The following screen shots show the RPN stack before and after running program CRMR These programs can be useful for statistical applications specifically to create the statistical matrix ΣDAT Examples of the use of these program are shown in a latter chapters Manipulating matrices by columns The calculator provides a menu with functions for manipulating ...

Страница 319: ...and decomposes it into vectors corresponding to its columns An application of function COL in ALG mode is shown below The matrix used has been stored earlier in variable A The matrix is shown in the figure to the left The figure to the right shows the matrix decomposed in columns To see the full result use the line editor triggered by pressing In RPN mode you need to list the matrix in the stack a...

Страница 320: ...matrix Here is an example in ALG mode The command used was COL 1 2 3 4 5 6 7 8 9 3 In RPN mode place the n vectors in stack levels n 1 n n 1 2 and the number n in stack level 1 With this set up function COL places the vectors as columns in the resulting matrix The following figure shows the RPN stack before and after using function COL Function COL Function COL takes as argument a matrix a vector ...

Страница 321: ...e is an example in the ALG mode using the matrix stored in A In RPN mode place the matrix in the stack first then enter the number representing a column location before applying function COL The following figure shows the RPN stack before and after applying function COL Function CSWP Function CSWP Column SWaP takes as arguments two indices say i and j representing two distinct columns in a matrix ...

Страница 322: ...2 and 3 have been swapped Swapping of columns and of rows see below is commonly used when solving systems of linear equations with matrices Details of these operations will be given in a subsequent Chapter Manipulating matrices by rows The calculator provides a menu with functions for manipulating matrices by operating in their rows This menu is available through the MTH MATRIX ROW sequence shown ...

Страница 323: ...is shown below The matrix used has been stored earlier in variable A The matrix is shown in the figure to the left The figure to the right shows the matrix decomposed in rows To see the full result use the line editor triggered by pressing In RPN mode you need to list the matrix in the stack and the activate function ROW i e A ROW The figure below shows the RPN stack before and after the applicati...

Страница 324: ...ack level 1 With this set up function ROW places the vectors as rows in the resulting matrix The following figure shows the RPN stack before and after using function ROW Function ROW Function ROW takes as argument a matrix a vector with the same length as the number of rows in the matrix and an integer number n representing the location of a row Function ROW inserts the vector in row n of the matr...

Страница 325: ...PN stack before and after applying function ROW Function RSWP Function RSWP Row SWaP takes as arguments two indices say i and j representing two distinct rows in a matrix and a matrix and produces a new matrix with rows i and j swapped The following example in ALG mode shows an application of this function We use the matrix stored in variable A for the example This matrix is listed first In RPN mo...

Страница 326: ...he right hand side figure shows the resulting matrix after function RCI is activated Function RCIJ Function RCIJ stands for take Row I and multiplying it by a constant C and then add that multiplied row to row J replacing row J with the resulting sum This type of row operation is very common in the process of Gaussian or Gauss Jordan elimination more details on this procedure are presented in a su...

Страница 327: ...he constant value then by the row to be multiplied by the constant value and finally enter the row that will be replaced The following figure shows the RPN stack before and after applying function RCIJ under the same conditions as in the ALG example shown above ...

Страница 328: ...matrix Details of these operations are presented next To illustrate the operations we will create a number of matrices that we will store in variables The generic name of the matrices will be Aij and Bij where i represents the number of rows and j the number of columns of the matrices The matrices to be used are generated by using function RANM random matrices If you try this exercise in your calc...

Страница 329: ...e 22 B22 22 B22 23 B23 23 B23 32 B32 32 B32 Translating the ALG examples to RPN is straightforward as illustrated here The remaining examples of matrix operations will be performed in ALG mode only Multiplication There are numerous multiplication operations that involve matrices These are described next Multiplication by a scalar Multiplication of the matrix A aij m n by a scalar k results in the ...

Страница 330: ...ers e g Matrix vector multiplication Matrix vector multiplication is possible only if the number of columns of the matrix is equal to the length of the vector This operation follows the rules of matrix multiplication as shown in the next section A couple of examples of matrix vector multiplication follow Vector matrix multiplication on the other hand is not defined This multiplication can be perfo...

Страница 331: ...ve i e in general A B B A Furthermore one of the multiplications may not even exist The following screen shots show the results of multiplications of the matrices that we stored earlier The matrix vector multiplication introduced in the previous section can be thought of as the product of a matrix m n with a matrix n 1 i e a column vector resulting in an m 1 matrix i e another vector To verify thi...

Страница 332: ...f the same dimensions This function is available through Function catalog N or through the MATRICES OPERATIONS sub menu Ø Applications of function HADAMARD are presented next Raising a matrix to a real power You can raise a matrix to any power as long as the power is either an integer or a real number with no fractional part The following example shows the result of raising matrix B22 created earl...

Страница 333: ...cker s delta function Identity matrices can be obtained by using function IDN described in Chapter 9 The identity matrix has the property that A I I A A To verify this property we present the following examples using the matrices stored earlier on The inverse matrix The inverse of a square matrix A is the matrix A 1 such that A A 1 A 1 A I where I is the identity matrix of the same dimensions as A...

Страница 334: ...ALIZE menu is accessed through the keystroke sequence system flag 117 set to CHOOSE boxes This menu contains the following functions These functions are described next Because many of these functions use concepts of matrix theory such as singular values rank etc we will include short descriptions of these concepts intermingled with the description of functions ...

Страница 335: ... vector then the Frobenius norm A F is simply the vector s magnitude Function ABS is accessible directly in the keyboard as Ê Try the following exercises in ALG mode using the matrices stored earlier for matrix operations Function SNRM Function SNRM calculates the Spectral NoRM of a matrix which is defined as the matrix s largest singular value also known as the Euclidean norm of the matrix For ex...

Страница 336: ...trices and S is a diagonal matrix The diagonal elements of S are called the singular values of A and are usually ordered so that si si 1 for i 1 2 n 1 The columns uj of U and vj of V are the corresponding singular vectors Orthogonal matrices are such that U UT I A diagonal matrix has non zero elements only along its main diagonal The rank of a matrix can be determined from its SVD by counting the ...

Страница 337: ...lating eigenvalues and eigenvectors are presented later in the chapter Condition number of a matrix The condition number of a square non singular matrix is defined as the product of the matrix norm times the norm of its inverse i e cond A A A 1 We will choose as the matrix norm A the maximum of its row norm RNRM and column norm CNRM while the norm of the inverse A 1 will be selected as the minimum...

Страница 338: ...NV A33 CNRM INV A33 0 261044 Thus the condition number is also calculated as CNRM A33 CNRM INV A33 COND A33 6 7871485 Function RANK Function RANK determines the rank of a square matrix Try the following examples The rank of a matrix The rank of a square matrix is the maximum number of linearly independent rows or columns that the matrix contains Suppose that you write a square matrix An n as A c1 ...

Страница 339: ...s the determinant of a square matrix For example where the values dj are constant we say that ck is linearly dependent on the columns included in the summation Notice that the values of j include any value in the set 1 2 n in any combination as long as j k If the expression shown above cannot be written for any of the column vectors then we say that all the columns are linearly independent A simil...

Страница 340: ... determinant is therefore A 3 3 determinant is calculated by augmenting the determinant an operation that consists on copying the first two columns of the determinant and placing them to the right of column 3 as shown in the diagram below The diagram also shows the elements to be multiplied with the corresponding sign to attach to their product in a similar fashion as done earlier for a 2 2 determ...

Страница 341: ...wer level with cofactors of order n 2 n 2 and so on until we are left only with a long sum of 2 2 determinants The 2 2 determinants are then calculated through the method shown above The method of calculating a determinant by cofactor expansion is very inefficient in the sense that it involves a number of operations that grows very fast as the size of the determinant increases A more efficient met...

Страница 342: ...OPER menu The matrix OPER OPERATIONS is available through the keystroke sequence Ø system flag 117 set to CHOOSE boxes The OPERATIONS menu includes the following functions Functions ABS CNRM COND DET RANK RNRM SNRM TRACE and TRAN are also found in the MTH MATRIX NORM menu the subject of the previous section Function SIZE was presented in Chapter 10 Function HADAMARD was presented earlier in the co...

Страница 343: ...s corresponding decimal or approximate form Function LCXM Function LCXM can be used to generate matrices such that the element aij is a function of i and j The input to this function consists of two integers n and m representing the number of rows and columns of the matrix to be generated and a program that takes i and j as input The numbers n m and the program occupy stack levels 3 2 and 1 respec...

Страница 344: ...ed in RPN mode Solution of linear systems A system of n linear equations in m variables can be written as a11 x1 a12 x2 a13 x3 a1 m 1 x m 1 a1 m x m b1 a21 x1 a22 x2 a23 x3 a2 m 1 x m 1 a2 m x m b2 a31 x1 a32 x2 a33 x3 a3 m 1 x m 1 a3 m x m b3 an 1 1 x1 an 1 2 x2 an 1 3 x3 an 1 m 1 x m 1 an 1 m x m bn 1 an1 x1 an2 x2 an3 x3 an m 1 x m 1 an m x m bn This system of linear equations can be written as...

Страница 345: ...the matrix A in the format a11 a12 in the A field Also enter the vector b in the B field When the X field is highlighted press SOLVE If a solution is available the solution vector x will be shown in the X field The solution is also copied to stack level 1 Some examples follow A square system The system of linear equations 2x1 3x2 5x3 13 x1 3x2 8x3 13 2x1 2x2 4x3 6 can be written as the matrix equa...

Страница 346: ...er while the A field is selected The following screen shows the Matrix Writer used for entering matrix A as well as the input form for the numerical solver after entering matrix A press in the Matrix Writer Press to select the B field The vector b can be entered as a row vector with a single set of brackets i e 13 13 6 OK After entering matrix A and vector b and with the X field highlighted we can...

Страница 347: ... represents a plane in the three dimensional Cartesian coordinate system x1 x2 x3 The solution to the system of equations shown above will be the intersection of two planes in space We know however that the intersection of two non parallel planes is a straight line and not a single point Therefore there is more than one point that satisfy the system In that sense the system is not uniquely determi...

Страница 348: ...nment press The procedure that we describe next can be used to copy the matrix A and the solution vector X into the stack To check that the solution is correct try the following Press to highlight the A field Press L CALC to copy matrix A onto the stack Press OK to return to the numerical solver environment Press CALC to copy solution vector X onto the stack Press OK to return to the numerical sol...

Страница 349: ...e This result indicates that x 15 10 3 10 is also a solution to the system confirming our observation that a system with more unknowns than equations is not uniquely determined under determined How does the calculator came up with the solution x 15 37 2 46 9 62 shown earlier Actually the calculator minimizes the distance from a point which will constitute the solution to each of the planes represe...

Страница 350: ...s not unique Some numerical algorithms can be used to force a solution to the system by minimizing the distance from the presumptive solution point to each of the lines in the system Such is the approach followed by the calculator numerical solver Let s use the numerical solver to attempt a solution to this system of equations Ï OK Enter matrix A and vector b as illustrated in the previous example...

Страница 351: ...esult in a variable X and the matrix into variable A as follows Press K x to store the solution vector into variable X Press ƒ ƒ ƒ to clear three levels of the stack Press K a to store the matrix into variable A Now let s verify the solution by using A X which results in the vector 8 6917 3 4109 1 1301 which is not equal to 15 5 22 the original vector b The solution is simply the point that is clo...

Страница 352: ...f equations LSQ returns the solution with the minimum residual value e A x b The system of equations may not have a solution therefore the value returned is not a real solution to the system just the one with the smallest residual Function LSQ takes as input vector b and matrix A in that order Function LSQ can be found in Function catalog N Next we use function LSQ to repeat the solutions found ea...

Страница 353: ... 3x2 5x3 10 x1 3x2 8x3 85 with The solution using LSQ is shown next Over determined system Consider the system x1 3x2 15 2x1 5x2 5 x1 x2 22 with The solution using LSQ is shown next 85 10 8 3 1 5 3 2 3 2 1 b x A and x x x 22 5 15 1 1 5 2 3 1 2 1 b x A and x x ...

Страница 354: ... x1 3x2 8x3 13 2x1 2x2 4x3 6 we can find the solution in the calculator as follows which is the same result found earlier Solution by division of matrices While the operation of division is not defined for matrices we can use the calculator s key to divide vector b by matrix A to solve for x in the matrix equation A x b This is an arbitrary extension of the algebraic division operation to matrices...

Страница 355: ...matrix Suppose that you want to solve the following three sets of equations X 2Y 3Z 14 2X 4Y 6Z 9 2X 4Y 6Z 2 3X 2Y Z 2 3X 2Y Z 5 3X 2Y Z 2 4X 2Y Z 5 4X 2Y Z 19 4X 2Y Z 12 We can write the three systems of equations as a single matrix equation A X B where The sub indices in the variable names X Y and Z determine to which equation system they refer to To solve this expanded system we use the followi...

Страница 356: ...pper triangular form allows for the solution of all n unknowns utilizing only one equation at a time in a procedure known as backward substitution Example of Gaussian elimination using equations To illustrate the Gaussian elimination procedure we will use the following system of 3 equations in 3 unknowns 2X 4Y 6Z 14 3X 2Y Z 3 4X 2Y Z 4 We can store these equations in the calculator in variables E1...

Страница 357: ... equation E2 by equation 2 3 equation 1 i e E1 3 E2 and the third by equation 3 4 equation 1 to get Next divide the second equation by 8 to get Next replace the third equation E3 with equation 3 6 equation 2 i e E2 6 E3 to get Notice that when we perform a linear combination of equations the calculator modifies the result to an expression on the left hand side of the equal sign i e ...

Страница 358: ...finding the values of the unknowns starting from the last equation and working upwards Thus we solve for Z first Next we substitute Z 2 into equation 2 E2 and solve E2 for Y Next we substitute Z 2 and Y 1 into E1 and solve E1 for X The solution is therefore X 1 Y 1 Z 2 Example of Gaussian elimination using matrices The system of equations used in the example above can be written as a matrix equati...

Страница 359: ...nu In your calculator use the following keystrokes First enter the augmented matrix and make an extra copy of the same in the stack This step is not necessary except as an insurance that you have an extra copy of the augmented matrix saved in case you make a mistake in the forward elimination procedure that we are about to undertake 2 4 6 14 3 2 1 3 4 2 1 4 Save augmented matrix in variable AAUG a...

Страница 360: ... 3Z 7 Y Z 3 7Z 14 which can now be solved one equation at a time by backward substitution as in the previous example Gauss Jordan elimination using matrices Gauss Jordan elimination consists in continuing the row operations in the upper triangular matrix resulting from the forward elimination process until an identity matrix results in place of the original A matrix For example for the case we jus...

Страница 361: ...nt is called a pivot element or simply a pivot In many situations it is possible that the pivot element become zero in which case we cannot divide the row by its pivot Also to improve the numerical solution of a system of equations using Gaussian or Gauss Jordan elimination it is recommended that the pivot be the element with the largest absolute value in a given column In such cases we exchange r...

Страница 362: ...g is also registered as a row or column exchange respectively in the permutation matrix When the solution is achieved then we multiply the permutation matrix by the unknown vector x to obtain the order of the unknowns in the solution In other words the final solution is given by P x b where b is the last column of the augmented matrix after the solution has been found Example of Gauss Jordan elimi...

Страница 363: ...ix now are Checking the pivot at position 1 1 we now find that 16 is a better pivot than 8 thus we perform a column swap as follows 1 2 N OK RSWP The augmented matrix and the permutation matrix now are Now we have the largest possible value in position 1 1 i e we performed full pivoting at 1 1 Next we proceed to divide by the pivot 16Y1L RCI The permutation matrix does not change but the augmented...

Страница 364: ...sing 8 25 2 L RCI Next we eliminate the 3 from position 3 2 by using 3 2 3 RCIJ Having filled with zeroes the position below the pivot we proceed to check the pivot at position 3 3 The current value of 2 is larger than or 0 thus we keep it unchanged We do divide the whole third row by 2 to convert the pivot to 1 by using 2Y3 RCI Next we proceed to eliminate the in position 1 3 by using 1 1 16 1 2 ...

Страница 365: ...P as The solution is given by P x b or Which results in Step by step calculator procedure for solving linear systems The example we just worked is of course the step by step user driven procedure to use full pivoting for Gauss Jordan elimination solution of linear equation systems You can see the step by step procedure used by the calculator to solve a system of equations without user intervention...

Страница 366: ... have corresponded to 2 1 1 RCIJ Press OK and follow the operations in your calculator s screen You will see the following operations performed L3 L3 8 L1 L1 2 L1 1 L2 L1 25 L1 3 L3 L2 25 L2 3 L3 and finally a message indicating Reduction result showing When you press OK the calculator returns the final result 1 2 1 Calculating the inverse matrix step by step The calculation of an inverse matrix c...

Страница 367: ...inverse is based on the augmented matrix Aaug n n A n n In n The calculator showed you the steps up to the point in which the left hand half of the augmented matrix has been converted to a diagonal matrix From there the final step is to divide each row by the corresponding main diagonal pivot In other words the calculator has transformed Aaug n n A n n In n into I A 1 Inverse matrices and determin...

Страница 368: ...mined a solution can be produced by using Function LSQ Least SQuares as shown earlier The calculator however offers other possibilities for solving linear systems of equations by using Functions included in the MATRICES LINEAR SYSTEMS menu accessible through Ø Set system flag 117 to CHOOSE boxes The functions included are LINSOLVE REF rref RREF and SYST2MAT Function LINSOLVE Function LINSOLVE take...

Страница 369: ...limination procedure is known as an echelon form Function REF Reduce to Echelon Form produces such a matrix given the augmented matrix in stack level 1 Consider the augmented matrix Representing a linear system of equations A x b where A 1 2 1 2 1 2 5 2 1 and b 0 3 12 Enter the augmented matrix and save it into variable AAUG in ALG mode 1 2 1 0 2 1 2 3 5 2 1 12 UG Application of function REF produ...

Страница 370: ...x resulting from a Gauss Jordan elimination without pivoting A row reduced echelon form for an augmented matrix can be obtained by using function rref This function produces a list of the pivots and an equivalent matrix in row reduced echelon form so that the matrix of coefficients is reduced to a diagonal matrix For example for matrix AAUG function rref produces the following result The second sc...

Страница 371: ... numerical method we produce as a first approximation the solution x 0 Evaluating f x 0 b A x 0 e 0 Thus e is a vector of residuals of Function for the vector x x 0 To use Function RSD you need the terms b A and x 0 as arguments The vector returned is e b A x 0 For example using A 2 1 0 2 x 0 1 8 2 7 and b 1 6 we can find the vector of residuals as follows The result is e b A x 0 0 1 0 6 Note If w...

Страница 372: ...raic equation involving a polynomial of order n for a square matrix An n The resulting equation is known as the characteristic polynomial of matrix A Solving the characteristic polynomial produces the eigenvalues of the matrix The calculator provides a number of functions that provide information regarding the eigenvalues and eigenvectors of a square matrix Some of these functions are located unde...

Страница 373: ...xercise produces an empty list as the solution Change mode to Approx and repeat the entry to get the following eigenvalues 1 38 2 22 1 38 2 22 1 76 0 Function EGV Function EGV EiGenValues and eigenvectors produces the eigenvalues and eigenvectors of a square matrix The eigenvectors are returned as the columns Note In some cases you may not be able to find an exact solution to the characteristic po...

Страница 374: ... The eigenvalues are In summary λ1 0 29 x1 1 00 0 79 0 91 T λ2 3 16 x2 1 00 0 51 0 65 T λ3 7 54 x1 0 03 1 00 0 84 T Function JORDAN Function JORDAN is intended to produce the diagonalization or Jordan cycle decomposition of a matrix In RPN mode given a square matrix A function JORDAN produces four outputs namely The minimum polynomial of matrix A stack level 4 The characteristic polynomial of matr...

Страница 375: ...ots Function MAD This function although not available in the EIGEN menu also provides information related to the eigenvalues of a matrix Function MAD is available through the MATRICES OPERATIONS sub menu Ø and is intended to produce the adjoint matrix of a matrix In RPN mode function MAD generates a number of properties of a square matrix namely the determinant stack level 4 the formal inverse sta...

Страница 376: ... 0 0 0 1 2 1 2 1 4 1 2 1 6 1 2 3 2 4 2 3 2 7 1 X 3 6 x 2 2 X 8 The same exercise in ALG mode will look as follows Matrix factorization Matrix factorization or decomposition consists of obtaining matrices that when multiplied result in a given matrix We present matrix decomposition through the use of Functions contained in the matrix FACT menu This menu is accessed through Ø Function contained in t...

Страница 377: ... let matrix U v1 v2 vn where the vi i 1 2 n are column vectors and if vi vj δij where δij is the Kronecker s delta function then U will be an orthogonal matrix This conditions also imply that U UT I The Singular Value Decomposition SVD of a rectangular matrix Am n consists in determining the matrices U S and V such that Am n U m m S m n V T n n where U and V are orthogonal matrices and S is a diag...

Страница 378: ...x A returning matrices Q and T in stack levels 2 and 1 respectively such that A Q T QT where Q is an orthogonal matrix and T is a triangular matrix For example in RPN mode 2 3 1 5 4 2 7 5 4 SCHUR results in 2 0 66 0 29 0 70 0 73 0 01 0 68 0 19 0 96 0 21 1 1 03 1 02 3 86 0 5 52 8 23 0 1 82 5 52 Function LQ The LQ function produces the LQ factorization of a matrix An m returning a lower Ln m trapezo...

Страница 379: ...rms A quadratic form from a square matrix A is a polynomial expression originated from x A xT For example if we use A 2 1 1 5 4 2 3 5 1 and x X Y Z T the corresponding quadratic form is calculated as Finally x A xT 2X2 4Y2 Z2 6XY 2XZ 7ZY The QUADF menu The calculator provides the QUADF menu for operations related to QUADratic Forms The QUADF menu is accessed through Ø Note Examples and definitions...

Страница 380: ...QXA takes as arguments a quadratic form in stack level 2 and a vector of variables in stack level 1 returning the square matrix A from which the quadratic form is derived in stack level 2 and the list of variables in stack level 1 For example X 2 Y 2 Z 2 4 X Y 16 X Z X Y Z QX returns 2 1 2 8 2 1 0 8 0 1 1 X Y Z Diagonal representation of a quadratic form Given a symmetric square matrix A it is pos...

Страница 381: ...resentation of a quadratic form Q x A xT taking as arguments the quadratic form in stack level 2 and the vector of variables in stack level 1 The result of this function call is the following An array of coefficients representing the diagonal terms of D stack level 4 A matrix P such that A PT D P stack level 3 The diagonalized quadratic form stack level 2 The list of variables stack level 1 For ex...

Страница 382: ...55 Information on the functions listed in this menu is presented below by using the calculator s own help facility The figures show the help facility entry and the attached examples Function IMAGE Function ISOM ...

Страница 383: ...Page 11 56 Function KER Function MKISOM ...

Страница 384: ...uence ô D Please notice that if you are using the RPN mode these two keys must be pressed simultaneously to activate any of the graph functions After activating the 2D 3D function the calculator will produce the PLOT SETUP window which includes the TYPE field as illustrated below Right in front of the TYPE field you will most likely see the option Function highlighted This is the default type of g...

Страница 385: ... grids Ps Contour for plotting contour plots of surfaces Y Slice for plotting a slicing view of a function f x y Gridmap for plotting real and imaginary part traces of a complex function Pr Surface for parametric surfaces given by x x u v y y u v z z u v Plotting an expression of the form y f x In this section we present an example of a plot of a function of the form y f x In order to proceed with...

Страница 386: ...lator display Θ Note You will notice that a new variable called PPAR shows up in your soft menu key labels This stands for Plot PARameters To see its contents press PPAR A detailed explanation of the contents of PPAR is provided later in this Chapter Press ƒ to drop this line from the stack Note Two new variables show up in your soft menu key labels namely EQ and Y1 To see the contents of EQ use E...

Страница 387: ...o see labels EDIT L LABEL MENU Θ To recover the first graphics menu LL PICT Θ To trace the curve TRACE X Y Then use the right and left arrow keys š to move about the curve The coordinates of the points you trace will be shown at the bottom of the screen Check that for x 1 05 y 0 231 Also check that for x 1 48 y 0 134 Here is picture of the graph in tracing mode Θ To recover the menu and return to ...

Страница 388: ... now Enter the PLOT WINDOW environment by entering ò press them simultaneously if in RPN mode Keep the range of 4 to 4 for H VIEW press AUTO to generate the V VIEW To plot the graph press ERASE DRAW Θ Once the graph is plotted press FCN to access the function menu With this menu you can obtain additional information about the plot such as intersects with the x axis roots slopes of the tangent line...

Страница 389: ...vailable in the first menu are AREA to calculate the area under the curve and SHADE to shade an area under the curve Press L to see more options The second menu includes one button called VIEW that flashes for a few seconds the equation plotted Press VIEW Alternatively you can press the button NEXQ NEXt eQuation to see the name of the function Y1 x Press L to recover the menu Θ The button gives th...

Страница 390: ...or future use If you want to save your graph to a variable get into the PICTURE environment by pressing š Then press EDIT LL PICT This captures the current picture into a graphics object To return to the stack press PICT CANCL In level 1 of the stack you will see a graphics object described as Graphic 131 64 This can be stored into a variable name say PIC1 To display your figure again recall the c...

Страница 391: ...to return to normal calculator display Next we ll resize the plot window First press simultaneously if in RPN mode the left shift key and the ñ A key to produce the PLOT FUNCTION window If there is any equation highlighted in this window press DEL as needed to clear the window completely When the PLOT FUNCTION window is empty you will get a prompt message that reads No Equ Press ADD Press the soft...

Страница 392: ...ve the cursor along the curve As you move the cursor along the curve the coordinates of the curve are displayed at the bottom of the screen Check that when Y 1 00E0 X 2 72E0 This is the point e 1 since ln e 1 Press L to recover the graphics menu Next we will find the intersection of the curve with the x axis by pressing FCN ROOT The calculator returns the value Root 1 confirming that ln 1 0 Press ...

Страница 393: ...o the graph press EDIT L LABEL Press MENU to remove the menu labels and get a full view of the graph Press LL PICT CANCL to return to the PLOT WINDOW FUNCTION Press to return to normal calculator display Next press X to see the contents of this variable A value of 10 275 is placed in the stack This value is determined by our selection for the horizontal display range We selected a range between 1 ...

Страница 394: ...lt value zero 0 which specifies increments in X corresponding to 1 pixel in the graphics display The next element in PPAR is a list containing first the coordinates of the point of intersection of the plot axes i e 0 0 followed by a list that specifies the tick mark annotation on the x and y axes respectively 10d 10d Next PPAR lists the type of plot that is to be generated i e FUNCTION and finally...

Страница 395: ...ertical range Press ERASE DRAW to produce the graph of y ln x y exp x and y x simultaneously if in RPN mode You will notice that only the graph of y exp x is clearly visible Something went wrong with the AUTO selection of the vertical range What happens is that when you press AUTO in the PLOT FUNCTION WINDOW screen the calculator produces the vertical range corresponding to the first function in t...

Страница 396: ...k and V Tick will be separated by that many pixels Θ The default value for both by H Tick and V Tick is 10 Soft key menu options Θ Use EDIT to edit functions of values in the selected field Θ Use CHOOS to select the type of plot to use when the Type field is highlighted For the current exercises we want this field set to FUNCTION Θ Press the AXES soft menu key to select or deselect the plotting of...

Страница 397: ... the graph according to the current contents of PPAR for the equations listed in the PLOT FUNCTION window Θ Press L to activate the second menu list Θ Use and to move the selected equation one location up or down respectively Θ Use CLEAR if you want to clear all the equations currently active in the PLOT FUNCTION window The calculator will verify whether or not you want to clear all the functions ...

Страница 398: ...options Θ Use EDIT to edit any entry in the window Θ Use AUTO as explained in Settings above Θ Use ERASE to erase any graph currently existing in the graphics display window Θ Use DRAW to produce the graph according to the current contents of PPAR for the equations listed in the PLOT FUNCTION window Θ Press L to activate the second menu list Θ Use RESET to reset the field selected i e where the cu...

Страница 399: ...ely or simultaneously can be used to plot any function of the form y f x It is left as an exercise to the reader to produce the plots of trigonometric and hyperbolic functions and their inverses The table below suggests the values to use for the vertical and horizontal ranges in each case You can include the function Y X when plotting simultaneously a function and its inverse to verify their refle...

Страница 400: ...ress OK Θ To accept the changes made to the PLOT SETUP screen press L OK You will be returned to normal calculator display Θ The next step is to access the Table Set up screen by using the keystroke combination õ i e soft key E simultaneously if in RPN mode This will produce a screen where you can select the starting value Start and the increment Step Enter the following 5 OK 0 5 OK 0 5 OK i e Zoo...

Страница 401: ... factor 0 5 to produce the new increment of 0 25 Thus the zoom in option is useful when you want more resolution for the values of x in your table Θ To increase the resolution by an additional factor of 0 5 press ZOOM select In once more and press OK The x increment is now 0 0125 Θ To recover the previous x increment press ZOOM OK to select the option Un zoom The x increment is increased to 0 25 Θ...

Страница 402: ...8 OK 8 OK and the V VIEW range to 6 to 2 by using 6 OK 2 OK Θ Change the Indep Low value to 0 and the High value to 6 28 2π by using 0 OK 6 28 OK Θ Press ERASE DRAW to plot the function in polar coordinates The result is a curve shaped like a hearth This curve is known as a cardiod cardios Greek for heart Θ Press EDIT L LABEL MENU to see the graph with labels Press L to recover the menu Press L PI...

Страница 403: ...s ERASE DRAW to see the two equations plotted in the same figure The result is two intersecting cardioids Press CANCL to return to normal calculator display Plotting conic curves The most general form of a conic curve in the x y plane is Ax2 By2 Cxy Dx Ey F 0 We also recognize as conic equations those given in the canonical form for the following figures Θ circle x xo 2 y yo 2 r2 Θ ellipse x xo 2 ...

Страница 404: ... in RPN mode and select Conic as the TYPE The list of equations will be listed in the EQ field Θ Make sure that the independent variable Indep is set to X and the dependent variable Depnd to Y Θ Press L OK to return to normal calculator display Θ Enter the PLOT WINDOW environment by pressing ò simultaneously if in RPN mode Θ Change the range for H VIEW to 3 to 3 by using 3 OK 3 OK Also change the ...

Страница 405: ...system of equations x x t and y y t where t is known as the parameter An example of such graph is the trajectory of a projectile x t x0 v0 COS θ0 t y t y0 v0 sin θ0 t g t2 To plot equations like these Note The H View and V View ranges were selected to show the intersection of the two curves There is no general rule to select those ranges except based on what we know about the curves For example fo...

Страница 406: ...ange TYPE to Parametric by pressing CHOOS OK Θ Press and type X t i Y t OK to define the parametric plot as that of a complex variable The real and imaginary parts of the complex variable correspond to the x and y coordinates of the curve Θ The cursor is now in the Indep field Press t OK to change the independent variable to t Θ Press L OK to return to normal calculator display Θ Press ò simultane...

Страница 407: ... any point on the graph Use and š to move the cursor about the curve At the bottom of the screen you will see the value of the parameter t and coordinates of the cursor as X Y Θ Press L CANCL to return to the PLOT WINDOW environment Then press or L OK to return to normal calculator display A review of your soft menu key labels shows that you now have the following variables t EQ PPAR Y X g θ0 V0 Y...

Страница 408: ...e of values X Y for an expression of the form Y f X i e a Function type of graph In this section we present the procedure for generating a table corresponding to a parametric plot For this purpose we ll take advantage of the parametric equations defined in the example above Θ First let s access the TABLE SETUP window by pressing õ simultaneously if in RPN mode For the independent variable change t...

Страница 409: ... the dependent variable default name Y will be plotted in the vertical axis Θ Press The cursor is now in the Indep field Press t OK to change the independent variable to t Θ Press L OK to return to normal calculator display Θ Press ò simultaneously if in RPN mode to access the PLOT window in this case it will be called PLOT WINDOW DIFF EQ Θ Change the H VIEW and V VIEW parameters to read H VIEW 15...

Страница 410: ...labels for the axes are shown as 0 horizontal and 1 vertical These are the definitions for the axes as given in the PLOT WINDOW screen see above i e H VAR t 0 and V VAR x 1 Θ Press LL PICT to recover menu and return to PICT environment Θ Press X Y to determine coordinates of any point on the graph Use and š to move the cursor in the plot area At the bottom of the screen you will see the coordinate...

Страница 411: ...ecause the calculator samples the entire plotting domain point by point it takes a few minutes to produce a truth plot The present plot should produce a shaded ellipse of semi axes 6 and 3 in x and y respectively centered at the origin Θ Press EDIT L LABEL MENU to see the graph with labels The window parameters are such that you only see half of the labels in the x axis Press L to recover the menu...

Страница 412: ...t only for these types of plots but also for all kind of statistical applications as will be shown in Chapter 18 As a matter of fact the use of histogram plots is postponed until we get to that chapter for the plotting of a histogram requires to perform a grouping of data and a frequency analysis before the actual plot In this section we will show how to load data in the variable ΣDAT and how to p...

Страница 413: ...This is the matrix we stored earlier into ΣDAT Θ Highlight the Col field This field lets you choose the column of ΣDAT that is to be plotted The default value is 1 Keep it to plot column 1 in ΣDAT Θ Press L OK to return to normal calculator display Θ Press ò simultaneously if in RPN mode to access the PLOT WINDOW screen Θ Change the V View to read V View 0 5 Θ Press ERASE DRAW to draw the bar plot...

Страница 414: ...ss ERASE DRAW Θ Press CANCL to return to the PLOT WINDOW screen then to return to normal calculator display Scatter plots We will use the same ΣDAT matrix to produce scatter plots First we will plot the values of y vs x then those of y vs z as follows Θ Press ô simultaneously if in RPN mode to access to the PLOT SETUP window Θ Change TYPE to Scatter Θ Press to highlight the Cols field Enter 1 OK 2...

Страница 415: ... to the PLOT SETUP window Θ Press to highlight the Cols field Enter 3 OK 2 OK to select column 3 as X and column 2 as Y in the Y vs X scatter plot Θ Press L OK to return to normal calculator display Θ Press ò simultaneously if in RPN mode to access the PLOT WINDOW screen Θ Change the plot window ranges to read H View 0 7 V View 0 7 Θ Press ERASE DRAW to draw the bar plot Press EDIT L LABEL MENU to...

Страница 416: ...e sure that X is selected as the Indep and Y as the Depnd variables Θ Press L OK to return to normal calculator display Θ Press ò simultaneously if in RPN mode to access the PLOT WINDOW screen Θ Change the plot window ranges to read X Left 5 X Right 5 Y Near 5 Y Far 5 Θ Press ERASE DRAW to draw the slope field plot Press EDIT L LABEL MENU to see the plot unencumbered by the menu and with identifyi...

Страница 417: ...NCL to return to the PLOT WINDOW environment Then press or L OK to return to normal calculator display Fast 3D plots Fast 3D plots are used to visualize three dimensional surfaces represented by equations of the form z f x y For example if you want to visualize z f x y x2 y2 we can use the following Θ Press ô simultaneously if in RPN mode to access to the PLOT SETUP window Θ Change TYPE to Fast3D ...

Страница 418: ...done press EXIT Θ Press CANCL to return to the PLOT WINDOW environment Θ Change the Step data to read Step Indep 20 Depnd 16 Θ Press ERASE DRAW to see the surface plot Sample views Θ When done press EXIT Θ Press CANCL to return to PLOT WINDOW Θ Press or L OK to return to normal calculator display Try also a Fast 3D plot for the surface z f x y sin x2 y2 Note The Step Indep and Depnd values represe...

Страница 419: ...Θ Change TYPE to Wireframe Θ Press and type X 2 Y 3 OK Θ Make sure that X is selected as the Indep and Y as the Depnd variables Θ Press L OK to return to normal calculator display Θ Press ò simultaneously if in RPN mode to access the PLOT WINDOW screen Θ Keep the default plot window ranges to read X Left 1 X Right 1 Y Near 1 Y Far 1 Z Low 1 Z High 1 XE 0 YE 3 ZE 0 Step Indep 10 Depnd 8 The coordin...

Страница 420: ... plot Θ Press EDIT L LABEL MENU to see the graph with labels and ranges This version of the graph occupies more area in the display than the previous one We can change the viewpoint once more to see another version of the graph Θ Press LL PICT CANCL to return to the PLOT WINDOW environment Θ Change the eye coordinate data to read XE 3 YE 3 ZE 3 Θ Press ERASE DRAW to see the surface plot This time ...

Страница 421: ...rfaces z constant on the x y plane For example to produce a Ps Contour plot for the surface z x2 y2 use the following Θ Press ô simultaneously if in RPN mode to access to the PLOT SETUP window Θ Change TYPE to Ps Contour Θ Press and type X 2 Y 2 OK Θ Make sure that X is selected as the Indep and Y as the Depnd variables Θ Press L OK to return to normal calculator display Θ Press ò simultaneously i...

Страница 422: ...the slope field plot Press EDITL LABEL MENU to see the plot unencumbered by the menu and with identifying labels Θ Press LL PICT to leave the EDIT environment Θ Press CANCL to return to the PLOT WINDOW environment Then press or L OK to return to normal calculator display Y Slice plots Y Slice plots are animated plots of z vs y for different values of x from the function z f x y For example to prod...

Страница 423: ...he animation Press CANCL to return to the PLOT WINDOW environment Θ Press or L OK to return to normal calculator display Try also a Ps Contour plot for the surface z f x y x y sin y Θ Press ô simultaneously if in RPN mode to access the PLOT SETUP window Θ Press and type X Y SIN Y OK Θ Press ERASE DRAW to produce the Y Slice animation Θ Press to stop the animation Θ Press CANCL to return to the PLO...

Страница 424: ...d of functions corresponding to the real and imaginary parts of the complex function Θ Press EDIT L LABEL MENU to see the graph with labels and ranges Θ Press LL PICT CANCL to return to the PLOT WINDOW environment Θ Press or L OK to return to normal calculator display Other functions of a complex variable worth trying for Gridmap plots are 1 SIN X Y i e F z sin z 2 X Y 2 i e F z z2 3 EXP X Y i e F...

Страница 425: ... Indep 10 Depnd 8 Θ Press ERASE DRAW to draw the three dimensional surface Θ Press EDIT L LABEL MENU to see the graph with labels and ranges Θ Press LL PICT CANCL to return to the PLOT WINDOW environment Θ Press or L OK to return to normal calculator display The VPAR variable The VPAR Volume Parameter variable contains information regarding the volume used to produce a three dimensional graph Ther...

Страница 426: ...o use these functions we will try the following exercise First we get the graphics screen corresponding to the following instructions Θ Press ô simultaneously if in RPN mode to access to the PLOT SETUP window Θ Change TYPE to Function if needed Θ Change EQ to X Θ Make sure that Indep is set to X also Θ Press L OK to return to normal calculator display Θ Press ò simultaneously if in RPN mode to acc...

Страница 427: ...n done to deselect this option MARK This command allows the user to set a mark point which can be used for a number of purposes such as Θ Start of line with the LINE or TLINE command Θ Corner for a BOX command Θ Center for a CIRCLE command Using the MARK command by itself simply leaves an x in the location of the mark Press L MARK to see it in action LINE This command is used to draw a line betwee...

Страница 428: ... This highlights the cursor Move the cursor with the arrow keys to a point away and in a diagonal direction from the current cursor position Press BOX again A rectangle is drawn whose diagonal joins the initial and ending cursor positions The initial position of the box is still marked with and x Moving the cursor to another position and pressing BOX will generate a new box containing the initial ...

Страница 429: ...ced in the stack Select the subset you want to extract by placing a MARK at a point in the graph moving the cursor to the diagonal corner of the rectangle enclosing the graphics subset and press SUB This feature can be used to move parts of a graphics object around the graph REPL This command places the contents of a graphic object currently in stack level 1 at the cursor location in the graphics ...

Страница 430: ... produces an input screen that allows you to change the current X and Y Factors The X and Y Factors relate the horizontal and vertical user defined unit ranges to their corresponding pixel ranges Change the H Factor to read 8 and press OK then change the V Factor to read 2 and press OK Check off the option Recenter on cursor and press OK Back in the graphics display press ZIN The graphic is re dra...

Страница 431: ...om which you started the zoom box operation ZDFLT ZAUTO Pressing ZDFLT re draws the current plot using the default x and y ranges i e 6 5 to 6 5 in x and 3 1 to 3 1 in y The command ZAUTO on the other hand creates a zoom window using the current independent variable x range but adjusting the dependent variable y range to fit the curve as when you use the function AUTO in the PLOT WINDOW input form...

Страница 432: ...y pressing the P key fourth key from the left in fourth row from the top of the keyboard This menu provides a list of menus related to the Computer Algebraic System or CAS these are All but one of these menus are available directly in the keyboard by pressing the appropriate keystroke combination as follows The Chapter of the user manual where the menus are described is also listed ALGEBRA the 4 k...

Страница 433: ...OTADD function adds this function to the list of functions to plot similar to ô Plot setup same as ô SIGNTAB function sign table of given function showing intervals of positive and negative variation zero points and infinite asymptotes TABVAL table of values for a function TABVAR variation table of a function Examples of some of these functions are provided next PLOT X 2 1 is similar to ô with EQ ...

Страница 434: ...ces the following table of variation A detailed interpretation of the table of variation is easier to follow in RPN mode The output is in a graphical format showing the original function F X the derivative F X right after derivation and after simplification and finally a table of variation The table consists of two rows labeled in the right hand side Thus the top row represents values of X and the...

Страница 435: ...infinity A plot of the graph is shown below to illustrate these observations Function DRAW3DMATRIX This function takes as argument a n m matrix Z zij and minimum and maximum values for the plot You want to select the values of vmin and vmax so that they contain the values listed in Z The general call to the function is therefore DRAW3DMATRIX Z vmin vmax To illustrate the use of this function we fi...

Страница 436: ...hics The functions in entries 1 and 2 will be presented in this Chapter Differential equations the subject of item 3 are presented in Chapter 16 Graphic functions the subject of item 4 were presented at the end of Chapter 12 Finally entries 5 DERVX and 6 INTVX are the functions to obtain a derivative and a indefinite integral for a function of the default CAS variable typically X Functions DERVX a...

Страница 437: ...r the function first then the expression x a and finally function lim Examples in ALG mode are shown next including some limits to infinity The keystrokes for the first example are as follows using Algebraic mode and system flag 117 set to CHOOSE boxes Ö2 OK 2 OK x 1 í x Å 1 The infinity symbol is associated with the 0 key i e è Note The functions available in the LIMITS SERIES menu are shown next...

Страница 438: ...erivatives using this limit are shown in the following screen shots Functions DERIV and DERVX The function DERIV is used to take derivatives in terms of any independent variable while the function DERVX takes derivatives with respect to the CAS default variable VX typically X While function DERVX is available directly in the CALC menu both functions are available in the DERIV INTEG sub menu within...

Страница 439: ... SIGMA and SIGMAVX to Fourier series FOURIER and to vector analysis CURL DIV HESS LAPL Next we discuss functions DERIV and DERVX the remaining functions are presented either later in this Chapter or in subsequent Chapters Calculating derivatives with The symbol is available as the T key This symbol can be used to enter a derivative in the stack or in the Equation Writer see Chapter 2 If you use th...

Страница 440: ...e stack The result in ALG mode is In the Equation Writer when you press the calculator provides the following expression The insert cursor will be located right at the denominator awaiting for the user to enter an independent variable say s s Then press the right arrow key to move to the placeholder between parentheses Next enter the function to be differentiated say s ln s ...

Страница 441: ...e expression above are abbreviations the calculator uses to indicate a first derivative when the independent variable in this case x is clearly defined Thus the latter result is interpreted as in the formula for the chain rule shown above Here is another example of a chain rule application Note The symbol is used formally in mathematics to indicate a partial derivative i e the derivative of a func...

Страница 442: ...not in the cases where function DERVX was used In these cases the equation was re written with all its terms moved to the left hand side of the equal sign Also the equal sign was removed but it is understood that the resulting expression is equal to zero Implicit derivatives Implicit derivatives are possible in expressions such as Application of derivatives Derivatives can be used for analyzing th...

Страница 443: ...for the function y tan x Θ Press ô simultaneously in RPN mode to access to the PLOT SETUP window Θ Change TYPE to FUNCTION if needed by using CHOOS Θ Press and type in the equation TAN X Θ Make sure the independent variable is set to X Θ Press L OK to return to normal calculator display Θ Press ò simultaneously to access the PLOT window Θ Change H VIEW range to 2 to 2 and V VIEW range to 5 to 5 Θ ...

Страница 444: ...efined while from 0 to the function is defined On the other hand indicates that the function is not defined between and 1 nor between 1 and The domain of this function is therefore 1 X 1 Function TABVAL This function is accessed through the command catalog or through the GRAPH sub menu in the CALC menu Function TABVAL takes as arguments a function of the CAS variable f X and a list of two numbers ...

Страница 445: ...his case SIGNTAB does not provide information in the intervals between and π 2 nor between π 2 and Thus SIGNTAB for this case provides information only on the main domain of TAN X namely π 2 X π 2 A second example of function SIGNTAB is shown below For this case the function is negative for X 1 and positive for X 1 Function TABVAR This function is accessed through the command catalog or through th...

Страница 446: ...using the function TABVAR Use the following keystrokes in RPN mode X 3 4 X 2 11 X 30 N t select TABVAR OK This is what the calculator shows in stack level 1 This is a graphic object To be able to the result in its entirety press The variation table of the function is shown as follows Press to recover normal calculator display Press ƒ to drop this last result from the stack Two lists corresponding ...

Страница 447: ...he graph of the function reaches a maximum or minimum Furthermore the value of the second derivative of the function f x at those points determines whether the point is a relative or local maximum f x 0 or minimum f x 0 These ideas are illustrated in the figure below In this figure we limit ourselves to determining extreme points of the function y f x in the x interval a b Within this interval we ...

Страница 448: ...hows that f 11 3 14 thus x 11 3 is a relative minimum For x 1 we have the following This result indicates that f 1 14 thus x 1 is a relative maximum Evaluate the function at those points to verify that indeed f 1 f 11 3 Higher order derivatives Higher order derivatives can be calculated by applying a derivative function several times e g ...

Страница 449: ...unctions INT RISCH and SIGMA work with functions of any variable while functions INTVX and SIGMAVX utilize functions of the CAS variable VX typically x Functions INT and RISCH require therefore not only the expression for the function being integrated but also the independent variable name Function INT requires also a value of x where the anti derivative will be evaluated Functions INTVX and SIGMA...

Страница 450: ...on Writer see Chapter 2 for an example Within the Equation Writer the symbol Á produces the integral sign and provides placeholders for the integration limits a b for the function f x and for the variable of integration x The following screen shots show how to build a particular integral The insert cursor is first located in the lower limit of integration enter a value and press the right arrow ke...

Страница 451: ...dows selected see Chapter 1 the evaluation of derivatives and integrals will be shown step by step For example here is the evaluation of a derivative in the Equation Writer ʳʳʳʳʳ Notice the application of the chain rule in the first step leaving the derivative of the function under the integral explicitly in the numerator In the second step the resulting fraction is rationalized eliminating the sq...

Страница 452: ...his integral First CAS identifies a square root integral next a rational fraction and a second rational expression to come up with the final result Notice that these steps make a lot of sense to the calculator although not enough information is provided to the user on the individual steps Integrating an equation Integrating an equation is straightforward the calculator simply integrates both sides...

Страница 453: ... want to calculate the integral If we use step by step calculation in the Equation Writer this is the sequence of variable substitutions This second step shows the proper substitution to use u x2 1 The last four steps show the progression of the solution a square root followed by a fraction a second fraction and the final result This result can be simplified by using function SIMP to read ...

Страница 454: ...ample the integral xex dx can be solved by integration by parts if we use u x dv ex dx since v ex With du dx the integral becomes xexdx udv uv vdu xex exdx xex ex The calculator provides function IBP under the CALC DERIV INTG menu that takes as arguments the original function to integrate namely u X v X and the function v X and returns u X v X and v X u X In other words function IBP returns the tw...

Страница 455: ...term by term For example to integrate we can decompose the fraction into its partial component fractions as follows The direct integration produces the same result with some switching of the terms Rigorous mode set in the CAS see Chapter 2 Improper integrals These are integrals with infinite limits of integration Typically an improper integral is dealt with by first calculating the integral as a l...

Страница 456: ... the CAS set to Approx mode The left hand side figure shows the integral typed in the line editor before pressing The right hand figure shows the result after pressing If you enter the integral with the CAS set to Exact mode you will be asked to change to Approx mode however the limits of the integral will be shown in a different format as shown here These limits represent 1 1_mm and 0 1_mm which ...

Страница 457: ...erwise the calculator simply returns the unevaluated integral For example 3 The integrand may have units too For example 4 If both the limits of integration and the integrand have units the resulting units are combined according to the rules of integration For example Infinite series An infinite series has the form The infinite series typically starts with indices n 0 or n 1 Each term in the serie...

Страница 458: ...n practice we cannot evaluate all terms in an infinite series instead we approximate the series by a polynomial of order k Pk x and estimate the order of a residual Rk x such that i e The polynomial Pk x is referred to as Taylor s polynomial The order of the residual is estimated in terms of a small quantity h x x0 i e evaluating the polynomial at a value of x very close to x0 The residual if give...

Страница 459: ... performs a Maclaurin series expansion i e about X 0 of an expression in the default independent variable VX typically X The expansion uses a 4 th order relative power i e the difference between the highest and lowest power in the expansion is 4 For example Function TAYLR produces a Taylor series expansion of a function of any variable x about a point x a for the order k specified by the user Thus...

Страница 460: ... Expression for the Taylor polynomial 4 Order of the residual or remainder Because of the relatively large amount of output this function is easier to handle in RPN mode For example Drop the contents of stack level 1 by pressing ƒ and then enter μ to decompose the list The results are as follows In the right hand side figure above we are using the line editor to see the series expansion in detail ...

Страница 461: ...the DEFINE function à To illustrate the concept of partial derivative we will define a couple of multi variate functions f x y x cos y and g x y z x2 y2 1 2sin z as follows We can evaluate the functions as we would evaluate any other calculator function e g Graphics of two dimensional functions are possible using Fast3D Wireframe Ps Contour Y Slice Gridmap and Pr Surface plots as described in Chap...

Страница 462: ...ary derivatives with respect to the variable of interest while considering all other variables as constant Thus for example which are the same results as found with the limits calculated earlier Consider another example In this calculation we treat y as a constant and take derivatives of the expression with respect to x Similarly you can use the derivative functions in the calculator e g DERVX DER...

Страница 463: ...The last two expressions represent cross derivatives the partial derivatives signs in the denominator shows the order of derivation In the left hand side the derivation is taking first with respect to x and then with respect to y and in the right hand side the opposite is true It is important to indicate that if a function is continuous and differentiable then 2 2 2 2 y f y y f x f x x f y f x y x...

Страница 464: ...hain rule for the derivative dz dt for this case is written as To see the expression that the calculator produces for this version of the chain rule use The result is given by d1y t d2z x t y t d1x t d1z x y y t The term d1y t is to be interpreted as the derivative of y t with respect to the 1st independent variable i e t or d1y t dy dt Similarly d1x t dx dt On the other hand d1z x t y t means the...

Страница 465: ... maximum if 2 f x2 0 or a relative minimum if 2 f x2 0 The value Δ is referred to as the discriminant If Δ 2f x2 2f y2 2f x y 2 0 we have a condition known as a saddle point where the function would attain a maximum in x if we were to hold y constant while at the same time attaining a minimum if we were to hold x constant or vice versa Example 1 Determine the extreme points if any of the function ...

Страница 466: ...e calculator and edited in the computer illustrates the existence of these two points Using function HESS to analyze extrema Function HESS can be used to analyze extrema of a function of two variables as shown next Function HESS in general takes as input a function of n independent variables φ x1 x2 xn and a vector of the functions x1 x2 xn Function HESS returns the Hessian matrix of the function ...

Страница 467: ...to analyze extrema in functions of two variables For example for the function f X Y X3 3X Y2 5 proceed as follows in RPN mode X 3 3 X Y 2 5 X Y Enter function and variables HESS Apply function HESS SOLVE Find critical points μ Decompose vector s1 K s2 K Store critical points The variables s1 and s2 at this point contain the vectors X 1 Y 0 and X 1 Y 0 respect H K Store Hessian matrix J H s1 SUBST ...

Страница 468: ...presenting the volume of the solid body contained under the surface f x y above the region R The region R can be described as R a x b f x y g x or as R c y d r y x s y Thus the double integral can be written as Calculating a double integral in the calculator is straightforward A double integral can be built in the Equation Writer see example in Chapter 2 An example follows This double integral is ...

Страница 469: ...xpression to use is where R is the region R expressed in u v coordinates Double integral in polar coordinates To transform from polar to Cartesian coordinates we use x r θ r cos θ and y r θ r sin θ Thus the Jacobian of the transformation is With this result integrals in polar coordinates are written as v y u y v x u x J J det det R R dudv J v u y v u x dydx y x φ φ r r r y r y x r x J cos sin sin ...

Страница 470: ...θ Double integrals in polar coordinates can be entered in the calculator making sure that the Jacobian J r is included in the integrand The following is an example of a double integral calculated in polar coordinates shown step by step β α θ θ θ θ φ θ φ g f R rdrd r dA r ...

Страница 471: ...rred to as a vector field The following operator referred to as the del or nabla operator is a vector based operator that can be applied to a scalar or vector function When this operator is applied to a scalar function we can obtain the gradient of the function and when applied to a vector function we can obtain the divergence and the curl of that function A combination of gradient and divergence ...

Страница 472: ...the CALC menu e g A program to calculate the gradient The following program which you can store into variable GRADIENT uses function DERIV to calculate the gradient of a scalar function of X Y Z Calculations for other base variables will not work If you work frequently in the X Y Z system however this function will facilitate calculations X Y Z 3 ARRY DERIV Type the program while in RPN mode After...

Страница 473: ...It follows that F grad φ φ The calculator provides function POTENTIAL available through the command catalog N to calculate the potential function of a vector field if it exists For example if F x y z xi yj zk applying function POTENTIAL we find Since function SQ x represents x2 this results indicates that the potential function for the vector field F x y z xi yj zk is φ x y z x2 y2 z2 2 Notice tha...

Страница 474: ...ector field For example for F X Y Z XY X2 Y2 Z2 YZ the divergence is calculated in ALG mode as follows Laplacian The divergence of the gradient of a scalar function produces an operator called the Laplacian operator Thus the Laplacian of a scalar function φ x y z is given by The partial differential equation 2 φ 0 is known as Laplace s equation Function LAPL can be used to calculate the Laplacian ...

Страница 475: ...his chapter we introduced function POTENTIAL to calculate the potential function φ x y z for a vector field F x y z f x y z i g x y z j h x y z k such that F grad φ φ We also indicated that the conditions for the existence of φ were f y g x f z h x and g z h y These conditions are equivalent to the vector expression curl F F 0 A vector field F x y z with zero curl is known as an irrotational field...

Страница 476: ...x y z j η x y z k such that F curl Φ Φ then function Φ x y z is referred to as the vector potential of F x y z The calculator provides function VPOTENTIAL available through the command catalog N to calculate the vector potential Φ x y z given the vector field F x y z f x y z i g x y z j h x y z k For example given the vector field F x y z yi zj xk function VPOTENTIAL produces i e Φ x y z x2 2j y2 ...

Страница 477: ...z k and those of the vector potential function Φ x y z φ x y z i ψ x y z j η x y z k are related by f η y ψ x g φ z η x and h ψ x φ y A condition for function Φ x y z to exists is that div F F 0 i e f x g y f z 0 Thus if this condition is not satisfied the vector potential function Φ x y z does not exist For example given F X Y X Y Z 2 function VPOTENTIAL returns an error message since function F ...

Страница 478: ...easiest way to enter a differential equation is to type it in the equation writer For example to type the following ODE x 1 dy x dx 2 2 x y x ex sin x use O Ü x 1 x y Ü x Q2 2 x y Ü x x S x The derivative dy dx is represented by x y x or by d1y x For solution or calculation purposes you need to specify y x in the expression i e the dependent variable must include its independent variable s in any ...

Страница 479: ...represents the partial differential equation x f t g t h t Because the order of the variable t is different in f x t g t y and h x y t derivatives with respect to t have different indices i e d2f x t d1g t y and d3h x y t All of them however represent derivatives with respect to the same variable Expressions for derivatives using the order of variable index notation do not translate into derivativ...

Страница 480: ... solution to the differential equation y f x y sin x cos y using a slope field plot To solve this problem follow the instructions in Chapter 12 for slopefield plots If you could reproduce the slope field plot in paper you can trace by hand lines that are tangent to the line segments shown in the plot This lines constitute lines of y x y constant for the solution of y f x y Thus slope fields are us...

Страница 481: ...n Otherwise the equation is said to be non linear Examples of linear differential equations are d2 x dt2 β dx dt ωo x A sin ωf t and C t u C x D 2 C x2 An equation whose right hand side not involving the function or its derivatives is equal to zero is called a homogeneous equation Otherwise it is called non homogeneous The solution to the homogeneous equation is known as a general solution A parti...

Страница 482: ...on While this result seems very complicated it can be simplified if we take K1 10 cC0 7 cC1 cC2 40 K2 6 cC0 cC1 cC2 24 and K3 15 cC0 2 cC1 cC2 15 Then the solution is y K1 e 3x K2 e5x K3 e2x The reason why the result provided by LDEC shows such complicated combination of constants is because internally to produce the solution LDEC utilizes Laplace transforms to be presented later in this chapter w...

Страница 483: ...330 x 241 13500 constitute a particular solution of the ODE To verify that yp 450 x2 330 x 241 13500 is indeed a particular solution of the ODE use the following d1d1d1Y X 4 d1d1Y X 11 d1Y X 30 Y X X 2 Y X 450 X 2 330 X 241 13500 SUBST EV L Allow the calculator about ten seconds to produce the result X 2 X 2 Example 3 Solving a system of linear differential equations with constant coefficients Con...

Страница 484: ...ator provides function DESOLVE Differential Equation SOLVEr to solve certain types of differential equations The function requires as input the differential equation and the unknown function and returns the solution to the equation if available You can also provide a vector containing the differential equation and the initial conditions instead of only a differential equation as input to DESOLVE T...

Страница 485: ...thus the ODE is now written d dx x dy dx exp x and x dy dx exp x C Next we can write dy dx C exp x x C x ex x In the calculator you may try to integrate d1y x C EXP x x y x DESOLVE The result is y x INT EXP xt C xt xt x C0 i e The variable ODETYPE You will notice in the soft menu key labels a new variable called ODETY ODETYPE This variable is produced with the call to the DESOL function and holds ...

Страница 486: ...use d1d1y t 5 y t 2 COS t 2 y 0 6 5 d1y 0 1 2 y t DESOLVE Notice that the initial conditions were changed to their Exact expressions y 0 6 5 rather than y 0 1 2 and d1y 0 1 2 rather than d1y 0 0 5 Changing to these Exact expressions facilitates the solution The solution is Press μμto simplify the result to y t 19 5 SIN 5 t 148 COS 5 t 80 COS t 2 190 Note To obtain fractional expressions for decima...

Страница 487: ...into the solution to the differential equation f t Definitions The Laplace transform for function f t is the function F s defined as The image variable s can be and it generally is a complex number Many practical applications of Laplace transforms involve an original function f t where t represents time e g control systems in electric or hydraulic circuits In most cases one is interested in the sy...

Страница 488: ...nd Exact Example 1 You can get the definition of the Laplace transform use the following f X L P in RPN mode or L P f X in ALG mode The calculator returns the result RPN left ALG right Compare these expressions with the one given earlier in the definition of the Laplace transform i e and you will notice that the CAS default variable X in the equation writer screen replaces the variable s in this d...

Страница 489: ...a2 The transform is interpreted as follows L cos a t b s cos b a sin b s2 a2 Laplace transform theorems To help you determine the Laplace transform of functions you can use a number of theorems some of which are listed below A few examples of the theorem applications are also included Θ Differentiation theorem for the first derivative Let fo be the initial condition for f t i e f 0 fo then L df dt...

Страница 490: ... then Example 2 As a follow up to Example 1 the acceleration a t is defined as a t d2 r dt2 If the initial velocity is vo v 0 dr dt t 0 then the Laplace transform of the acceleration can be written as A s L a t L d2r dt2 s2 R s s ro v o Example 3 Let f t e at using the calculator with EXP a X LAP you get 1 X a or F s 1 s a The third derivative of this expression can be calculated by using X X X μ ...

Страница 491: ...nction of period T Limit theorem for the initial value Let F s L f t then Limit theorem for the final value Let F s L f t then Example 4 Using the convolution theorem find the Laplace transform of f g t if f t sin t and g t exp t To find F s L f t and G s L g t use SIN X LAP μ Result 1 X 2 1 i e F s 1 s2 1 Also EXP X LAP Result 1 X 1 i e G s 1 s 1 Thus L f g t F s G s 1 s2 1 1 s 1 1 s 1 s2 1 1 s3 ...

Страница 492: ...nt The formal definition of Dirac s delta function δ x is δ x 0 for x 0 and Also if f x is a continuous function then An interpretation for the integral above paraphrased from Friedman 1990 is that the δ function picks out the value of the function f x at x x0 Dirac s delta function is typically represented by an upward arrow at the point x x0 indicating that the function has a non zero value only...

Страница 493: ...p function H t is simply referred to as 1 To check the transform in the calculator use 1 LAP The result is 1 X i e L 1 1 s Similarly U0 LAP produces the result U0 X i e L U0 U0 s You can obtain Dirac s delta function in the calculator by using 1 ILAP The result is Delta X This result is simply symbolic i e you cannot find a numerical value for say Delta 5 This result can be defined the Laplace tra...

Страница 494: ...aplace transform on F s The theorems on derivatives of a function i e L df dt s F s fo L d2f dt2 s2 F s s fo df dt o and in general L dnf dtn sn F s sn 1 fo s f n 2 o f n 1 o are particularly useful in transforming an ODE into an algebraic equation Example 1 To solve the first order equation dh dt k h t a e t by using Laplace transforms we can write L dh dt k h t L a e t L dh dt k L h t a L e t Wi...

Страница 495: ...n to the ODE would be if you use the function LDEC a EXP X X k LDEC μ The result is i e h t a k 1 e t k 1 cCo a k 1 e kt Thus cC0 in the results from LDEC represents the initial condition h 0 Example 2 Use Laplace transforms to solve the second order linear equation d2 y dt2 2y sin 3t Using Laplace transforms we can write L d2 y dt2 2y L sin 3t L d2 y dt2 2 L y t L sin 3t Note When using the funct...

Страница 496: ...X 2 18 To find the solution to the ODE y t we need to use the inverse Laplace transform as follows OBJ ƒ ƒ Isolates right hand side of last expression ILAPμ Obtains the inverse Laplace transform The result is i e y t 1 7 sin 3x yo cos 2x 2 7y1 3 14 sin 2x Check what the solution to the ODE would be if you use the function LDEC SIN 3 X X 2 2 LDEC μ The result is i e the same as before with cC0 y0 a...

Страница 497: ... by writing X 2 Y X y0 y1 Y EXP 3 X Y ISOL The result is Y X y0 y1 EXP 3 X X 2 1 To find the solution to the ODE y t we need to use the inverse Laplace transform as follows OBJ ƒ ƒ Isolates right hand side of last expression ILAP μ Obtains the inverse Laplace transform The result is y1 SIN X y0 COS X SIN X 3 Heaviside X 3 Note Using the two examples shown here we can confirm what we indicated earl...

Страница 498: ...te L 1 yo s s2 1 y1 s2 1 e 3s s2 1 yo L 1 s s2 1 y1 L 1 1 s2 1 L 1 e 3s s2 1 Then we use the calculator to obtain the following X X 2 1 ILAP Result COS X i e L 1 s s2 1 cos t 1 X 2 1 ILAP Result SIN X i e L 1 1 s2 1 sin t EXP 3 X X 2 1 ILAP Result SIN X 3 Heaviside X 3 2 The very last result i e the inverse Laplace transform of the expression EXP 3 X X 2 1 can also be calculated by using the secon...

Страница 499: ...e of the ODE describing the system In this example we want to use Heaviside s step function H t In the calculator we can define this function as H X IFTE X 0 1 0 à This definition will create the variable H in the calculator s soft menu key Example 1 To see a plot of H t 2 for example use a FUNCTION type of plot see Chapter 12 Press ô simultaneously in RPN mode to access to the PLOT SETUP window C...

Страница 500: ...de to access to the PLOT SETUP window Change TYPE to FUNCTION if needed Change EQ to 0 5 COS X 0 25 SIN X SIN X 3 H X 3 Make sure that Indep is set to X H VIEW 0 20 V VIEW 3 2 Press ERASE DRAW to plot the function Press EDIT L LABEL to see the plot The resulting graph will look like this Notice that the signal starts with a relatively small amplitude but suddenly at t 3 it switches to an oscillato...

Страница 501: ...tion to the ODE y t we need to use the inverse Laplace transform as follows OBJ ƒ ƒ Isolates right hand side of last expression ILAP Obtains the inverse Laplace transform The result is y1 SIN X 1 y0 COS X 1 COS X 3 1 Heaviside X 3 Thus we write as the solution y t yo cos t y1 sin t H t 3 1 sin t 3 Check what the solution to the ODE would be if you use the function LDEC H X 3 ENTER X 2 1 LDEC The r...

Страница 502: ... the solution for t 3 The Heaviside step function can be combined with a constant function and with linear functions to generate square triangular and saw tooth finite pulses as follows Θ Square pulse of size Uo in the interval a t b f t Uo H t a H t b Θ Triangular pulse with a maximum value Uo increasing from a t b decreasing from b t c f t Uo t a b a H t a H t b 1 t b b c H t b H t c Θ Saw tooth...

Страница 503: ...d T if f x T f t For example because sin x 2π sin x and cos x 2π cos x the functions sin and cos are 2π periodic functions If two functions f x and g x are periodic of period T then their linear combination h x a f x b g x is also periodic of period T A T periodic function f t can be expanded into a series of sine and cosine functions known as a Fourier series given by where the coefficients an an...

Страница 504: ...ents a0 a1 and b1 for the corresponding Fourier series we proceed as follows First define function f t t2 t Next we ll use the Equation Writer to calculate the coefficients Thus the first three terms of the function are f t 1 3 4 π2 cos π t 2 π sin π t A graphical comparison of the original function with the Fourier expansion using these three terms shows that the fitting is acceptable for t 1 or ...

Страница 505: ...he function FOURIER is available in the DERIV sub menu within the CALC menu Ö Fourier series for a quadratic function Determine the coefficients c0 c1 and c2 for the function f t t2 t with period T 2 Note Because the integral used by function FOURIER is calculated in the interval 0 T while the one defined earlier was calculated in the interval T 2 T 2 we need to shift the function in the t axis by...

Страница 506: ...sub directory where you defined functions f and g and calculate the coefficients Accept change to Complex mode when requested Thus c0 1 3 c1 π i 2 π2 c2 π i 1 2π2 The Fourier series with three elements will be written as g t Re 1 3 π i 2 π2 exp i π t π i 1 2π2 exp 2 i π t A plot of the shifted function g t and the Fourier series fitting follows ...

Страница 507: ...given by figures show normal font and small font displays The general expression turns out to be after simplifying the previous result We can simplify this expression even further by using Euler s formula for complex numbers namely e2inπ cos 2nπ i sin 2nπ 1 i 0 1 since cos 2nπ 1 and sin 2nπ 0 for n integer Using the calculator you can simplify the expression in the equation writer O by replacing e...

Страница 508: ...c n representing the general term cn in the complex Fourier series Θ Next define the finite complex Fourier series F X k where X is the independent variable and k determines the number of terms to be used Ideally we would like to write this finite complex Fourier series as However because the function c n is not defined for n 0 we will be better advised to re write the expression as 2 exp X T n i ...

Страница 509: ...ng t as the independent variable we can evaluate F t 2 1 3 to get This result shows only the first term c0 and part of the first exponential term in the series The decimal display format was changed to Fix with 2 decimals to be able to show some of the coefficients in the expansion and in the exponent As expected the coefficients are complex numbers The function F thus defined is fine for obtainin...

Страница 510: ... 0 F 0 5 3 1 3 0 192401031886 0 F 0 5 4 1 3 0 167070735979 0 F 0 5 5 1 3 0 294394690453 0 F 0 5 6 1 3 0 305652599743 0 To compare the results from the series with those of the original function load these functions into the PLOT FUNCTION input form ñ simultaneously if using RPN mode Change the limits of the Plot Window ò as follows Press the soft menu keys ERASE DRAW to produce the plot Notice tha...

Страница 511: ...1 X 2 X If you started this example after finishing example 1 you already have a value of 2 stored in CAS variable PERIOD If you are not sure check the value of this variable and store a 2 in it if needed The coefficient c0 for the Fourier series is calculated as follows The calculator will request a change to Approx mode because of the integration of the function IFTE included in the integrand Ac...

Страница 512: ...s definition in the calculator i e where T 2 is the period The value of T can be stored using Typing the first integral above in the Equation Writer selecting the entire expression and using EVAL will produce the following Recall the einπ cos nπ i sin nπ 1 n Performing this substitution in the result above we have π dX T X n 2 i EXP X 2 1 1 0 π 2 1 dX T X n 2 i EXP X 2 2 1 ...

Страница 513: ...ining the coefficient cn namely Once again replacing einπ 1 n and using e2inπ 1 we get Press to copy this second result to the screen Now add ANS 1 and ANS 2 to get the full expression for cn Pressing will place this result in the Equation Writer where we can simplify SIMP it to read Once again replacing einπ 1 n results in ...

Страница 514: ... c0 Σ n 1 k c n EXP 2 i π n X T c n EXP 2 i π n X T To compare the original function and the Fourier series we can produce the simultaneous plot of both functions The details are similar to those of example 1 except that here we use a horizontal range of 0 to 2 and a vertical range from 0 to 1 and adjust the equations to plot as shown here The resulting graph is shown below for k 5 the number of e...

Страница 515: ...n the series shows not so good a fitting The Fourier series can be used to generate a periodic triangular wave or saw tooth wave by changing the horizontal axis range for example from 2 to 4 The graph shown below uses k 5 Fourier series for a square wave A square wave can be generated by using the function 4 3 0 3 1 1 1 0 0 x if x if x if x g ...

Страница 516: ...n the calculator by using DEFINE g X IFTE X 1 AND X 3 1 0 The function plotted as follows horizontal range 0 to 4 vertical range 0 to 1 2 Using a procedure similar to that of the triangular shape in example 2 above you can find that and We can simplify this expression by using einπ 2 in and e3inπ 2 i n to get 5 0 1 1 3 1 0 dX T c ...

Страница 517: ...ents the approximation is shown below A better approximation is obtained by using k 10 i e For k 20 the fitting is even better but it takes longer to produce the graph Fourier series applications in differential equations Suppose we want to use the periodic square wave defined in the previous example as the excitation of an undamped spring mass system whose homogeneous equation is d2 y dX2 0 25y 0...

Страница 518: ...in the Equation writer Exploring the equation in the Equation Writer reveals the existence of two constants of integration cC0 and cC1 These values would be calculated using initial conditions Suppose that we use the values cC0 0 5 and cC1 0 5 we can replace those values in the solution above by using function SUBST see Chapter 5 For this case use SUBST ANS 1 cC0 0 5 followed by SUBST ANS 1 cC1 0 ...

Страница 519: ...tegration of the form The function κ s t is known as the kernel of the transformation The use of an integral transform allows us to resolve a function into a given spectrum of components To understand the concept of a spectrum consider the Fourier series representing a periodic function with a period T This Fourier series can be re written as where for n 1 2 b a dt t f t s s F κ sin cos 1 0 n n n ...

Страница 520: ...r a function The discrete spectrum will show that the function has components at angular frequencies ωn which are integer multiples of the fundamental angular frequency ω0 Suppose that we are faced with the need to expand a non periodic function into sine and cosine components A non periodic function can be thought of as having an infinitely large period Thus for a very large value of T the fundam...

Страница 521: ...fined below Example 1 Determine the coefficients C ω S ω and the continuous spectrum A ω for the function f x exp x for x 0 and f x 0 x 0 In the calculator set up and evaluate the following integrals to calculate C ω and S ω respectively CAS modes are set to Exact and Real Their results are respectively The continuous spectrum A ω is calculated as sin 2 1 dx x x f S ω π ω 2 2 ω ω ω S C A ...

Страница 522: ... transforms and their inverses used in this Chapter Fourier sine transform Inverse sine transform Fourier cosine transform Inverse cosine transform Fourier transform proper Inverse Fourier transform proper Example 1 Determine the Fourier transform of the function f t exp t for t 0 and f t 0 for t 0 0 sin 2 dt t t f F t f s ω π ω F 0 1 sin dt t F t f F s ω ω ω F 0 cos 2 dt t t f F t f c ω π ω F 0 1...

Страница 523: ...he magnitude or absolute value of the Fourier transform F ω is the frequency spectrum of the original function f t For the example shown above F ω 1 2π 1 ω2 1 2 The plot of F ω vs ω was shown earlier Some functions such as constant values sin x exp x x2 etc do not have Fourier transform Functions that go to zero sufficiently fast as x goes to infinity do have Fourier transforms ε ω ε ω π π 0 1 0 1...

Страница 524: ... efficiently a discrete Fourier transform DFT This algorithm has applications in the analysis of different types of time dependent signals from turbulence measurements to communication signals The discrete Fourier transform of a sequence of data values xj j 0 1 2 n 1 is a new finite sequence Xk defined as The direct calculation of the sequence Xk involves n2 products which would involve enormous a...

Страница 525: ... time dependent signal The calculator can be fed that data say from a computer or a data logger for processing Or you can generate your own data by programming a function and adding a few random numbers to it Example 1 Define the function f x 2 sin 3x 5 cos 5x 0 5 RAND where RAND is the uniform random number generator provided by the calculator Generate 128 data points by using values of x in the ...

Страница 526: ...urns an array of complex numbers that are the arrays of coefficients Xk of the DFT The magnitude of the coefficients Xk represents a frequency spectrum of the original data To obtain the magnitude of the coefficients you could transform the array into a list and then apply function ABS to the list This is accomplished by using OBJ μ ƒ LIST Ê Finally you can convert the list back to a column vector...

Страница 527: ...s case is 0 to 64 while the vertical range is 1 to 10 To reproduce the signal whose spectrum is shown use function IFFT Since we left a copy of the spectrum in the stack a row vector all you need to do if find function IFFT in the MTH FFT menu or through the command catalog N As an alternative you could simply type the function name i e type ifft The signal is shown as an array row vector with com...

Страница 528: ...on as x 2 d1d1y x a x d1y x b y x 0 Then type and substitute the suggested solution y x x n SUBST The result is x 2 n x n 1 1 n 1 a x n x n 1 b x n 0 which simplifies to n n 1 x n a n x n b x n 0 Dividing by x n results in an auxiliary algebraic equation n n 1 a n b 0 or Θ If the equation has two different roots say n1 and n2 then the general solution of this equation is y x K1 x n 1 K2 x n 2 Θ If...

Страница 529: ...X 3 3 X 2 i e P3 x 5x3 3x 2 4 LEGENDRE result 35 X 4 30 X 2 3 8 i e P4 x 35x4 30x2 3 8 5 LEGENDRE result 63 X 5 70 X 3 15 X 8 i e P5 x 63x5 70x3 15x 8 The ODE 1 x2 d2 y dx2 2 x dy dx n n 1 m2 1 x2 y 0 has for solution the function y x Pn m x 1 x2 m 2 dmPn dxm This function is referred to as an associated Legendre function Bessel s equation The ordinary differential equation x2 d2 y dx2 x dy dx x2 ...

Страница 530: ...his will create the variable J in the soft menu keys For example to evaluate J3 0 1 using 5 terms in the series calculate J 0 1 3 5 i e in RPN mode 1 3 5 J The result is 2 08203157E 5 If you want to obtain an expression for J0 x with say 5 terms in the series use J x 0 5 The result is 1 0 25 x 2 0 015625 x 4 4 3403777E 4 x 6 6 782168E 6 x 8 6 78168 x 10 For non integer values ν the solution to the...

Страница 531: ...l functions of the third kind of order ν as Hn 1 x Jν x i Yν x and Hn 2 x Jν x i Yν x These functions are also known as the first and second Hankel functions of order ν In some applications you may also have to utilize the so called modified Bessel functions of the first kind of order ν defined as Iν x i ν Jν i x where i is the unit imaginary number These functions are solutions to the differentia...

Страница 532: ...ation 1 x2 d2 y dx2 x dy dx n2 y 0 In the calculator the function TCHEBYCHEFF generates the Chebyshev or Tchebycheff polynomial of the first kind of order n given a value of n 0 If the integer n is negative n 0 the function TCHEBYCHEFF generates a Tchebycheff polynomial of the second kind of order n whose definition is Un x sin n arccos x sin arccos x You can access the function TCHEBYCHEFF throug...

Страница 533: ...ons of n elements taken m at a time This function is available in the calculator as function COMB in the MTH PROB menu see also Chapter 17 You can define the following function to calculate Laguerre s polynomials When done typing it in the equation writer press use function DEFINE to create the function L x n into variable L To generate the first four Laguerre polynomials use L x 0 L x 1 L x 2 L x...

Страница 534: ...1 HERMITE result 2 X i e H1 2x 2 HERMITE result 4 X 2 2 i e H2 4x2 2 3 HERMITE result 8 X 3 12 X i e H3 8x3 12x Numerical and graphical solutions to ODEs Differential equations that cannot be solved analytically can be solved numerically or graphically as illustrated below Numerical solution of first order ODE Through the use of the numerical solver Ï you can access an input form that lets you sol...

Страница 535: ...EDIT The result is 0 2499 0 25 Press OK Solution presented as a table of values Suppose we wanted to produce a table of values of v for t 0 00 0 25 2 00 we will proceed as follows First prepare a table to write down your results Write down in your table the step by step results Next within the SOLVE environment change the final value of the independent variable to 0 25 use 25 OK SOLVE wait EDIT So...

Страница 536: ...t order ODE When we can not obtain a closed form solution for the integral we can always plot the integral by selecting Diff Eq in the TYPE field of the PLOT environment as follows suppose that we want to plot the position x t for a velocity function v t exp t2 with x 0 at t 0 We know there is no closed form expression for the integral however we know that the definition of v t is dx dt exp t2 The...

Страница 537: ...ep Default Tol 0 0001 Init Soln 0 Θ To plot the graph use ERASE DRAW When you observe the graph being plotted you ll notice that the graph is not very smooth That is because the plotter is using a time step that may be a bit large for a smooth graph To refine the graph and make it smoother use a step of 0 1 Press CANCL and change the Step value to 0 1 then use ERASE DRAW once more to repeat the gr...

Страница 538: ...mplished by defining the solution as a vector As an example suppose that a spring mass system is subject to a damping force proportional to its speed so that the resulting differential equation is or x 18 75 x 1 962 x subject to the initial conditions v x 6 x 0 at t 0 We want to find x x at t 2 Re write the ODE as w Aw where w x x T and A is the 2 x 2 matrix shown below The initial conditions are ...

Страница 539: ... at a given time t If we wanted to produce a table of values of x and x for t 0 00 0 25 2 00 we will proceed as follows First prepare a table to write down your results Next within the SOLVE environment change the final value of the independent variable to 0 25 use 25 OK SOLVE wait EDIT Solves for w at t 0 25 w 0 968 1 368 OK INIT 5 OK SOLVE wait EDIT Changes initial value of t to 0 25 and final v...

Страница 540: ...activating the differential equation numerical solver Ï OK The SOLVE screen should look like this Notice that the initial condition for the solution Soln w Init 0 includes the vector 0 6 Press L OK Next press ô simultaneously if in RPN mode to enter the PLOT environment Highlight the field in front of TYPE using the keys Then press CHOOS and highlight Diff Eq using the keys Press OK Modify the res...

Страница 541: ...lot of x vs t looks like this To plot the second curve we need to use the PLOT SETUP input form once more To reach this form from the graph above use CANCL L OK ô simultaneously if in RPN mode Change the value of the V Var field to 2 and press DRAW do not press ERASE or you would loose the graph produced above Use EDIT L LABEL MENU to see axes labels and range Notice that the x axis label is the n...

Страница 542: ...lt is t 1 EXP 100 t Next we add an integration constant by using C Then we divide by FI x by using EXP 100 t The result is t 1 EXP 100 t C EXP 100 t i e y t 1 t C e100t Use of the initial condition y 0 1 results in 1 1 0 C e0 or C 0 the particular solution being y t 1 t Numerical solution If we attempt a direct numerical solution of the original equation dy dt 100y 100t 101 using the calculator s ...

Страница 543: ...is particular case the general solution y t 1 t C e100t contains the components t and C e100t which vary at very different rates except for the cases C 0 or C 0 e g for C 1 t 0 1 C e100t 22026 The calculator s ODE numerical solver allows for the solution of stiff ODEs by selecting the option _Stiff in the SOLVE Y T F T Y screen With this option selected you need to provide the values of f y and f ...

Страница 544: ... problem for a first order differential equation using the Runge Kutta Fehlbert 4th 5th order solution scheme Suppose that the differential equation to be solved is given by dy dx f x y with y 0 at x 0 and that you will allow a convergence criteria ε for the solution You can also specify an increment in the independent variable Δx to be used by the function To run this function you will prepare yo...

Страница 545: ... variable y contains the value 4 3880 Function RRK This function is similar to the RKF function except that RRK Rosenbrock and Runge Kutta methods requires as the list in stack level 3 for input not only the names of the independent and dependent variables and the function defining the differential equation but also the expressions for the first and second derivatives of the expression Thus the in...

Страница 546: ... list similar to that of function RKF as well as the tolerance for the solution and a possible step Δx and returns the same input list followed by the tolerance and an estimate of the next step in the independent variable The function returns the input list the tolerance and the next step in the independent variable that satisfies that tolerance Thus the input stack looks as follows ˆˍʳʳʳʳʳʳ x y f...

Страница 547: ... the tolerance an estimate of the next step in the independent variable and the current method CURRENT used to arrive at the next step Thus the input stack looks as follows 4 x y f x y 3 ε 2 Δx 1 LAST After running this function the stack will show the lines 4 x y f x y 3 ε 2 Δx next 1 CURRENT Thus this function is used to determine the appropriate size of a time step Δx next to satisfy the requir...

Страница 548: ...x y 3 ε 2 Δy 1 error Thus this function is used to determine the increment in the solution Δy as well as the absolute error error The following screen shots show the RPN stack before and after application of function RKFERR These result show that Δy 0 827 and error 1 89 10 6 Function RSBERR This function performs similarly to RKERR but with the input elements listed for function RRK Thus the input...

Страница 549: ...at if Dx is reduced to 0 01 Δy 0 00307 and error 0 000547 Note As you execute the commands in the DIFF menu values of x and y will be produced and stored as variables in your calculator The results provided by the functions in this section will depend on the current values of x and y Therefore some of the results illustrated above may differ from what you get in your calculator ...

Страница 550: ...d the PROBABILITY option option 7 to show the following functions see right hand side figure below In this section we discuss functions COMB PERM factorial RAND and RDZ Factorials combinations and permutations The factorial of an integer n is defined as n n n 1 n 2 3 2 1 By definition 0 1 Factorials are used in the calculation of the number of permutations and combinations of objects For example t...

Страница 551: ...lator provides a random number generator that returns a uniformly distributed random real number between 0 and 1 The generator is able to produce sequences of random numbers However after a certain number of times a very large number indeed the sequence tends to repeat itself For that reason the random number generator is more properly referred to as a pseudo random number generator To generate a ...

Страница 552: ...ND Second random number 0 51109 RAND Third random number 0 085429 Re start the sequence RDZ 0 25 Use 0 25 as the seed RAND First random number 0 75285 RAND Second random number 0 51109 RAND Third random number 0 085429 To generate a sequence of random numbers use function SEQ For example to generate a list of 5 random numbers you can use in ALG mode SEQ RAND j 1 5 1 In RPN mode use the following p...

Страница 553: ...tion for the binomial and Poisson distributions Binomial distribution The probability mass function of the binomial distribution is given by where n x C n x is the combination of n elements taken x at a time The values n and p are the parameters of the distribution The value n represents the number of repetitions of an experiment or observation that can have one of two outcomes e g success and fai...

Страница 554: ...ne the following probability mass functions pmf and cumulative distribution functions cdf DEFINE pmfb n p x COMB n x p x 1 p n x DEFINE cdfb n p x Σ k 0 x pmfb n p k DEFINE pmfp λ x EXP λ λ x x DEFINE cdfp λ x Σ k 0 x pmfp λ x The function names stand for Θ pmfb probability mass function for the binomial distribution Θ cdfb cumulative distribution function for the binomial distribution Θ pmfp prob...

Страница 555: ... gamma exponential beta and Weibull distributions These distributions are described in any statistics textbook Some of these distributions make use of a the Gamma function defined earlier which is calculated in the calculator by using the factorial function as Γ x x 1 for any real number x The gamma distribution The probability distribution function pdf for the gamma distribution is given by The c...

Страница 556: ...s given by Functions for continuous distributions To define a collection of functions corresponding to the gamma exponential beta and Weibull distributions first create a sub directory called CFUN Continuous FUNctions and define the following functions change to Approx mode Gamma pdf gpdf x x α 1 EXP x β β α GAMMA α Gamma cdf gcdf x 0 x gpdf t t Beta pdf βpdf x GAMMA α β x α 1 1 x β 1 GAMMA α GAMM...

Страница 557: ...s defined in this section do not include their parameters α and or β in their definitions Therefore you don t need to enter them in the display to calculate the functions However those parameters must be previously defined by storing the corresponding values in the variables α and β Once all functions and the values α and β have been stored you can order the menu labels by using function ORDER The...

Страница 558: ...χ2 distribution and the F distribution The functions provided by the calculator to evaluate probabilities for these distributions are contained in the MTH PROBABILITY menu introduced earlier in this chapter The functions are NDIST UTPN UTPT UTPC and UTPF Their application is described in the following sections To see these functions activate the MTH menu and select the PROBABILITY option Normal di...

Страница 559: ...nd the value x e g UTPN μ σ2 x For example check that for a normal distribution with μ 1 0 σ2 0 5 UTPN 0 75 0 638163 Use UTPN 1 0 0 5 0 75 0 638163 Different probability calculations for normal distributions X is N μ σ2 can be defined using the function UTPN as follows Θ P X a 1 UTPN μ σ2 a Θ P a X b P X b P X a 1 UTPN μ σ2 b 1 UTPN μ σ2 a UTPN μ σ2 a UTPN μ σ2 b Θ P X c UTPN μ σ2 c Examples Using...

Страница 560: ...istribution can be defined using the function UTPT as follows Θ P T a 1 UTPT ν a Θ P a T b P T b P T a 1 UTPT ν b 1 UTPT ν a UTPT ν a UTPT ν b Θ P T c UTPT ν c Examples Given ν 12 determine P T 0 5 1 UTPT 12 0 5 0 68694 P 0 5 T 0 5 UTPT 12 0 5 UTPT 12 0 5 0 3738 P T 1 2 UTPT 12 1 2 0 8733 The Chi square distribution The Chi square χ2 distribution has one parameter ν known as the degrees of freedom...

Страница 561: ...the Chi squared distribution can be defined using the function UTPC as follows Θ P X a 1 UTPC ν a Θ P a X b P X b P X a 1 UTPC ν b 1 UTPC ν a UTPC ν a UTPC ν b Θ P X c UTPC ν c Examples Given ν 6 determine P X 5 32 1 UTPC 6 5 32 0 4965 P 1 2 X 10 5 UTPC 6 1 2 UTPC 6 10 5 0 8717 P X 20 UTPC 6 20 2 769 E 3 The F distribution The F distribution has two parameters νN numerator degrees of freedom and ν...

Страница 562: ... P F 2 1 UTPF 10 5 2 0 7700 P 5 F 10 UTPF 10 5 5 UTPF 10 5 10 3 4693 E 2 P F 5 UTPF 10 5 5 4 4808 E 2 Inverse cumulative distribution functions For a continuous random variable X with cumulative density function cdf F x P X x p to calculate the inverse cumulative distribution function we need to find the value of x such that x F 1 p This value is relatively simple to find for the cases of the expo...

Страница 563: ...al results change the CAS setting to Approx The function to plot for the Gamma distribution is Y X 0 X z α 1 exp z β β α GAMMA α z p For the Beta distribution the function to plot is Y X 0 X z α 1 1 z β 1 GAMMA α β GAMMA α GAMMA β z p To produce the plot it is necessary to store values of α β and p before attempting the plot For example for α 2 β 3 and p 0 3 the plot of Y X for the Gamma distribut...

Страница 564: ...Alternatively you can use function TRACE X Y to estimate the roots by tracing the curve near its intercepts with the x axis Two estimates are shown below These estimates suggest solutions x 1 9 and x 3 3 You can verify these solutions by evaluating function Y1 X for X 1 9 and X 3 3 i e For the normal Student s t Chi square χ2 and F distributions which are represented by functions UTPN UTPT UPTC an...

Страница 565: ...ution with μ 10 σ2 2 with p 0 25 store the equation p 1 UTPN μ σ2 x into variable EQ figure in the left hand side below Then launch the numerical solver to get the input form in the right hand side figure The next step is to enter the values of μ σ2 and p and solve for x This input form can be used to solve for any of the four variables involved in the equation for the normal distribution To facil...

Страница 566: ...in all the examples shown above we are working with p P X x In many statistical inference problems we will actually try to find the value of x for which P X x α Furthermore for the normal distribution we most likely will be working with the standard normal distribution in which μ 0 and σ2 1 The standard normal variable is typically referred to as Z so that the problem to solve will be P Z z α For ...

Страница 567: ...Page 17 18 With these four equations whenever you launch the numerical solver you have the following choices Examples of solution of equations EQNA EQTA EQCA and EQFA are shown below ʳʳʳʳʳ ...

Страница 568: ... in this Chapter First however we demonstrate how to enter data for statistical analysis Entering data For the analysis of a single set of data a sample we can use applications number 1 2 and 4 from the list above All of these applications require that the data be available as columns of the matrix ΣDAT This can be accomplished by entering the data in columns using the matrix writer This operation...

Страница 569: ... when you exit the Matrix Writer the data you have entered is automatically saved in ΣDAT Calculating single variable statistics Assuming that the single data set was stored as a column vector in variable ΣDAT To access the different STAT programs press Ù Press OK to select 1 Single var There will be available to you an input form labeled SINGLE VARIABLE STATISTICS with the data currently in your ...

Страница 570: ...tity can take an infinite number of values the population of x in this case is infinite If you select a sub set of a population represented by the n data values x1 x2 xn we say you have selected a sample of values of x Samples are characterized by a number of measures or statistics There are measures of central tendency such as the mean median and mode and measures of spreading such as the range v...

Страница 571: ...m in RPN mode see Chapter 21 for more information on programming in User RPL language nC RCLΣ DUP SIZE 2 GET IF 1 THEN nC COL SWAP DROP OBJ 1 ARRY END OBJ OBJ DROP DROP DUP n LIST SORT IF n MOD 2 0 THEN DUP n 2 EVAL GET SWAP n 1 2 EVAL GET 2 ELSE n 1 2 EVAL GET END Median TAG Store this program under the name MED An example of application of this program is shown next Example 2 To run the program ...

Страница 572: ...ce and standard deviation which are calculated using n in the denominator of the variance rather than n 1 Example 3 If you were to repeat the exercise in Example 1 of this section using Population rather than Sample as the Type you will get the same values for the mean total maximum and minimum The variance and standard deviation however will be given by Variance 0 852 Std Dev 0 923 Obtaining freq...

Страница 573: ... the class mark and is defined as xMi xBi xB i 1 2 for i 1 2 k If the classes are chosen such that the class size is the same then we can define the class size as the value Bin Width Δx xmax xmin k and the class boundaries can be calculated as xBi xbot i 1 Δx Any data point xj j 1 2 n belongs to the i th class if xBi xj xB i 1 The application 2 Frequencies in the STAT menu will perform this freque...

Страница 574: ...rogram 2 Frequencies by using Ù OK The data is already loaded in ΣDAT and the option Col should hold the value 1 since we have only one column in ΣDAT Θ Change X Min to 10 Bin Count to 8 and Bin Width to 10 then press OK Using the RPN mode the results are shown in the stack as a column vector in stack level 2 and a row vector of two components in stack level 1 The vector in stack level 1 is the nu...

Страница 575: ...the cumulative frequency is 33 16 49 and so on The cumulative frequency represents the frequency of those numbers that are smaller than or equal to the upper boundary of any given class Given the column vector of frequencies generated by the calculator you can obtain a cumulative frequency vector by using the following program in RPN mode Class No Class Bound Class mark Frequency Cumulative i XBi ...

Страница 576: ...l value of x the number of bins and the bin width to generate the histogram Alternatively you can generate the column vector containing the frequency count as performed in the example above store this vector into ΣDAT and select Barplot as your graph type In the next example we show you how to use the first method to generate a histogram Example 1 Using the 200 data points generated in the example...

Страница 577: ...terplots that simulate these two plots by entering the proper data in columns 1 and 2 of a new ΣDAT matrix and changing the Type to SCATTER in the PLOT SETUP window Fitting data to a function y f x The program 3 Fit data available as option number 3 in the STAT menu can be used to fit linear logarithmic exponential and power functions to data sets x y stored in columns of the ΣDAT matrix In order ...

Страница 578: ...itting press OK The output from this program shown below for our particular data set consists of the following three lines in RPN mode 3 0 195238095238 2 00857142857 X 2 Correlation 0 983781424465 1 Covariance 7 03 Level 3 shows the form of the equation In this case y 0 06924 0 00383 x Level 2 shows the sample correlation coefficient and level 1 shows the covariance of x y Definitions For a sample...

Страница 579: ...e linearized as described in the table below The sample covariance of ξ η is given by Also we define the sample variances of ξ and η respectively as The sample correlation coefficient rξη is Indep Depend Type of Actual Linearized variable Variable Covar Fitting Model Model ξ η sξη Linear y a bx same x y sxy Log y a b ln x same ln x y sln x y Exp y a ebx ln y ln a bx x ln y sx ln y Power y a xb ln ...

Страница 580: ...rix into the statistical matrix ΣDAT by using function STOΣ Finally launch the data fit application by using Ù OK The display shows the current ΣDAT already loaded Change your set up screen to the following parameters if needed Press OK to get 3 3 99504833324 EXP 579206831203 X 2 Correlation 0 996624999526 1 Covariance 6 23350666124 The best fit for the data is therefore y 3 995 e 0 58 x Obtaining...

Страница 581: ... summary stats option use Ù OK Θ Select the column numbers corresponding to the x and y data i e X Col 1 and Y Col 2 Θ Using the CHK key select all the options for outputs i e _ΣX _ΣY etc Θ Press OK to obtain the following results ΣX 24 2 ΣY 11 72 ΣX2 148 54 ΣY2 26 6246 ΣXY 12 602 NΣ 8 Calculation of percentiles Percentiles are measures that divide a data set into 100 parts The basic procedure to ...

Страница 582: ...the 27 percentile of the list 2 1 0 1 3 5 1 2 3 6 7 9 In RPN mode enter 0 27 2 1 0 1 3 5 1 2 3 6 7 9 TILE In ALG mode enter TILE 0 27 2 1 0 1 3 5 1 2 3 6 7 9 The result is 1 The STAT soft menu All the pre programmed statistical functions described above are accessible through a STAT soft menu The STAT soft menu can be accessed by using in RPN mode the command 96 MENU You can create your own progra...

Страница 583: ... 1 of stack into ΣDATA matrix The ΣPAR sub menu The ΣPAR sub menu contains functions used to modify statistical parameters The parameters shown correspond to the last example of data fitting The parameters shown in the display are Xcol indicates column of ΣDATA representing x Default 1 Ycol indicates column of ΣDATA representing y Default 2 Intercept shows intercept of most recent data fitting Def...

Страница 584: ...ch column in ΣDATA matrix SDEV shows standard deviation of each column in ΣDATA matrix MAXΣ shows maximum value of each column in ΣDATA matrix MINΣ shows average of each column in ΣDATA matrix BINS used as xs Δx n BINS provides frequency distribution for data in Xcol column in ΣDATA matrix with the frequency bins defined as xs xs Δx xs xs 2Δx xs xs nΔx VAR shows variance of each column in ΣDATA ma...

Страница 585: ...he data in columns Xcol and Ycol of the ΣDATA matrix The functions available in this sub menu are ΣLINE provides the equation corresponding to the most recent fitting LR provides intercept and slope of most recent fitting PREDX used as y PREDX given y find x for the fitting y f x PREDY used as x PREDY given x find y for the fitting y f x CORR provides the correlation coefficient for the most recen...

Страница 586: ...tack by using the Matrix Writer Θ To store the matrix into ΣDATA use DATA DAT Θ Calculate statistics of each column STAT 1VAR TOT produces 38 5 87 5 82799 8 MEAN produces 5 5 12 5 11828 54 SDEV produces 3 39 6 78 21097 01 MAX produces 10 21 5 55066 MIN produces 1 1 3 7 7 8 L VAR produces 11 52 46 08 445084146 33 PSDEV produces 3 142 6 284 19532 04 PVAR produces 9 87 39 49 381500696 85 Θ Data Θ Gen...

Страница 587: ... fitting equation and some of its statistics STAT FIT LINE produces 1 5 2 X LR produces Intercept 1 5 Slope 2 3 PREDX produces 0 75 1 PREDY produces 3 50 CORR produces 1 0 COV produces 23 04 L PCOV produces 19 74 Θ Obtain summary statistics for data in columns 1 and 2 STAT SUMS X produces 38 5 Y produces 87 5 X2 produces 280 87 Y2 produces 1370 23 XY produces 619 49 N produces 7 ...

Страница 588: ...uce scattergram of y vs x STATL show line for log fitting Obviously the log fit is not a good choice CANCL returns to normal display Select the best fitting by using STAT PAR MODL BESTF shows EXPFIT as the best fit for these data L STAT FIT LINE produces 2 6545 EXP 0 9927 X CORR produces 0 99995 good correlation 2300 PREDX produces 6 8139 5 2 PREDY produces 463 33 ...

Страница 589: ...nt to the concept of random sampling Θ Population collection of all conceivable observations of a process or attribute of a component Θ Sample sub set of a population Θ Random sample a sample representative of the population Θ Random variable real valued function defined on a sample space Could be discrete or continuous If the population follows a certain probability distribution that depends on a...

Страница 590: ...the estimate of μ is the sample statistic x 2 2 2 5 2 1 2 3 2 2 5 2 26 This single value of X namely x 2 26 constitutes a point estimation of the population parameter μ Estimation of Confidence Intervals The next level of inference from point estimation is interval estimation i e instead of obtaining a single value of an estimator we provide two statistics a and b which define an interval containi...

Страница 591: ... as X zα σ n and an upper one sided confidence interval as X zα σ n Notice that in these last two intervals we use the value zα rather than zα 2 In general the value zk in the standard normal distribution is defined as that value of z whose probability of exceedence is k i e Pr Z zk k or Pr Z zk 1 k The normal distribution was described in Chapter 17 Confidence intervals for the population mean wh...

Страница 592: ...ed then an estimate of p is given by p k n while the standard error of p is σp p 1 p n In practice the sample estimate for p i e p replaces p in the standard error formula For a large sample size n 30 and n p 5 and n 1 p 5 the sampling distribution is very nearly normal Therefore the 100 1 α central two sided confidence interval for the population mean p is p zα 2 σp p zα 2 σp For a small sample n...

Страница 593: ...ntervals for the difference and sum of the mean values of the populations i e μ1 μ2 are given by For large samples i e n1 30 and n2 30 and unknown but equal population variances σ1 2 σ2 2 the confidence intervals for the difference and sum of the mean values of the populations i e μ1 μ2 are given by If one of the samples is small i e n1 30 or n2 30 and with unknown but equal population variances σ...

Страница 594: ...ey have the same population variance However if we have reason to believe that the two unknown population variances are different we can use the following confidence interval where the estimated standard deviation for the sum or difference is and n the degrees of freedom of the t variate are calculated using the integer value closest to Determining confidence intervals The application 6 Conf Inter...

Страница 595: ...Confidence interval for the difference of two proportions p1 p2 for large samples with unknown population variances 5 T INT 1 μ Single sample confidence interval for the population mean μ for small samples with unknown population variance 6 T INT μ1 μ2 Confidence interval for the difference of the population means μ1 μ2 for small samples with unknown population variances Example 1 Determine the ce...

Страница 596: ... in the screen above corresponds to the values zα 2 in the confidence interval formula X zα 2 σ n X zα 2 σ n The values μ Min and μ Max are the lower and upper limits of this interval i e μ Min X zα 2 σ n and μ Max X zα 2 σ n Press GRAPH to see a graphical display of the confidence interval information The graph shows the standard normal distribution pdf probability density function the location o...

Страница 597: ...alculator Press OK to select option 2 Z INT μ 1 μ2 Enter the following values When done press OK The results as text and graph are shown below The variable Δμ represents μ 1 μ2 Example 3 A survey of public opinion indicates that in a sample of 150 people 60 favor increasing property taxes to finance some public projects Determine the 99 confidence interval for the population proportion that would ...

Страница 598: ...ference between two proportions if sample 1 shows 20 successes out of 120 trials and sample 2 shows 15 successes out of 100 trials Press Ù OK to access the confidence interval feature in the calculator Press OK to select option 4 Z INT p1 p2 Enter the following values When done press OK The results as text and graph are shown below ...

Страница 599: ...es When done press OK The results as text and graph are shown below The figure shows the Student s t pdf for ν 50 1 49 degrees of freedom Example 6 Determine the 99 confidence interval for the difference in means of two populations given the sample data x1 157 8 x2 160 0 n1 50 n2 55 The populations standard deviations are s1 13 2 s 2 24 5 Press Ù OK to access the confidence interval feature in the...

Страница 600: ...rmula for the confidence interval for the variance first we introduce the sampling distribution of the variance Consider a random sample X1 X2 Xn of independent normally distributed variables with mean μ variance σ2 and sample mean X The statistic is an unbiased estimator of the variance σ2 The quantity has a χn 1 2 chi square distribution with ν n 1 degrees of freedom The 1 α 100 two sided confid...

Страница 601: ...l solver to solve the equation α UTPC γ x In this program γ represents the degrees of freedom n 1 and α represents the probability of exceeding a certain value of x χ2 i e Pr χ2 χα 2 α For the present example α 0 05 γ 24 and α 0 025 Solving the equation presented above results in χ2 n 1 α 2 χ2 24 0 025 39 3640770266 On the other hand the value χ2 n 1 α 2 χ2 24 0 975 is calculated by using the valu...

Страница 602: ...eps 1 Declare a null hypothesis H0 This is the hypothesis to be tested For example H0 μ1 μ2 0 i e we hypothesize that the mean value of population 1 and the mean value of population 2 are the same If H0 is true any observed difference in means is attributed to errors in random sampling 2 Declare an alternate hypothesis H1 For the example under consideration it could be H1 μ1 μ2 0 Note this is what...

Страница 603: ...hesis Pr Not Type II error Pr T R H1 1 β The complement of β is called the power of the test of the null hypothesis H0 vs the alternative H1 The power of a test is used for example to determine a minimum sample size to restrict errors Selecting values of α and β A typical value of the level of significance or probability of Type I error is α 0 05 i e incorrect rejection once in 20 times on the ave...

Страница 604: ...f confidence 1 α 100 or significance level α using a sample of size n with a mean x and a standard deviation s This test is referred to as a two sided or two tailed test The procedure for the test is as follows First we calculate the appropriate statistic for the test to or zo as follows Θ If n 30 and the standard deviation of the population σ is known use the z statistic Θ If n 30 and σ is known ...

Страница 605: ...n standard deviation therefore we calculate a t statistic as follows The corresponding P value for n 25 1 24 degrees of freedom is P value 2 UTPT 24 0 7142 2 0 7590 1 518 since 1 518 0 05 i e P value α we cannot reject the null hypothesis Ho μ 22 0 One sided hypothesis The problem consists in testing the null hypothesis Ho μ μo against the alternative hypothesis H1 μ μο or H1 μ μο at a level of co...

Страница 606: ...ue UTPN 0 1 zo Θ If using t P value UTPT ν to Example 2 Test the null hypothesis Ho μ 22 0 μo against the alternative hypothesis H1 μ 22 5 at a level of confidence of 95 i e α 0 05 using a sample of size n 25 with a mean x 22 0 and a standard deviation s 3 5 Again we assume that we don t know the value of the population standard deviation therefore the value of the t statistic is the same as in th...

Страница 607: ... H1 μ1 μ2 δ The P value for this test is calculated as Θ If using z P value 2 UTPN 0 1 zo Θ If using t P value 2 UTPT ν to with the degrees of freedom for the t distribution given by ν n1 n2 2 The test criteria are Θ Reject Ho if P value α Θ Do not reject Ho if P value α One sided hypothesis If the alternative hypothesis is a two sided hypothesis i e H1 μ1 μ2 δ or H1 μ1 μ2 δ the P value for this t...

Страница 608: ...erences concerning one proportion Suppose that we want to test the null hypothesis H0 p p0 where p represents the probability of obtaining a successful outcome in any given repetition of a Bernoulli trial To test the hypothesis we perform n repetitions of the experiment and find that k successful outcomes are recorded Thus an estimate of p is given by p k n The variance for the sample will be esti...

Страница 609: ...sents the probability of obtaining a successful outcome in any given repetition of a Bernoulli trial for two populations 1 and 2 To test the hypothesis we perform n1 repetitions of the experiment from population 1 and find that k1 successful outcomes are recorded Also we find k2 successful outcomes out of n2 trials in sample 2 Thus estimates of p1 and p2 are given respectively by p1 k1 n1 and p2 k...

Страница 610: ...s A z0 zα 2 One tailed test If using a one tailed test we will find the value of za from Pr Z zα 1 Φ zα α or Φ z α 1 α Reject the null hypothesis H0 if z0 zα and H1 p1 p2 p0 or if z0 zα and H1 p1 p2 p0 Hypothesis testing using pre programmed features The calculator provides with hypothesis testing procedures under application 5 Hypoth tests can be accessed by using Ù OK As with the calculation of ...

Страница 611: ...two proportions p1 p2 for large samples with unknown population variances 5 T Test 1 μ Single sample hypothesis testing for the population mean μ for small samples with unknown population variance 6 T Test μ1 μ2 Hypothesis testing for the difference of the population means μ1 μ2 for small samples with unknown population variances Try the following exercises Example 1 For μ0 150 σ 10 x 158 n 50 for...

Страница 612: ...raphically by pressing the soft menu key GRAPH Example 2 For μ0 150 x 158 s 10 n 50 for α 0 05 test the hypothesis H0 μ μ0 against the alternative hypothesis H1 μ μ0 The population standard deviation σ is not known Press Ù OK to access the hypothesis testing feature in the calculator Press OK to select option 5 T Test 1 μ Enter the following data and press OK Select the alternative hypothesis H1 μ...

Страница 613: ... see the results graphically as follows Example 3 Data from two samples show that x1 158 x1 160 s1 10 s2 4 5 n1 50 and n2 55 For α 0 05 and a pooled variance test the hypothesis H0 μ1 μ2 0 against the alternative hypothesis H1 μ1 μ2 0 Press Ù OK to access the hypothesis testing feature in the calculator Press OK to select option 6 T Test μ1 μ2 Enter the following data and press OK Select the alter...

Страница 614: ...ed is Ho σ2 σo 2 at a level of confidence 1 α 100 or significance level α using a sample of size n and variance s2 The test statistic to be used is a chi squared test statistic defined as Depending on the alternative hypothesis chosen the P value is calculated as follows Θ H1 σ2 σo 2 P value P χ2 χo 2 1 UTPC ν χo 2 Θ H1 σ2 σo 2 P value P χ2 χo 2 UTPC ν χo 2 Θ H1 σ2 σo 2 P value 2 min P χ2 χo 2 P χ...

Страница 615: ... 0 2587 Since 0 2587 0 05 i e P value α we cannot reject the null hypothesis Ho σ2 25 σo 2 Inferences concerning two variances The null hypothesis to be tested is Ho σ1 2 σ2 2 at a level of confidence 1 α 100 or significance level α using two samples of sizes n1 and n2 and variances s1 2 and s2 2 The test statistic to be used is an F test statistic defined as where sN 2 and sD 2 represent the nume...

Страница 616: ...________________________ nM is the value of n corresponding to the sM and nm is the value of n corresponding to sm ____________________________________________________________________ The P value is calculated in all cases as P value P F Fo UTPF νN νD Fo The test criteria are Θ Reject Ho if P value α Θ Do not reject Ho if P value α Example1 Consider two samples drawn from normal populations such t...

Страница 617: ...riable The regression curve of Y on x is defined as the relationship between x and the mean of the corresponding distribution of the Y s Assume that the regression curve of Y on x is linear i e mean distribution of Y s is given by Α Βx Y differs from the mean Α Β x by a value ε thus Y Α Β x ε where ε is a random variable To visually check whether the data follows a linear trend draw a scattergram ...

Страница 618: ...inear regression The summary statistics such as Σx Σx2 etc can be used to define the following quantities Notes Θ a b are unbiased estimators of Α Β Θ The Gauss Markov theorem of probability indicates that among all unbiased estimators for Α and Β the least square estimators a b are the most efficient n i i n i i x b n a y 1 1 n i i n i i n i i i x b x a y x 1 2 1 1 n i i n i i x n i i xx x n x s ...

Страница 619: ...bles with mean Α Β xi and the common variance σ2 εi independent normally distributed random variables with mean zero and the common variance σ2 Let yi actual data value yi a b xi least square prediction of the data Then the prediction error is ei yi yi yi a b xi An estimate of σ2 is the so called standard error of the estimate Confidence intervals and hypothesis testing in linear regression Here a...

Страница 620: ...nd it turns out that the test suggests that you do not reject the null hypothesis H0 Β 0 then the validity of a linear regression is in doubt In other words the sample data does not support the assertion that Β 0 Therefore this is a test of the significance of the regression model Θ Hypothesis testing on the intercept Α Null hypothesis H0 Α Α0 tested against the alternative hypothesis H1 Α Α0 The ...

Страница 621: ... either confidence intervals or two tailed tests obtain tα 2 with 1 α 100 confidence from t distribution with ν n 2 7 For one or two tailed tests find the value of t using the appropriate equation for either Α or Β Reject the null hypothesis if P value α 8 For confidence intervals use the appropriate formulas as shown above Example 1 For the following x y data determine the 95 confidence interval ...

Страница 622: ...the slope Β and intercept A Θ First we obtain t n 2 α 2 t3 0 025 3 18244630528 See chapter 17 for a program to solve for tν a Θ Next we calculate the terms t n 2 α 2 se Sxx 3 182 0 1826 2 5 1 2 0 8602 t n 2 α 2 se 1 n x2 Sxx 1 2 3 1824 0 1826 1 5 32 2 5 1 2 2 65 Θ Finally for the slope B the 95 confidence interval is 0 86 0 860242 0 86 0 860242 1 72 0 00024217 For the intercept A the 95 confidence...

Страница 623: ...example the value of the level of significance is α 0 05 g 3 and tn 2 α 2 t3 0 025 Also for γ 3 and α 0 025 tn 2 α 2 t3 0 025 3 18244630528 Because t0 tn 2 α 2 we cannot reject the null hypothesis H0 Α 0 against the alternative hypothesis H1 Α 0 at the level of significance α 0 05 This result suggests that taking A 0 for this linear regression should be acceptable After all the value we found for ...

Страница 624: ... by putting together the matrix X _ _ _ _ Then the vector of coefficients is obtained from b XT X 1 XT y where y is the vector y y1 y2 ym T For example use the following data to obtain the multiple linear fitting y b0 b1 x1 b2 x2 b3 x3 x1 x2 x3 xn y x11 x21 x31 xn1 y1 x12 x22 x32 xn2 y2 x13 x32 x33 xn3 y3 x1 m 1 x 2 m 1 x 3 m 1 x n m 1 ym 1 x1 m x 2 m x 3 m x n m ym 1 x11 x21 x31 xn1 1 x12 x22 x32...

Страница 625: ...sion Next enter the matrices X and b into the stack 1 1 2 3 1 2 1 2 5 3 1 2 5 1 3 5 4 5 2 5 1 4 4 5 3 1 6 5 3 5 keep an extra copy 5 7 8 2 5 0 8 2 9 5 Press J MTREG The result is 2 1649 0 7144 1 7850 7 0941 i e y 2 1649 0 7144 x1 1 7850 10 2 x2 7 0941 x3 You should have in your calculator s stack the value of the matrix X and the vector b the fitted values of y are obtained from y X b thus just pr...

Страница 626: ... 1 XT y where y is the vector y y1 y2 yn T In Chapter 10 we defined the Vandermonde matrix corresponding to a vector x x1 x2 xm The Vandermonde matrix is similar to the matrix X of interest to the polynomial fitting but having only n rather than p 1 columns We can take advantage of the VANDERMONDE function to create the matrix X if we observe the following rules If p n 1 X Vn If p n 1 then remove ...

Страница 627: ...lready developed for multiple linear fitting We need to add to this program the steps 1 through 3 listed above The algorithm for the program therefore can be written as follows Enter vectors x and y of the same dimension as lists Note since the function VANDERMONDE uses a list as input it is more convenient to enter the x y data as a list Also enter the value of p Θ Determine n size of vector x Θ ...

Страница 628: ...P loop ELSE IF p n 1 THEN n 1 Calculate n 1 p 1 Calculate p 1 FOR j Start a loop with j n n 1 p 1 x j Calculate xj as a list OBJ ARRY Convert list to array j COL Add column to matrix NEXT Close FOR NEXT loop END Ends second IF clause END Ends first IF clause Its result is X y OBJ ARRY Convert list y to an array MTREG X and y used by program MTREG NUM Convert to decimal format Close sub program 2 C...

Страница 629: ...742 23x2 xx yy 3 POLY Result 998 05 1303 21 505 27 79 23 i e y 998 05 1303 21x 505 27x2 79 23x3 xx yy 4 POLY Result 20 92 2 61 1 52 6 05 3 51 i e y 20 92 2 61x 1 52x2 6 05x3 3 51x4 xx yy 5 POLY Result 19 08 0 18 2 94 6 36 3 48 0 00 i e y 19 08 0 18x 2 94x2 6 36x3 3 48x4 0 0011x5 xx yy 6 POLY Result 16 73 67 17 48 69 21 11 1 07 0 19 0 00 i e y 16 72 67 17x 48 69x2 21 11x3 1 07x4 0 19x5 0 0058x6 Sel...

Страница 630: ...tor of polynomial coefficients b We can calculate a vector of fitted data y by using y X b An error vector is calculated by e y y The sum of square errors is equal to the square of the magnitude of the error vector i e SSE e 2 e e Σ ei 2 Σ yi y i 2 To calculate the correlation coefficient we need to calculate first what is known as the sum of squared totals SST defined as SST Σ yi y 2 where y is t...

Страница 631: ...NEXT Close FOR NEXT loop END Ends second IF clause END Ends first IF clause Produces X y OBJ ARRY Convert list y to an array X yv Enter matrix and array as X and y Open subprogram 3 X yv MTREG X and y used by program MTREG NUM If needed converts to floating point b Resulting vector passed as b Open subprogram 4 b yv Place b and yv in stack X b Calculate X b Calculate e y X b ABS SQ DUP Calculate S...

Страница 632: ...duce the following table of values of the correlation coefficient r and the sum of square errors SSE While the correlation coefficient is very close to 1 0 for all values of p in the table the values of SSE vary widely The smallest value of SSE corresponds to p 4 Thus you could select the preferred polynomial data fitting for the original x y data as y 20 92 2 61x 1 52x2 6 05x3 3 51x4 p r SSE 2 0 ...

Страница 633: ...ystem the relative position of digits determines its value In general a number n in base b can be written as a series of digits n a1a2 an c1c2 cm b The point separates n integer digits from m decimal digits The value of the number converted to our customary decimal system is calculated by using n a1 bn 1 a2 bn 2 anb0 c1 b 1 c2 b 2 cm b m For example 15 234 10 1 101 5 100 2 10 1 3 10 2 4 10 3 and 1...

Страница 634: ... â the 3 key To select which number system current base will be used for numbers preceded by select one of the following functions in the first BASE menu i e HEX adecimal DEC imal OCT al or BIN ary For example if HEX is selected any number written in the calculator that starts with will be a hexadecimal number Thus you can write numbers such as 53 A5B etc in this system As different systems are se...

Страница 635: ...cimal number preceded by into a decimal number while the function R B works in the opposite direction Try the following exercises HEX is the current base The following examples show conversions when the base is the octal system We also present transformations using the binary system as the current base Notice that every time you enter a number starting with you get as the entry the number you ente...

Страница 636: ... will affect the way that binary integer operations are performed For example if a binary integer exceeds the current wordsize the leading bits will be dropped before any operation can be performed on such number Operations with binary integers The operations of addition subtraction change of sign multiplication and division are defined for binary integers Some examples of addition and subtraction...

Страница 637: ...be either true 1 or false 0 Some examples of logical statements are shown below Functions AND OR XOR and NOT can be applied to comparison statements under the following rules These functions can be used to build logical statements for programming purposes In the context of this Chapter they will by used to provide the result of bit by bit operations along the lines of the rules provided above In t...

Страница 638: ... menu are used to manipulate bits in a binary integer The definition of these functions are shown below RL Rotate Left one bit e g 1100b 11000b SL Shift Left one bit e g 1101b 11010b ASR Arithmetic Shift Right one bit e g 1100010b 110001b SR Shift Right one bit e g 11011b 1101b RR Rotate Right one bit e g 1101b 10000000000000000000000000000000000000000000000000 000000000001b ...

Страница 639: ...ne byte e g 1101b 110100000000b SRB Shift Right one byte e g 11011b 0b RRB Rotate Right one byte e g 1101b 1101000000000000000000000000000000000000000000000 00000000000b Hexadecimal numbers for pixel references Many plot option specifications use pixel references as input e g 332h A23h Ah 0 360 ARC to draw an arc of a circle We use functions C PX and PX C to convert quickly between user unit coord...

Страница 640: ... CST Thus to create a menu you must put together this variable with the features that you want to display in your menu and the actions required by the soft menu keys To show examples of customizing menus we need to set system flag 117 to SOFT menu Make sure you do this before continuing See Chapter 2 for instructions on setting system flags The PRG MODES MENU menu Commands useful in customizing me...

Страница 641: ...t menu Custom menus MENU and TMENU functions Suppose that you need to activate four functions for a particular application Say that you need to be able to quickly access the functions EXP LN GAMMA and 2 and you want to place them in a soft menu that you will keep active for a while You could do this by creating a temporary menu with function TMENU or a more permanent menu with function MENU The ma...

Страница 642: ...n the stack so that the argument to the function can be typed at the prompt e g EXP We need not worry about the closing parenthesis because the calculator will complete the parentheses before executing the function The implementation of function TMENU in ALG mode with the argument list shown above is as follows First we enter the list then we produce the temporary menu see menu key labels by using...

Страница 643: ...mand The examples above illustrate the difference The general form of the argument list for commands TMENU or MENU in ALG mode is label1 function1 ls1 rs1 label2 function2 ls2 rs2 While in RPN mode the argument list has this format label1 function1 ls1 rs1 label2 function2 ls2 rs2 In these specifications function1 function 2 etc represent the main operation of the key while ls1 ls2 etc represent t...

Страница 644: ... combined with 61 key simultaneous with Thus the VAR function will be referred to as key 31 0 or 31 1 while the UPDIR function will be key 31 2 the COPY function will be key 31 3 the upper case J is key 31 4 and lower case j is key 31 5 Key 31 6 is not defined In general a key will be described by the arrangement XY Z where X row number Y column number Z shifting We can combine a given key with th...

Страница 645: ... a user defined key Suppose that you want to have access to the old fashioned PLOT command first introduced with the HP 48G series calculator but currently not directly available from the keyboard The menu number for this menu is 81 01 You can see this menu active by using ALG mode MENU 81 01 RPN mode 81 01 MENU If you want to have a quick way to activate this menu from the keyboard you could assi...

Страница 646: ...line To unlock the keyboard press Ì once more Un assigning a user defined key To remove the assignment performed above use function DELKEYS as follows ALG mode DELKEYS 13 0 RPN mode 13 0 DELKEYS Assigning multiple user defined keys The simplest way to assign several user defined is to provide a list of commands and key specifications For example suppose that we assign the three trigonometric funct...

Страница 647: ...Page 20 8 To un assign all user defined keys use ALG mode DELKEYS 0 RPN mode 0 DELKEYS Check that the user key definitions were removed by using function RCLKEYS ...

Страница 648: ...ramming Throughout the previous Chapters in this guide we have presented a number of programs that can be used for a variety of applications e g programs CRMC and CRMT used to create a matrix out of a number of lists were presented in Chapter 10 In this section we present a simple program to introduce concepts related to programming the calculator The program we will write will be used to define t...

Страница 649: ...ble name x to store the value placed in level 1 of stack through the programming steps x STO The variable x while the program is executing is stored in your variable menu as any other variable you had previously stored After calculating the function the program purges erases the variable x so it will not show in your variable menu after finishing evaluating the program If we were not to purge the ...

Страница 650: ...trol is then passed back to the main program but there are no more commands between the first set of closing programming symbols and the second one therefore the program terminates The last value in the stack i e SINH x 1 x2 is returned as the program output The variable x in the last version of the program never occupies a place among the variables in your variable menu It is operated upon within...

Страница 651: ...OME directory will be accessible from any directory within HOME unless redefined within a directory or sub directory Θ If you re define the variable within a directory or sub directory this definition takes precedence over any other definition in directories above the current one Θ When running a program that references a given global variable the program will use the value of the global variable ...

Страница 652: ...keystroke combination Within the PRG menu we identify the following sub menus press L to move to the next collection of sub menus in the PRG menu Here is a brief description of the contents of these sub menus and their sub menus STACK Functions for manipulating elements of the RPN stack MEM Functions related to memory manipulation DIR Functions related to manipulating directories ARITH Functions t...

Страница 653: ...GLE To change angle measure and coordinate systems FLAG To set and un set flags and check their status KEYS To define and activate user defined keys Chapter 20 MENU To define and activate custom menus Chapter 20 MISC Miscellaneous mode changes beep clock etc IN Functions for program input OUT Functions for program output TIME Time related functions ALRM Alarm manipulation ERROR Functions for error...

Страница 654: ...H END STR ROT CRDIR TEST TAG UNROT PGDIR BRCH CASE UNIT ROLL VARS CASE C R ROLLD TVARS THEN R C PICK ORDER END NUM UNPICK CHR PICK3 MEM ARITH BRCH START DTAG DEPTH STO START AND EQ DUP2 STO NEXT OR TYPE DUPN STOx STEP XOR VTYPE DROP2 STO NOT DROPN INCR BRCH FOR SAME LIST DUPDU DECR FOR TYPE OBJ NIP SINV NEXT SF LIST NDUPN SNEG STEP CF SUB SCONJ FS REPL MEM BRCH DO FC PURGE BRCH DO FS C MEM IFT UNT...

Страница 655: ...D STR STOF SIZE HEAD RCLF IN LIST PROC ANIMATE TAIL RESET INFORM DOLIST SREPL NOVAL DOSUB PICT MODES KEYS CHOOSE NSUB PICT MODES FMT ASN INPUT ENDSUB PDIM STD STOKEYS KEY STREAM LINE FIX RECLKEYS WAIT REVLIST TLINE SCI DELKEYS PROMPT SORT BOX ENG SEQ ARC FM MODES MENU OUT PIXON ML MENU PVIEW PIXOF CST TEXT PIX MODES ANGLE TMENU CLLCD PVIEW DEG RCLMENU DISP PX C RAD FREEZE C PX GRAD MSGBOX RECT BEE...

Страница 656: ...and RCL in MEM DIR sub menu are available in the keyboard through the keys K and Θ Functions RCL and PURGE in MEM DIR sub menu are available through the TOOL menu I Θ Within the BRCH sub menu pressing the left shift key or the right shift key before pressing any of the sub menu keys will create constructs related to the sub menu key chosen This only works with the calculator in RPN mode Examples a...

Страница 657: ...ailable after the key word for each construct so you can start typing at the right location Keystroke sequence for commonly used commands The following are keystroke sequences to access commonly used commands for numerical programming within the PRG menu The commands are first listed by menu ...

Страница 658: ...CH IF IF THEN BRCH IF THEN ELSE BRCH IF ELSE END BRCH IF END BRCH CASE CASE BRCH CASE CASE THEN BRCH CASE THEN END BRCH CASE END BRCH START START BRCH START START NEXT BRCH START NEXT STEP BRCH START STEP BRCH FOR FOR BRCH FOR FOR NEXT BRCH FOR NEXT STEP BRCH FOR STEP BRCH DO DO BRCH DO DO UNTIL BRCH DO UNTIL END BRCH DO END ...

Страница 659: ...ME SF TEST L L SF CF TEST L L CF FS TEST L L FS FC TEST L L FC FS C TEST L L FS C FC C TEST L L FC C TYPE OBJ TYPE OBJ ARRY TYPE ARRY LIST TYPE LIST STR TYPE STR TAG TYPE TAG NUM TYPE L NUM CHR TYPE L CHR TYPE TYPE L TYPE LIST ELEM GET LIST ELEM GET GETI LIST ELEM GETI PUT LIST ELEM PUT PUTI LIST ELEM PUTI SIZE LIST ELEM SIZE HEAD LIST ELEM L HEAD TAIL LIST ELEM L TAIL ...

Страница 660: ...N INFORM L IN INFOR INPUT L IN INPUT MSGBOX L OUT MSGBO PVIEW L OUT PVIEW RUN DBUG LL RUN DBG SST LL RUN SST SST LL RUN SST HALT LL RUN HALT KILL LL RUN KILL Programs for generating lists of numbers Please notice that the functions in the PRG menu are not the only functions that can be used in programming As a matter of fact almost all functions in the calculator can be included in a program Thus ...

Страница 661: ...AP TAIL NEXT 1 GET n LIST REVLIST n PURGE The operation of these programs is as follows 1 LISC creates a list of n elements all equals to a constant c Operation enter n enter c press LISC Example 5 6 5 LISC creates the list 6 5 6 5 6 5 6 5 6 5 2 CRLST creates a list of numbers from n1 to n2 with increment Δn i e n1 n1 Δn n1 2 Δn n1 N Δn where N floor n2 n1 Δn 1 Operation enter n1 enter n2 enter Δn...

Страница 662: ... by using function DEFINE à with an argument of the form function_name x1 x2 expression containing variables x1 x2 The program is stored in a variable called function_name When the program is recalled to the stack by using function_name The program shows up as follows x1 x2 expression containing variables x1 x2 To evaluate the function for a set of input variables x1 x2 in RPN mode enter the varia...

Страница 663: ...is point there will be a variable called q in your soft menu key labels To see the contents of q use q The program generated by defining the function q Cu n y0 S0 is shown as Cu n y0 S0 Cu n y0 5 3 S0 This is to be interpreted as enter Cu n y0 S0 in that order then calculate the expression between quotes For example to calculate q for Cu 1 0 n 0 012 y0 2 m and S0 0 0001 use in RPN mode 1 0 012 2 0...

Страница 664: ...e of programs is with an example Example Velocity head for a rectangular channel Suppose that we want to calculate the velocity head hv in a rectangular channel of width b with a flow depth y that carries a discharge Q The specific energy is calculated as hv Q2 2g by 2 where g is the acceleration of gravity g 9 806 m s2 in S I units or g 32 2 ft s2 in E S units If we were to calculate hv for Q 23 ...

Страница 665: ...ype the following y b º g 2 Q º and keeping only the operations shown below do not type the following 2 º Unlike the interactive use of the calculator performed earlier we need to do some swapping of stack levels 1 and 2 within the program To write the program we use therefore å Opens program symbols Multiply y with b º Square b y Multiply b y 2 times g 2 Enter a 2 and multiply it with g b y 2 STA...

Страница 666: ...re sequential programs in the sense that the program flow follows a single path i e INPUT OPERATION OUTPUT Branching of the program flow is possible by using the commands in the menu BRCH More detail on program branching is presented below Interactive input in programs In the sequential program examples shown in the previous section it is not always clear to the user the order in which the variabl...

Страница 667: ...r s CAS Calculator Algebraic System must be set to symbolic and exact modes This is accomplished by using H CAS and ensuring that the check marks in the options _Numeric and _Approx are removed Press OK OK to return to normal calculator display Press J to display your variables menu We will use this latter approach to check what formula results from using the program hv as follows We know that the...

Страница 668: ...ed by the User RPL language the simplest is to use an input string and the function INPUT L IN INPUT to load your input data The following program prompts the user for the value of a variable a and places the input in stack level 1 Enter a a 2 0 V INPUT OBJ This program includes the symbol tag and return available through the keystroke combinations ê and ë both associated with the key The tag symb...

Страница 669: ... FUNCtion of a Run the program by pressing FUNCa When prompted to enter the value of a enter for example 2 and press The result is simply the algebraic 2a2 3 which is an incorrect result The calculator provides functions for debugging programs to identify logical errors during program execution as shown below Debugging the program To figure out why it did not work we use the DBUG function in the c...

Страница 670: ...a 2 0 V SST Result user is prompted to enter value of a 2 Enter a value of 2 for a Result a 2 SST Result a 2 SST Result empty stack executing a SST Result empty stack entering subprogram At this point we are within the subprogram 2 a 2 3 which uses the local variable a To see the value of a use aμ This indeed shows that the local variable a 2 Let s kill the debugger at this point since we already ...

Страница 671: ... taking care of changing the variable names according to the needs of each program Let s get started by creating a sub directory called PTRICKS Programming TRICKS to hold programming tidbits that we can later borrow from to use in more complex programming exercises To create the sub directory first make sure that you move to the HOME directory Within the HOME directory use the following keystrokes...

Страница 672: ... can define the pressure p as a function of two variables V and T as p V T nRT V for a given mass of gas since n will also remain constant Assume that n 0 2 gmol then the function to program is We can define the function by typing the following program V T 1 662902_J K T V and storing it into variable p The next step is to add the input string that will prompt the user for the values of V and T To...

Страница 673: ...ms as a reference and copy and modify them to fulfill the requirements of new programs you write Application evaluating a function of three variables Suppose that we want to program the ideal gas law including the number of moles n as an additional variable i e we want to define the function and modify it to include the three variable input string The procedure to put together this function is ver...

Страница 674: ...ications of the form label helpInfo type0 type1 typen The label is a field label The helpInfo is a character string describing the field label in detail and the type specifications is a list of types of variables allowed for the field see Chapter 24 for object types 3 Field format information a single number col or a list col tabs In this specification col is the number of columns in the input box...

Страница 675: ... channel through Chezy s formula Q C R S 1 2 where C is the Chezy coefficient a function of the channel surface s roughness typical values 80 150 R is the hydraulic radius of the channel a length and S is the channel bed s slope a dimensionless numbers typically 0 01 to 0 000001 The following program defines an input form through function INFORM CHEZY S EQN C Chezy s coefficient 0 R Hydraulic radi...

Страница 676: ...lues loaded is as follows To see the effect of resetting these values use L RESET select Reset all to reset field values Now enter different values for the three fields say C 95 R 2 5 and S 0 003 pressing OK after entering each of these new values After these substitutions the input form will look like this Now to enter these values into the program press OK once more This activates the function I...

Страница 677: ...e of Q and put a tag or label to it On the other hand if the value in stack level 1 is 0 which happens when a CANCEL is entered while using the input box the program control is passed to the commands Operation cancelled MSGBOX These commands will produce a message box indicating that the operation was cancelled Example 2 To illustrate the use of item 3 Field format information in the arguments of ...

Страница 678: ...ws the user to create a choose box in a program This function requires three arguments 1 A prompt a character string describing the choose box 2 A list of choice definitions c1 c2 cn A choice definition ci can have any of two formats a An object e g a number algebraic etc that will be displayed in the choose box and will also be the result of the choice b A list object_displayed object_result so t...

Страница 679: ...its 1 486 1 CHOOSE Running this program press CHP1 shows the following choose box Depending on whether you select S I units or E S units function CHOOSE places either a value of 1 or a value of 1 486 in stack level 2 and a 1 in level 1 If you cancel the choose box CHOICE returns a zero 0 The values returned by function CHOOSE can be operated upon by other program commands as shown in the modified ...

Страница 680: ...l 2 then use the TAG function TYPE TAG For example to produce the tagged result B 5 use 5 Õ b TYPE TAG Decomposing a tagged numerical result into a number and a tag To decompose a tagged result into its numerical value and its tag simply use function OBJ TYPE OBJ The result of decomposing a tagged number with OBJ is to place the numerical value in stack level 2 and the tag in stack level 1 If you ...

Страница 681: ...r a value of 2 when prompted and press The result is now the tagged result F 11 Example 2 tagging input and output from function FUNCa In this example we modify the program FUNCa so that the output includes not only the evaluated function but also a copy of the input with a tag Use FUNCa to recall the contents of FUNCa to the stack Enter a a 2 0 V INPUT OBJ a 2 a 2 3 NUM F TAG Modify it to read No...

Страница 682: ...nter a value of 2 for a Result a 2 SST Result a 2 SST Result empty stack executing a SST Result empty stack entering subprogram SST Result 2 a 2 3 SST Result empty stack calculating SST Result 11 SST Result F SST Result F 11 SST Result a 2 SST Result swap levels 1 and 2 SST leaving subprogram SST leaving main program Note Because we use an input string to get the input data value the local variabl...

Страница 683: ...s This is necessary because without the program symbol separating the two listings of input variables V T N V T n the program will assume that the input command V T N V T n requires six input values while only three are available The result would have been the generation of an error message and the interruption of the program execution To include the subprogram mentioned above in the modified defi...

Страница 684: ...n the calculator is obtained by using L OUT MSGBO The message box command requires that the output string to be placed in the box be available in stack level 1 To see the operation of the MSGBOX command try the following exercise Õ t ê1 2 Ý r a d L OUT MSGBO In summary The common thread in the three examples shown here is the use of tags to identify input and output variables If we use an input st...

Страница 685: ... STR available at TYPE STR Using a message box for program output The function p from the last example can be modified to read Enter V T and n V T n 2 0 V INPUT OBJ V T n V T n 8 31451_J K mol n T V EVAL p TAG STR MSGBOX Store the program back into variable p by using p Run the program by pressing p Enter values of V 0 01_m 3 T 300_K and n 0 8_mol when prompted As in the earlier version of p befor...

Страница 686: ...n 2 0 V INPUT OBJ V T n V STR T STR n STR 8 31451_J K mol n T V EVAL p TAG STR MSGBOX Notice that you need to add the following piece of code after each of the variable names V T and n within the sub program STR To get this piece of code typed in the first time use TYPE STR Õ ë Because the functions for the TYPE menu remain available in the soft menu keys for the second and third occurrences of th...

Страница 687: ...g Θ Store the program back into variable p by using p Θ Run the program by pressing p Θ Enter values of V 0 01_m 3 T 300_K and n 0 8_mol when prompted As in the earlier version of p before pressing ENTER for input the stack will look like this The first program output is a message box containing the string Press OK to cancel message box output Note The plus sign in this program is used to concaten...

Страница 688: ...1_mol V T n V V TAG STR T T TAG STR n n TAG STR 8 31451_J K mol n T V EVAL p TAG STR MSGBOX This new version of the program includes an additional level of sub programming i e a third level of program symbols and some steps using lists i e V 1_m 3 T 1_K n 1_mol EVAL V T n The interpretation of this piece of code is as follows We use input string values of V 0 01 T 300 and n 0 8 1 V The value of V ...

Страница 689: ...culating value of n including units 6 V T n The values of V T and n located respectively in stack levels 3 2 and 1 are passed on to the next level of sub programming To see this version of the program in action do the following Θ Store the program back into variable p by using p Θ Run the program by pressing p Θ Enter values of V 0 01 T 300 and n 0 8 when prompted no units required now Before pres...

Страница 690: ... box output Press OK to cancel the message box output Relational and logical operators So far we have worked mainly with sequential programs The User RPL language provides statements that allow branching and looping of the program flow Many of these make decisions based on whether a logical statement is true or not In this section we present some of the elements used to construct such logical stat...

Страница 691: ...q is less than or equal to 7 12 _____________________________________________________ All of the operators except which can be created by typing Å Å are available in the keyboard They are also available in TEST Two numbers variables or algebraics connected by a relational operator form a logical expression that can take value of true 1 false 0 or could simply not be evaluated To determine whether ...

Страница 692: ... false depending on the truth value of the logical statements affected The operator NOT negation applies to a single logical statements All of the others apply to two logical statements Tabulating all possible combinations of one or two statements together with the resulting value of applying a certain logical operator produces what is called the truth table of the operator The following are truth...

Страница 693: ...f a program flow implies that the program makes a decision among two or more possible flow paths The User RPL language provides a number of commands that can be used for program branching The menus containing these commands are accessed through the keystroke sequence BRCH This menu shows sub menus for the program constructs The program constructs IF THEN ELSE END and CASE THEN END will be referred...

Страница 694: ...ollows 1 Evaluate logical_statement 2 If logical_statement is true perform program _statements and continue program flow after the END statement 3 If logical_statement is false skip program_statements and continue program flow after the END statement To type in the particles IF THEN ELSE and END use BRCH IF The functions IF THEN ELSE END are available in that menu to be typed selectively by the us...

Страница 695: ...he function f1 x x2 if x 3 and not output otherwise The IF THEN ELSE END construct The IF THEN ELSE END construct permits two alternative program flow paths based on the truth value of the logical_statement The general format of this construct is IF logical_statement THEN program_statements_if_true ELSE program_statements_if_false END The operation of this construct is as follows 1 Evaluate logica...

Страница 696: ...Nested IF THEN ELSE END constructs In most computer programming languages where the IF THEN ELSE END construct is available the general format used for program presentation is the following IF logical_statement THEN program_statements_if_true ELSE program_statements_if_false END In designing a calculator program that includes IF constructs you could start by writing by hand the pseudo code for the...

Страница 697: ...onstructs to deal with function with three or more branches For example consider the function Here is a possible way to evaluate this function using IF THEN ELSE END constructs IF x 3 THEN x2 ELSE IF x 5 THEN 1 x ELSE IF x 3π THEN sin x ELSE IF x 15 THEN exp x ELSE 2 END END END END elsewhere x if x x if x x if x x if x x f 2 15 3 exp 3 5 sin 5 3 1 3 2 3 π π ...

Страница 698: ... 1 x 5 6 f3 Result 0 631266 i e sin x with x in radians 12 f3 Result 162754 791419 i e exp x 23 f3 Result 2 i e 2 The CASE construct The CASE construct can be used to code several possible program flux paths as in the case of the nested IF constructs presented earlier The general format of this construct is as follows CASE Logical_statement1 THEN program_statements1 END Logical_statement2 THEN pro...

Страница 699: ...prompts CASE THEN END END Θ CASE Completes a CASE line by adding the particles THEN END Example program f3 x using the CASE statement The function is defined by the following 5 expressions Using the CASE statement in User RPL language we can code this function as x CASE x 3 THEN x 2 END x 5 THEN 1 x END x 3 π THEN SIN x END x 15 THEN EXP x END 2 END EVAL Store the program into a variable called f3...

Страница 700: ...mits and expression for the summation examples of summations are presented in Chapters 2 and 13 However in order to illustrate the use of programming loops we will calculate this summation with our own User RPL codes There are four different commands that can be used to code a program loop in User RPL these are START FOR DO and WHILE The commands START and FOR use an index or counter to determine ...

Страница 701: ...der for the loop to end you should ensure that start_value end_value Otherwise you will produce what is called an infinite never ending loop Example calculating of the summation S defined above The START NEXT construct contains an index whose value is inaccessible to the user Since for the calculation of the sum the index itself k in this case is needed we must create our own index k that we will ...

Страница 702: ...ted by k2 in the piece of code that reads k SQ S 7 The index k is incremented by 1 in the piece of code that reads 1 k 8 At this point the updated values of S and k are available in stack levels 2 and 1 respectively The piece of code k STO stores the value from stack level 1 into local variable k The updated value of S now occupies stack level 1 9 The piece of code S STO stores the value from stac...

Страница 703: ... SST SL1 0 SQ k k2 SST SL1 0 S SL2 0 k2 SST SL1 0 S k2 SST SL1 1 SL2 0 S k2 SST SL1 0 k SL2 1 SL3 0 S k2 SST SL1 1 k 1 SL2 0 S k2 SST SL1 k SL2 1 SL3 0 S k2 SST SL1 0 S k2 Stores value of SL2 1 into SL1 k SST SL1 S SL2 0 S k2 SST Empty stack Stores value of SL2 0 into SL1 S SST Empty stack NEXT end of loop loop execution number 2 for k 1 SST SL1 1 k SST SL1 1 SQ k k2 SST SL1 0 S SL2 1 k2 SST SL1 1...

Страница 704: ...3 SL3 5 S k2 SST SL1 5 S k2 Stores value of SL2 3 into SL1 k SST SL1 S SL2 5 S k2 SST Empty stack Stores value of SL2 0 into SL1 S SST Empty stack NEXT end of loop for n 2 the loop index is exhausted and control is passed to the statement following NEXT SST SL1 5 S is recalled to the stack SST SL1 S SL2 5 S is placed in the stack SST SL1 S 5 tagging output value SST SL1 S 5 leaving sub program SST...

Страница 705: ...ample generating a list of values Suppose that you want to generate a list of values of x from x 0 5 to x 6 5 in increments of 0 5 You can write the following program xs xe dx xs DUP xe START DUP dx dx STEP DROP xe xs dx ABS 1 LIST and store it in variable GLIST In this program xs starting value of the loop xe ending value of the loop dx increment value for loop The program places values of xs xs ...

Страница 706: ...e FOR command does require that we provide a name for the loop index e g j k n We need not to worry about incrementing the index ourselves as done in the examples using START The value corresponding to the index is available for calculations Commands involved in the FOR construct are available through BRCH FOR Within the BRCH menu BRCH the following keystrokes are available to generate FOR constru...

Страница 707: ...that the program is much simpler than the one stored in S1 There is no need to initialize k or to increment k within the program The program itself takes care of producing such increments The FOR STEP construct The general form of this statement is start_value end_value FOR loop_index program_statements increment STEP The start_value end_value and increment of the loop index can be positive or neg...

Страница 708: ...e the detailed operation of each command The DO construct The general structure of this command is DO program_statements UNTIL logical_statement END The DO command starts an indefinite loop executing the program_statements until the logical_statement returns FALSE 0 The logical_statement must contain the value of an index whose value is changed in the program_statements Example 1 This program prod...

Страница 709: ...ample 3 generate a list using a DO UNTIL END construct Type in the following program xs xe dx xe xs dx ABS 1 xs n x xs DO x dx EVAL DUP x STO UNTIL x xe END n LIST and store it in variable GLIS3 Θ Check out that the program call 0 5 2 5 0 5 GLIS3 produces the list 0 5 1 1 5 2 2 5 Θ To see step by step operation use the program DBUG for a short list for example J1 1 5 0 5 Enter parameters 1 1 5 0 5...

Страница 710: ...luation of logical_statement is false the loop is never executed Example 1 calculate the summation S using a WHILE REPEAT END construct The following program calculates the summation Using a WHILE REPEAT END loop 0 n S WHILE n 0 REPEAT n SQ S S STO n 1 n STO END S S TAG Store this program in a variable S4 Verify the following exercises J 3 S4 Result S 14 4 S4 Result S 30 5 S4 Result S 55 8 S4 Resu...

Страница 711: ...ions of the PRG ERROR sub menu provide ways to manipulate errors in the calculator and trap errors in programs The PRG ERROR sub menu available through LL ERROR contains the following functions and sub menus DOERR This function executes an user define error thus causing the calculator to behave as if that particular error has occurred The function can take as argument either an integer number a bi...

Страница 712: ...ry 0Y ERRM you get the following string Infinite Result ERR0 This function clears the last error number so that executing ERRN afterwards in Approx mode will return 0h For example if you try 0Y ERR0 ERRN you get 0h Also if you try 0Y ERR0 ERRM you get the empty string LASTARG This function returns copies of the arguments of the command or function executed most recently For example in RPN mode if ...

Страница 713: ...o that of the IF THEN END and of the IF THEN ELSE END constructs If an error is detected during the execution of the trap clause then the error clause is executed Otherwise the normal clause is executed As an example consider the following program ERR1 that takes as input two matrices A and b and checks if there is an error in the trap clause A b RPN mode i e A b If there is an error then the prog...

Страница 714: ...entheses attached to their name The RPL function is not exception except that the parentheses must be removed before we type a program in the screen Use the arrow keys š and the delete key ƒ to eliminate the parentheses from the RPL statement At this point you will be ready to type the RPL program The following figures show the RPL command with the program before and after pressing the key To stor...

Страница 715: ...Page 21 68 Whereas using RPL there is no problem when loading this program in algebraic mode ...

Страница 716: ...th graphics To accomplish such tasks we first introduce function in the PLOT menu The PLOT menu Commands for setting up and producing plots are available through the PLOT menu You can access the PLOT menu by using 81 01 L MODES MENU MENU The menu thus produced provides the user access to a variety of graphics functions For application in subsequent examples let s user define the C GRAPH key to pro...

Страница 717: ...s Ì C You will get the following menu press L to move to second menu Description of the PLOT menu The following diagram shows the menus in PLOT The number accompanying the different menus and functions in the diagram are used as reference in the subsequent description of those objects The soft menu key labeled 3D STAT FLAG PTYPE and PPAR produce additional menus which will be presented in more det...

Страница 718: ...is scale equal to the x axis scale Θ POLAR based on the values of the independent variable typically θ it samples the function in EQ and determines minimum and maximum values of both x and y Θ PARAMETRIC produces a similar result as POLAR based on the values of the parameter defining the equations for x and y Θ TRUTH produces no action Θ BAR the x axis range is set from 0 to n 1 where n is the num...

Страница 719: ...unction DRAX draws the axes in the current plot if any is visible DRAW 6 The function DRAW draws the plot defined in PPAR The PTYPE menu under PLOT 1 The PTYPE menu lists the name of all two dimensional plot types pre programmed in the calculator The menu contains the following menu keys These keys correspond to the plot types Function Conic Polar Parametric Truth and Diff Eq presented earlier Pre...

Страница 720: ...This information indicates that X is the independent variable Indep Y is the dependent variable Depnd the x axis range goes from 6 5 to 6 5 Xrng the y axis range goes from 3 1 to 3 2 Yrng The last piece of information in the screen the value of Res resolution determines the interval of the independent variable used for generating the plot The soft menu key labels included in the PPAR 2 menu repres...

Страница 721: ...ications for the DEPND variable are the same as those for the INDEP variable XRNG c and YRNG d The command XRNG specifies the plotting range for the x axis while the command YRNG specifies the plotting range for the y axis The input for any of these commands is two numbers representing the minimum and maximum values of x or y The values of the x and y axis ranges are stored as the ordered pairs xm...

Страница 722: ... factor xfactor the command SCALEW multiplies the horizontal scale by that factor The W in SCALEW stands for width The execution of SCALEW changes the values of xmin and xmax in PPAR SCALEH j Given a factor yfactor the command SCALEH multiplies the vertical scale by that factor The H in SCALEH stands for height The execution of SCALEW changes the values of ymin and ymax in PPAR ATICK l The command...

Страница 723: ...only an ordered pair is given as input to AXES only the axes origin is altered The argument to the command AXES whether an ordered pair or a list of values is stored as the fifth parameter in PPAR To return to the PLOT menu press PLOT Press L to reach the second menu of the PLOT menu set RESET f This button will reset the plot parameters to default values The 3D menu within PLOT 7 The 3D menu cont...

Страница 724: ... VPAR in the 3D menu you will get the following functions Press L to move to the next menu Next we describe the meaning of these functions INFO S and VPAR W When you press INFO S you get the information shown in the left hand side screen shot above The ranges in Xvol Yvol and Zvol describe the extent of the parallelepiped in space where the graph will be generated Xrng and Yrng describe the range ...

Страница 725: ...on EYEPT takes as input real values x y and z representing the location of the viewpoint for a three dimensional graph The viewpoint is a point in space from which the three dimensional graph is observed Changing the viewpoint will produce different views of the graph The figure below illustrates the idea of the viewpoint with respect to the actual graphic space and its projection in the plane of ...

Страница 726: ...lots related to statistical analysis Within this menu we find the following menus The diagram below shows the branching of the STAT menu within PLOT The numbers and letters accompanying each function or menu are used for reference in the descriptions that follow the figure ...

Страница 727: ... Σ E add or remove data rows from the matrix ΣDAT CLΣ F clears the ΣDAT G matrix and the soft menu key labeled ΣDAT is just used as a reference for interactive applications More details on the use of these functions are presented in a later chapter on statistical applications Press STAT to return to the STAT menu The ΣPAR menu within STAT III The ΣPAR menu provides the following functions INFO M a...

Страница 728: ...se functions correspond to Linear Fit Logarithmic Fit Exponential Fit Power Fit or Best Fit Data fitting is described in more detail in a later chapter Press PAR to return to the ΣPAR menu ΣPAR K ΣPAR is just a reference to the variable ΣPAR for interactive use RESET L This function resets the contents of ΣPAR to its default values Press L STAT to return to the STAT menu Press PLOT to return to th...

Страница 729: ...s Following we describe the general format for the variables necessary to produce the different types of plots available in the calculator Two dimensional graphics The two dimensional graphics generated by functions namely Function Conic Parametric Polar Truth and Differential Equation use PPAR with the format xmin ymin xmax ymax indep res axes ptype depend The two dimensional graphics generated f...

Страница 730: ...arameters shown above The variable EQ All plots except those based on ΣDAT also require that you define the function or functions to be plotted by storing the expressions or references to those functions in the variable EQ In summary to produce a plot in a program you need to load EQ if required Then load PPAR PPAR and ΣPAR or PPAR and VPAR Finally use the name of the proper plot type FUNCTION CON...

Страница 731: ...t PARAMETRIC as the plot type SIN t i SIN 2 t Define complex function X iY EQ Store complex function into EQ PPAR Show plot parameters t 0 6 29 INDEP Define t as the indep variable y DEPND Define Y as the dependent variable 2 2 2 2 XRNG Define 2 2 2 2 as the x range 1 1 1 1 YRNG L Define 1 1 1 1 as the y range 0 0 4 2 X t Y t Axes definition list AXES Define axes center ticks labels L PLOT Return ...

Страница 732: ...2 Store function to plot in variable EQ using the proper format e g X t iY t for PARAMETRIC 3 Enter name and range if necessary of independent and dependent variables 4 Enter axes specifications as a list center atick x label y label 5 Use ERASE DRAX LABEL DRAW to produce a fully labeled graph with axes This same approach can be used to produce plots with a program except that in a program you nee...

Страница 733: ... To run it press J if needed then press PLOT1 Example 2 A parametric plot Enter the following program Start program RAD PPAR EQ PURGE Change to radians purge vars SIN t i SIN 2 t STEQ Store X t iY t into EQ t 0 6 29 INDEP Set indep variable to r with range Y DEPND Set dependent variable to Y PARAMETRIC Select PARAMETRIC as the plot type 0 0 5 5 X t Y t AXES Set axes information 2 2 2 2 XRNG Set x ...

Страница 734: ...mmands in programs They just scratch the surface of programming applications of plots I invite the reader to try their own exercises on programming plots Drawing commands for use in programming You can draw figures in the graphics window directly from a program by using commands such as those contained in the PICT menu accessible by L PICT The functions available in this menu are the following Pre...

Страница 735: ...current graph can be thought of as a two dimensional graph with a minimum size of 131 pixels wide by 64 pixels high The maximum width of PICT is 2048 pixels with no restriction on the maximum height A pixel is each one of the dots in the calculator s screen that can be turned on dark or off clear to produce text or graphs The calculator screen has 131 pixels by 64 pixels i e the minimum size for P...

Страница 736: ...ordinates in the input ARC This command is used to draw an arc ARC takes as input the following objects Θ Coordinates of the center of the arc as x y in user coordinates or n m in pixels Θ Radius of arc as r user coordinates or k pixels Θ Initial angle θ1 and final angle θ2 PIX PIXON and PIXOFF These functions take as input the coordinates of point in user coordinates x y or in pixels n m ...

Страница 737: ... C converts pixel coordinates n m to user unit coordinates x y C PX The function C PX converts user unit coordinates x y to pixel coordinates n m Programming examples using drawing functions In this section we use the commands described above to produce graphics with programs Program listing are provided in the attached diskette or CD ROM Example 1 A program that uses drawing commands The followin...

Страница 738: ...r cross section is surveyed and a series of points representing coordinates x and y with respect to an arbitrary set of coordinates axes These points can be plotted and a sketch of the cross section produced for a given water surface elevation The figure below illustrate the terms presented in this paragraph The program available in the diskette or CD ROM that comes with your calculator utilizes f...

Страница 739: ...names such as XYD1 X Y Data set 1 and XYD2 X Y Data set 2 To run the program place one of the data sets in the stack e g J XYD1 then type in a water surface elevation say 4 0 and press XSECT The calculator will show an sketch of the cross section with the corresponding water surface To exit the graph display press Try the following examples XYD1 2 XSECT XYD1 3 XSECT XYD1 4 XSECT XYD1 6 XSECT Pleas...

Страница 740: ... 5 10 5 3 4 11 0 5 0 Note The program FRAME as originally programmed see diskette or CD ROM does not maintain the proper scaling of the graph If you want to maintain proper scaling replace FRAME with the following program STOΣ MINΣ MAXΣ 2 COL DUP COL DROP AXL ABS AXL 20 DUP NEG SWAP 2 COL ROW DROP SWAP yR xR 131 DUP R B SWAP yR OBJ DROP xR OBJ DROP FLOOR R B PDIM yR OBJ DROP YRNG xR OBJ DROP XRNG ...

Страница 741: ...nimation by using the Y Slice plot type Suppose that you want to animate the traveling wave f X Y 2 5 sin X Y We can treat the X as time in the animation producing plots of f X Y vs Y for different values of X To produce this graph use the following Θ ô simultaneously Select Y Slice for TYPE 2 5 SIN X Y for EQ X for INDEP Press L OK Θ ò simultaneously in RPN mode Use the following values Θ Press E...

Страница 742: ...ts to radians 131 R B 64 R B PDIM Set PICT to 131 64 pixels 0 100 XRNG 0 100 YRNG Set x and y ranges to 0 100 1 11 FOR j Start loop with j 1 11 ERASE Erase current PICT 50 50 5 j 1 NUM Centers of circles 50 50 0 2 π NUM ARC Draw circle center r 5 j 1 PICT RCL Place current PICT on stack NEXT End FOR NEXT loop 11 ANIMATE Animate End program Store this program in a variable called PANIM Plot ANIMati...

Страница 743: ...he following program Start program WLIST Place list WLIST in stack OBJ Decompose list stack level 1 11 ANIMATE Start animation End program Save this program in a variable called RANIM Re ANIMate To run it press RANIM The following program will animate the graphics in WLIST forward and backwards Start program WLIST DUP Place list WLIST in stack make extra copy REVLIST Reverse order concatenate 2 li...

Страница 744: ... in which the five functions will be plotted quickly one after the other To stop the animation press More information on the ANIMATE function The ANIMATE function as used in the two previous examples utilized as input the graphics to be animated and their number You can use additional information to produce the animation such as the time interval between graphics and the number of repetitions of t...

Страница 745: ...aph contained in level 1 is shown in the calculator s graphics display Press CANCL to return to normal calculator display The graph in level 1 is still not in GROB format although it is by definition a graphics object To convert a graph in the stack into a GROB use 3 L GROB GROB Now we have the following information in level 1 The first part of the description is similar to what we had originally ...

Страница 746: ...cessible through L GROB GROB contains the following functions Press L to move to the next menu GROB Of these functions we have already used SUB REPL from the graphics EDIT menu ANIMATE ANIMA and GROB PRG is simply a way to return to the programming menu While using GROB in the two previous examples you may have noticed that I used a 3 while converting the graph into a GROB while I used a 1 when I ...

Страница 747: ...b1 and grob2 GXOR The function GXOR Graphics XOR performs the same operation as GOR but using XOR to determine the final status of pixels in the overlapping area between graphic objects grob1 and grob2 LCD Takes a specified GROB and displays it in the calculator s display starting at the upper left corner LCD Copies the contents of the stack and menu display into a 131 x 64 pixels GROB SIZE The fu...

Страница 748: ...T RCL Place contents of PICT on stack SINE FUNCTION Place graph label string in stack 1 GROB Convert string into a small GROB 6 1 5 SWAP Coordinates to place label GROB GOR Combine PICT with the label GROB PICT STO Save combined GROB into PICT PVIEW Bring PICT to the stack End program Save the program under the name GRPR GROB PRogram Press GRPR to run the program The output will look like this A p...

Страница 749: ...s τ Locate the points A σxx τxy and B σyy τxy and draw the segment AB The point C where the segment AB crosses the σn axis will be the center of the circle Notice that the coordinates of point C are σyy σxy 0 When constructing the circle by hand you can use a compass to trace the circle since you know the location of the center C and of two points A and B Let the segment AC represent the x axis in...

Страница 750: ...ween segments AC and D C measures 2φn The stress condition for which the shear stress τ xy is a maximum is given by segment F G Under such conditions both normal stresses σ xx σ yy are equal The angle corresponding to this rotation is φs The angle between segment AC and segment F C in the figure represents 2φs Modular programming To develop the program that will plot Mohr s circle given a state of...

Страница 751: ...rcle construct Θ PCIRC Uses σc r and φn as input draw s Mohr s circle by producing a PARAMETRIC plot Θ DDIAM Uses σL as input draws the segment AB see Mohr s circle figure above joining the input data points in the Mohr s circle Θ σLBL Uses σL as input places labels to identify points A and B with labels σx and σy Θ σAXS Places the labels σ and τ in the x and y axes respectively Θ PTTL Places the ...

Страница 752: ...5 00E1 Press the right arrow key to increment the value of φ and see the corresponding value of σ xx τ xy For example for φ 45o we have the values σ xx τ xy 1 00E2 2 50E1 100 25 The value of σ yy will be found at an angle 90o ahead i e where φ 45 90 135o Press the key until reaching that value of φ we find σ yy τ xy 1 00E 10 2 5E1 0 25 To find the principal normal values press š until the cursor r...

Страница 753: ...am PRNST PRiNcipal STresses INDAT Enter data as in program MOHRCIRC CC r Calculate σc r and fn as in MOHRCIRC φn TAG Tag angle for principal stresses 3 ROLLD Move tagged angle to level 3 R C DUP Convert σc and r to σc r duplicate C R σPx TAG Calculate principal stress σPx tag it SWAP C R σPy TAG Swap calculate stress σPy tag it End program PRNST To run the program use J PRNST Start program PRNST 2...

Страница 754: ... the first two variables in the menu as we expected A second example of Mohr s circle calculations Determine the principal stresses for the stress state defined by σxx 12 5 kPa σyy 6 25 kPa and τxy 5 0 kPa Draw Mohr s circle and determine from the figure the values of σ xx σ yy and τ xy if the angle φ 35o To determine the principal stresses use the program PRNST as follows J PRNST Start program PR...

Страница 755: ...are 1 63E0 1 05E1 i e at φ 35o σ xx 1 63 kPa and σ yy 10 5kPa An input form for the Mohr s circle program For a fancier way to input data we can replace sub program INDAT with the following program that activates an input form MOHR S CIRCLE σx Normal stress in x 0 σy Normal stress in y 0 τxy Shear stress 0 1 1 1 1 1 1 INFORM DROP With this program substitution running MOHRC will produce an input f...

Страница 756: ...Page 22 41 Since program INDAT is used also for program PRNST PRiNcipal STresses running that particular program will now use an input form for example The result after pressing OK is the following ...

Страница 757: ...g related functions in the TYPE sub menu The TYPE sub menu is accessible through the PRG programming menu i e The functions provided in the TYPE sub menu are also shown below Among the functions in the TYPE menu that are useful for manipulating strings we have OBJ Converts string to the object it represents STR Converts an object to its string representation TAG Tags a quantity DTAG Removes the ta...

Страница 758: ...gs is a practical way to create output in programs For example concatenating YOU ARE AGE YEAR OLD creates the string YOU ARE 25 YEAR OLD where 25 is stored in the variable called AGE The CHARS menu The CHARS sub menu is accessible through the PRG programming menu i e The functions provided by the CHARS sub menu are the following ...

Страница 759: ...cter in a string SUB extract sub string given starting and ending position REPL replace characters in a string with a sub string starting at given position SREPL replaces a sub string by another sub string in a string To see those effects on action try the following exercises Store the string MY NAME IS CYRILLE into variable S1 We ll use this string to show examples of the functions in the CHARS m...

Страница 760: ...ed at the lower left side of the screen indicating the use of m On the other hand m shows the keystroke combination α M or m Greek characters such as σ will show the code α S or s Some characters like ρ do not have a keystroke sequence associated with them Therefore the only way to obtain such characters is through the character list by highlighting the desired character and pressing ECHO1 or ECHO...

Страница 761: ...______________________________________________________________ Number Type Example _________________________________________________________________ 0 Real Number 1 23E 5 1 Complex Number 1 2 2 3 2 String Hello world 3 Real Array 1 2 3 4 4 Complex Array 1 2 3 4 5 6 7 8 5 List 3 1 PI 6 Global Name X 7 Local Name y 8 Program a a 2 9 Algebraic object a 2 b 2 10 Binary Integer 2F1E h 11 Graphic Object...

Страница 762: ...ternal Object External 30 External Object External ____________________________________________________________________ Function TYPE This function available in the PRG TYPE sub menu or through the command catalog is used to determine the type of an object The function argument is the object of interest The function returns the object type as indicated by the numbers specified above Function VTYPE...

Страница 763: ...tem flag setting is that of system flags 60 and 61 that relate to the constant library CONLIB see Chapter 3 These flags operate in the following manner Θ user flag 60 clear default SI units set ENGL units Θ user flag 61 clear default use units set value only Functions for setting and changing flags These functions can be used to set un set or check on the status of user flags or system flags When ...

Страница 764: ...rns 1 if flag is clear not set 0 if flag is set FS C Tests flag as FS does then clears it FC C Tests flag as FC does then clears it STOF Stores new system flag settings RCLF Recalls existing flag settings RESET Resets current field values could be used to reset a flag User flags For programming purposes flags 1 through 256 are available to the user They have no meaning to the calculator operation ...

Страница 765: ...r set an alarm The input form looks like in the following figure The Message input field allows you to enter a character string identifying the alarm The Time field lets you enter the time for activating the alarm The Date field is used to set the date for the alarm or for the first time of activation if repetition is required For example you could set the following alarm The left hand side figure...

Страница 766: ...ed alarm providing an alarm set input form NEW For programming a new alarm PURG For deleting an alarm OK Returns to normal display Setting time and date Option 3 Set time date provides the following input form that let s the user set the current time and date Details were provided in Chapter 1 TIME Tools Option 4 Tools provides a number of functions useful for clock operation and calculations with...

Страница 767: ...te DDAYS x y Returns number of days between dates x and y HMS Converts time from decimal to HH MMSS HMS Converts time from HH MMSS to decimal HMS Add two times in HH MMSS format HMS Subtract two times in HH MMSS format TSTR time date Converts time date to string format CLKADJ x Adds x ticks to system time 1 tick 1 8192 sec Functions DATE TIME CLKADJ are used to adjust date and time There are no ex...

Страница 768: ...larm x into system alarm list RCLALARM x Recalls specified alarm x from system alarm list DELALARM x Deletes alarm x from system alarm list FINDALARM x Returns first alarm due after specified time The argument x in function STOALARM is a list containing a date reference mm ddyyy time of day in 24 hr format hh mm a string containing the text of the alarm and the number of repetitions of the alarm F...

Страница 769: ...ata storage user s memory Users do not have access to the system memory component To see the way in which the user s memory is partitioned use the FILES function A possible result is shown below This screen indicates the existence of three memory ports besides the memory corresponding to the HOME directory see Chapter 2 in this guide The memory ports available are Θ Port 0 labeled IRAM Θ Port 1 la...

Страница 770: ... Flash ROM segment Port 2 can store up to 1085 KB of data A fourth port Port 3 is available for use with an SD flash memory card An example is shown below The port appears in File Manager only when an SD card is inserted The HOME directory When using the calculator you may be creating variables to store intermediate and final results Some calculator operations such as graphics and statistical oper...

Страница 771: ... given directory sub directory or port is selected press OK to see the contents of the selected object Another way to access port memory is by using the LIB menu á associated with the 2 key This action produces the following screen If you have any library active in your calculator it will be shown in this screen One such library is the HP49D demo library shown in the screen above Pressing the corr...

Страница 772: ...ory When you create a backup object in port memory the calculator obtains a cyclic redundancy check CRC or checksum value based on the binary data contained in the object This value is stored with the backup object and is used by the calculator to monitor the integrity of the backup object When you restore a backup object into the HOME directory the calculator recalculates the CRC value and compar...

Страница 773: ...ic mode enter the command ARCHIVE Port_Number Backup_Name Here Port_Number is 0 1 2 or 3 if an SD memory card is available see below and Backup_Name is the name of the backup object that will store the contents of HOME The container is entered by using the keystroke sequence ê For example to back up HOME into HOME1 in Port 1 use To back up the HOME directory in RPN mode use the command Port_Number...

Страница 774: ...a variable in the HOME directory see Chapter 2 Θ Use the PURGE command as follows In algebraic mode use PURGE Port_Number Backup_Name In RPN mode use Port_Number Backup_Name PURGE To restore a backup object Θ Use the File Manager to copy the backup object from Port memory to the HOME directory Θ When a backup object is restored the calculator performs an integrity check on the restored object by c...

Страница 775: ...t s Port_Number Backup_Name EVAL To recall a backup object to the command line enter Port_Number Backup_Name RCL Using SD cards The calculator has a memory card port into which you can insert an SD flash card for backing up calculator objects or for downloading objects from other sources The SD card in the calculator will appear as port number 3 Inserting and removing an SD card The SD slot is loc...

Страница 776: ...teries Note formatting an SD card deletes all the data that is currently on it 1 Insert the SD card into the card slot as explained in the previous section 2 Hold down the key and then press the D key Release the D key and then release the key The system menu is displayed with several choices 3 Press 0 for FORMAT The formatting process begins 4 When the formatting is finished the HP 50g displays t...

Страница 777: ... characters in the suffix The type of each object will be displayed unless it is a PC object or an object of unknown type In these cases its type is listed as String In addition to using the File Manager operations you can use functions STO and RCL to store objects on and recall objects from the SD card as shown below You can also use the PURGE command to erase backup objects in the SD card Long n...

Страница 778: ...ether specify the position of the variable within a directory tree However some variables stored within a backup object cannot be recalled by specifying a path In this case the full backup object e g a directory will have to be recalled and the individual variables then accessed in the screen Note that in the case of objects with long files names you can specify the full name of the object or its ...

Страница 779: ...URGE command Purging all objects on the SD card by reformatting You can purge all objects from the SD card by reformatting it When an SD card is inserted FORMA appears an additional menu item in File Manager Selecting this option reformats the entire card a process which also deletes every object on the card Specifying a directory on an SD card You can store recall evaluate and purge objects that ...

Страница 780: ...provide all the functionality of the Equation Library Libraries can be downloaded into the calculator as a regular variable and then installed and attached to the HOME directory Installing and attaching a library To install a library list the library contents in the stack use variable soft menu key or function RCL and store it into port 0 or 1 For example to install a library variable into a port ...

Страница 781: ...you are thinking of deleting these libraries but there is some likelihood that you will need to use the Equation Library in the future you should copy them to a PC using the HP 48 50 Calculator Connectivity Kit before deleting them on the calculator You will then be able to re install the libraries later when you need to use the Equation Library Creating libraries A library can be written in Assem...

Страница 782: ...Page 26 14 will indicate when this battery needs replacement The diagram below shows the location of the backup battery in the top compartment at the back of the calculator ...

Страница 783: ...n port 2 constitute the Equation Library and they can be deleted just like any user created library However if you are thinking of deleting these libraries but there is some likelihood that you will need to use the Equation Library in the future you should copy them to a PC using the HP 48 49 Calculator Connectivity Kit before deleting them on the calculator You will then be able to re install the...

Страница 784: ...n you press SOLV in the Equation Library the application does the following The set of equations is stored in the appropriate variable EQ for one equation EQ and Mpar for more than one equation Mpar is a reserved variable used by the Multiple Equation Solver Note because EQ and Mpar are variables you can have a different EQ and Mpar for each directory in memory Each variable is created and set to ...

Страница 785: ... if required In addition each solver has special menu keys which are described in the following table You can tell which solver is started by looking at the special menu labels Actions for Solver Menu Keys Operation SOLVE application Multiple Equation Solver Store value X X Solve for value X X X Recall value X X X Evaluate equation EXPR Next equation if applicable NXEQ Undefine all ALL Solve for a...

Страница 786: ...lation form The display form gives the equation in its basic form the form you would see in books The calculation form includes computational refinements If an equation has a computational form an appears in the upper left corner of the equation display Operations for viewing Equations and Pictures Key Action Example EQN NXEQ Shows the display form of current or next equation in EquationWriter for...

Страница 787: ... has a picture To see the picture press PIC While the picture is displayed you can Key Action L Toggles between the catalog of descriptions and the catalog of units SI ENGL Makes SI or English units active unless this conflicts with the units already defined for an existing global variable Purge existing variables or enter the specific units to eliminate conflicts UNITS Toggles between units used ...

Страница 788: ...ion set the list of variables and additional information in Mpar Mpar is then used to set up the Solver menu for the current equation set Note that although you can view and edit EQ directly like any other variable Mpar can only be edited indirectly by executing commands that modify it as it is structured as library data dedicated to the Multiple Equation Solver application The following table sum...

Страница 789: ...k that variables have proper states when you supply guesses and find solutions Notice that marks the variables that were used in the last solution their values are compatible with each other Other variables may not have compatible values because they aren t involved in the solution Undefined all ALL Makes all variables not user defined but does not specify their values Solve for all ALL Creates va...

Страница 790: ...ble you want You should choose your equations to allow likely unknown variables to occur individually in equations You must avoid having two or more unknown variables in all equations You can also specify equations in an order that s best for your problems Label Meaning X0 Value x0 is not defined by you and not used in the last solution It can change with the next solution X0 ëëëë Value x0 is not ...

Страница 791: ...their variables won t necessarily be detected by the Multiple Equation Solver Σ QUOTE APPLY TVROOT and CONST The list of equations in EQ may contain menu definitions but those definitions are ignored by MINIT when it creates Mpar However you can reorder the menu labels using MITM described below To create a set of equations for the Multiple Equation Solver 1 Enter each equation in the set onto the...

Страница 792: ...hat s unknown not user defined and not found by the solver during this solution It then uses the root finder to find that value It continues eliminating unknown variables until it solves for the variable you specified or until it can t solve for any more variables Each time the Multiple Equation Solver starts solving for a variable only the variables with black menu labels are known During the sol...

Страница 793: ...ts you want No units If you are not using variables your implied units may not be compatible among your variables or with the implied units of constants or functions The current angle mode sets the implied units for angles Multiple roots An equation may have multiple roots and the solver may have found an inappropriate one Supple a guess for the variable to focus the search in the appropriate rang...

Страница 794: ...k in the label so it is not compatible with the variables that were involved Wrong direction The initial value of a variable may be leading the root finder in the wrong direction Supply a guess in the opposite direction from a critical value If negative values are valid try one ...

Страница 795: ...toggle the check mark on and off Θ Press the CANCL soft menu key to close an input form and return to the stack display You can also press the key or the key to close the input form Example Using input forms in the NUM SLV menu Before discussing these items in detail we will present some of the characteristics of the input forms by using input forms from the financial calculation application in th...

Страница 796: ...r n 10 8 5 OK Enter I YR 8 5 10000 OK Enter PV 10000 1000 OK Enter FV 1000 š SOLVE Select and solve for PMT The resulting screen is In this input form you will notice the following soft menu key labels EDIT Press to edit highlighted field AMOR Amortization menu option specific to this application SOLVE Press to solve for highlighted field Pressing L we see the following soft menu key labels RESET ...

Страница 797: ...ault value If instead you select Rest all all the fields will be reset to their default values typically 0 At this point you can accept your choice press OK or cancel the operation press CANCL Press CANCL in this instance Press CALC to access the stack The resulting screen is the following At this point you have access to the stack and the value last highlighted in the input form is provided for y...

Страница 798: ...owing specification This indicates that the value in the PMT field must be a real number Press OK to return to the input form and press L to recover the first menu Next press the key or the key to return to the stack In this instance the following values will be shown The top result is the value that was solved for PMT in the first part of the exercise The second value is the calculation we made t...

Страница 799: ...numbering of its rows and columns The figure shows 10 rows of keys combined with 3 5 or 6 columns Row 1 has 6 keys rows 2 and 3 have 3 keys each and rows 4 through 10 have 5 keys each There are 4 arrow keys located on the right hand side of the keyboard in the space occupied by rows 2 and 3 Each key has three four or ...

Страница 800: ... in the figure below To operate this main key functions simply press the corresponding key We will refer to the keys by the row and column where they are located in the sketch above thus key 10 1 is the ON key Main key functions in the calculator s keyboard ...

Страница 801: ...rectory The HIST function allows you access to the algebraic mode history i e the collection of recent command entries in that mode The EVAL key is used to evaluate algebraic and numeric expressions the apostrophe key is used to enter a set of apostrophes for algebraic expressions The SYMB activates the symbolic operations menu The delete key ƒ is used to delete characters in a line The yx key cal...

Страница 802: ...f the other keys to activate the alternative functions shown in the keyboard For example the P key key 4 4 has the following six functions associated with it P Main function to activate the SYMBolic menu Left shift function to activate the MTH Math menu N Right shift function to activate the CATalog function p ALPHA function to enter the upper case letter P p ALPHA Left Shift function to enter the...

Страница 803: ...ons in the calculator s Algebraic mode of operation press the left shift key first and then any of the keys in Row 1 When using these functions in the calculator s RPN mode you need to press the left shift key simultaneously with the key in Row 1 of your choice Function Y is used to enter functions of the form y f x for plotting function WIN is used to set parameters of the plot window function GR...

Страница 804: ...on activates the Matrix Writer Left shift functions of the calculator s keyboard The CMD function shows the most recent commands The PRG function activates the programming menus The MTRW function activates the Matrix Writer The MTH function activates a menu of mathematical function The DEL key is used to delete variables ...

Страница 805: ...ions The CALC function activates a menu of calculus functions The MATRICES function activates a menu for creating and manipulation of matrices The CONVERT function activates a menu for conversion of units and other expressions The ARITH function activates a menu of arithmetic functions The DEF key is used to define a simple function as a variable in the calculator menu The CONT key is used to cont...

Страница 806: ... menus associated with the different calculator keys when the right shift key is activated The functions BEGIN END COPY CUT and PASTE are used for editing purposes The UNDO key is used to undo the last calculator operation The CHARS function activates the special characters menu The EQW function is used to start the Equation Writer ...

Страница 807: ...vates the statistical operations menu The UNITS function activates the menu for units of measurement The CMPLX function activates the complex number functions menu The LIB function activates the library functions The BASE function activates the numeric base conversion menu The OFF key turns the calculator off the NUM key produces a numeric or floating point value of an expression The key enters a ...

Страница 808: ...the space SPC are the same as the main functions of these keys The function produces an asterisk when combined with the times key i e Alpha functions of the calculator s keyboard Alpha left shift characters The following sketch shows the characters associated with the different calculator keys when the ALPHA is combined with the left shift key ...

Страница 809: ... of the English alphabet A through Z The numbers mathematical symbols decimal point and the space SPC are the same as the main functions of these keys The ENTER and CONT keys also work as their main function even when the combination is used Alpha functions of the calculator s keyboard ...

Страница 810: ...s combined with the right shift key Alpha functions of the calculator s keyboard Notice that the combination is used mainly to enter a number of special characters from into the calculator stack The CLEAR OFF comma key enters and OFF keys also work as their main function even when the combination is used The special characters generated by the ...

Страница 811: ...Page B 13 combination include Greek letters α β Δ δ ε ρ μ λ σ θ τ ω and Π other characters generated by the combination are __ and ...

Страница 812: ...s FLAGS Provides menus for manipulating calculator flags CHOOS Lets the user chose options in the different fields in the form CAS Provides an input form to change CAS settings DISP Provides an input form to change display settings CANCL Closes this input form and returns to normal display OK Use this key to accept settings Pressing the L key shows the remaining options in the CALCULATOR MODES inp...

Страница 813: ...d options will show no check mark in the underline preceding the option of interest e g the _Numeric _Approx _Complex _Verbose _Step Step _Incr Pow options above Θ After having selected and unselected all the options that you want in the CAS MODES input form press the OK soft menu key This will take you back to the CALCULATOR MODES input form To return to normal calculator display at this point pr...

Страница 814: ... variable VX in your programs or equations so as to not get it confused with the CAS VX If you need to refer to the x component of velocity for example you can use vx or Vx Selecting the modulus The Modulo option of the CAS MODES input box represents a number default value 13 used in modular arithmetic More details about modular arithmetic are presented elsewhere Numeric vs symbolic CAS mode When ...

Страница 815: ...ic expressions whenever possible The following screen shows a couple of symbolic expressions entered with an active exact mode in Algebraic operating mode In Algebraic mode the object entered by the user is shown in the left hand side of the screen followed immediately by a result in the right hand side of the screen The results shown above show the symbolic expressions for ln 2 i e the natural lo...

Страница 816: ...t shift key and pressing the ENTER key simultaneously i e hold Real numbers vs integer numbers CAS operations utilize integer numbers in order to keep full precision in the calculations Real numbers are stored in the form of a mantissa and an exponent and have limited precision In APPROX mode however whenever you enter an integer number it is automatically transformed into a real number as illustr...

Страница 817: ...operation results in a complex number then the result will be shown in the form a bi or in the form of an ordered pair a b On the other hand if the _Complex CAS option is unset i e the Real CAS option is active and an operation results in a complex number you will be asked to switch to Complex mode If you decline the calculator will report an error Please notice that in COMPLEX mode the CAS is abl...

Страница 818: ...Verbose CAS option is selected certain calculus applications are provided with comment lines in the main display If the _Verbose CAS option is not selected then those calculus applications will show no comment lines The comment lines will appear momentarily in the top lines of the display while the operation is being calculated Step by step CAS mode When the _Step step CAS option is selected certa...

Страница 819: ...n A X3 5X2 3X 2 and B X 2 These polynomials are represented in the screen by lists of their coefficients For example the expression A 1 5 3 2 represents the polynomial A X3 5X2 3X 2 B 1 2 represents the polynomial B X 2 Q 1 represents the polynomial Q X and R 3 3 2 represents the polynomial R 3X2 3X 2 At this point press for example the key Continue pressing the key to produce additional steps Thu...

Страница 820: ...variable An example is shown next in Algebraic mode In the first case the polynomial X 3 5 is expanded in increasing order of the powers of X while in the second case the polynomial shows decreasing order of the powers of X The keystrokes in both cases are the following Üx 3 Q5 In the first case the _Incr pow option was selected while in the second it was not selected The same example in RPN notat...

Страница 821: ...ot selected non rational expressions will not be automatically simplified Using the CAS HELP facility Turn on the calculator and press the I key to activate the TOOL menu Next press the Bsoft menu key followed by the key the key in the lowest right corner of the keyboard to activate the HELP facility The display will look as follows At this point you will be provided with a list of all CAS command...

Страница 822: ... CAS commands HELPB 10 times Then press the OK F key to obtain information about the command ATAN2S The help facility indicates that the command or function ATAN2S replaces the value of atan x the arc tangent of a value x by its equivalent in terms of the function asin arcsine i e The fourth and fifth lines in the display provide an example of application of the function ATAN2S Line four namely AT...

Страница 823: ...display is the following There are now four lines of the display occupied with output The first two lines from the top correspond to the first exercise with the HELP facility in which we cancel the request for help The third line from the top shows the most recent call to the HELP facility while the last line shows the ECHO of the example command To activate the command press the key The result is...

Страница 824: ... CAS Computer Algebraic System There is a large number of other functions and commands that were originally developed for the HP 48G series calculators that are not included in the help facility Good references for those commands are the HP 48G Series User s Guide HP Part No 00048 90126 and the HP 48G Series Advanced User s Reference Manual HP Part No 00048 90136 both published by Hewlett Packard ...

Страница 825: ... limited to loss of data or data being rendered inaccurate or losses sustained by you or third parties or a failure of the CAS Software to operate with any other programs even if such holder or other party has been advised of the possibility of such damages If required by applicable law the maximum amount payable for damages by the copyright holder shall not exceed the royalty amount paid by Hewle...

Страница 826: ...ted with the EVAL key The result is the following screen By using the arrow keys š we can navigate through the collection of characters For example moving downwards in the screen produces more characters in the display Moving farther down we see these characters There will be one character highlighted at all times The lower line in the display will show the short cut for the highlighted character ...

Страница 827: ...quation writer EQW but the cursor remains in the character set screen to allow the user to select additional characters i e echoes a string of characters to the stack To exit the character set screen press For example suppose you have to type the expression λ2 2μ 5 Here is a suggested approach using the stack in either Algebraic or RPN mode Use the keystrokes to get to the characters screen Next u...

Страница 828: ...tau u ω omega v Δ upper case delta c Π upper case pi p Other characters tilde 1 factorial 2 question mark 3 backward slash 5 angle symbol 6 at Some characters commonly used that do not have simple keystroke shortcuts are x x bar γ gamma η eta Ω upper case omega These characters can be echoed from the CHARS screen ...

Страница 829: ... Execution from left to right means that if two operations of the same hierarchy say two multiplications exist in an expression the first multiplication to the left will be executed before the second and so on Consider for example the expression shown below in the equation writer The insertion cursor at this point is located to the right of the 2 in the argument of the SIN function in the denomina...

Страница 830: ...on It is important to point out that the multiplication in Step A5 includes the first term y 3 x 5 with a second term x2 4 which is already calculated To see the steps in calculating these second term press the down arrow key continuously until the clear editing cursor is triggered around the y once more Then press the right arrow key until these cursor is over the x in the second term in the nume...

Страница 831: ...function in the denominator Press the down arrow key continuously until the clear editing cursor is triggered around the y once more Then press the right arrow key until these cursor is over the 4 in the denominator Then press the upper arrow key to select this 4 The steps in the evaluation of the expression starting from this point are shown below Step C1 Step C2 ...

Страница 832: ... Step A6 The expression tree for the expression presented above is shown next The steps in the evaluation of the three terms A1 through A6 B1 through B5 and C1 through C5 are shown next to the circle containing numbers variables or operators ...

Страница 833: ...ns The different applications are described next Plot functions Selecting option 1 Plot functions in the APPS will produce the following menu list of graph related options The six options shown are equivalent to the keystroke sequences listed below Equation entry ñ Plot window ò Graph display ó Plot setup ô Table setup õ Table display ö These applications are presented in detail in Chapter 12 ...

Страница 834: ...ter Print Print selected object from calculator Transfer Transfer data to other device Start Server Calculator set as a server for communication with computers You can connect to another calculator or to a PC via infrared or via a cable A USB cable is provided with the calculator for a USB connection You can also use a serial cable to connect to the RS232 port on the calculator This cable is avail...

Страница 835: ...erical solver menu This operation is equivalent to the keystroke sequence Ï The numerical solver menu is presented in detail in Chapters 6 and 7 Time date Selecting option 5 Time date in the APPS menu produces the time and date menu This operation is equivalent to the keystroke sequence Ó The time and date menu is presented in detail in Chapter 26 ...

Страница 836: ...roke sequence O The equation writer is introduced in detail in Chapter 2 Examples that use the equation writer are available throughout this guide File manager Selecting option 7 File manager in the APPS menu launches the file manager application This operation is equivalent to the keystroke sequence The file manager is introduced in Chapter 2 ...

Страница 837: ...ecting option 9 Text editor in the APPS menu launches the line text editor The text editor can be started in many cases by pressing the down arrow key If the object in the display is an algebraic object pressing will most likely start the Equation Writer The text editor is introduced in Chapter 2 and presented in detail in Appendix L Math menu Selecting option 10 Math menu in the APPS menu produce...

Страница 838: ...AS menu in the APPS menu produces the CAS or SYMBOLIC menu This operation is also available by pressing the Pkey The CAS or SYMBOLIC menu is introduced in Chapter 5 algebraic and arithmetic operations Other functions from the CAS menu are presented in Chapters 4 complex numbers 6 equations solutions 10 matrix creation 11 matrix operation 13 calculus 14 multivariate calculus and 15 vector analysis ...

Страница 839: ... set if you are going to use the Equation Library Note too that the Equation Library will only appear on the APPS menu if the two Equation Library files are stored on the calculator The Equation Library is explained in detail in chapter 27 ...

Страница 840: ... mode H FLAGS CHK Θ In ALG mode CF 95 selects RPN mode Θ In RPN mode 95 SF selects ALG mode Θ A keyboard short cut to toggle between APPROX and EXACT mode is by holding the right shift key and pressing the ENTER key simultaneously i e hold Θ Set clear system flag 105 EXACT vs APPROX CAS mode H FLAGS CHK Θ In ALG mode SF 105 selects APPROX CAS mode CF 105 selects EXACT CAS mode Θ In RPN mode 105 SF...

Страница 841: ... measure o To degrees deg o To radian rad Θ Special characters o Angle symbol 6 o Factorial symbol 2 o Degree symbol o hold 6 Θ Lock unlock alpha keyboard o Lock alpha keyboard upper case o Unlock alpha keyboard upper case o Lock alpha keyboard lower case o Unlock alpha keyboard lower case Θ Greek letters Alpha α a Beta β b DELTA Δ c Delta d d Epsilon ε e Rho ρ f Mu μ m Lambda λ n PI Π p Sigma σ s...

Страница 842: ... not accessible through keyboard In RPN enter menu_number type MENU In ALG mode type MENU menu_number Menu_number is one of the following o STAT soft menu 96 o PLOT soft menu 81 o SOLVE soft menu 74 or use hold 7 o UTILITY soft menu 113 Θ Other menus o MATHS menu maths o MAIN menu main Θ Other keyboard short cuts o hold 7 SOLVE menu menu 74 o hold H PRG MODES menu Chapter 21 o hold Starts text edi...

Страница 843: ...kely the command of interest will not be selected at this point you may overshoot or undershoot it However you can use the vertical keys one stroke at a time to locate the command you want and then press OK Θ If while holding down the down arrow key you overshoot the command of interest you can hold down the up arrow key to move back towards that command Refine the selection with the vertical keys...

Страница 844: ...ha keyboard and use the vertical arrow keys to locate the command if needed Press OK to locate the to activate the command For example to locate the command PROPFRAC you can use one of the following keystroke sequences I L HELP pr OK I L HELP pro OK I L HELP prop OK See Appendix C for more information on the CAS Computer Algebraic System Appendix C includes other examples of application of the CAS...

Страница 845: ... given command if the soft menu key HELP shows up when you highlight that particular command Press this soft menu key to get the CAS help facility entry for the command The first few screens of the catalog are shown below User installed library commands would also appear on the command catalog list using italic font If the library includes a help item then the soft menu key HELP shows up when you ...

Страница 846: ... contains the following sub menus The CMPLX sub menu The CMPLX sub menu contains functions pertinent to operations with complex numbers These functions are described in Chapter 4 The CONSTANTS sub menu The CONSTANTS sub menu provides access to the calculator mathematical constants These are described in Chapter 3 ...

Страница 847: ...scribed in Chapter 3 The INTEGER sub menu The INTEGER sub menu provides functions for manipulating integer numbers and some polynomials These functions are presented in Chapter 5 The MODULAR sub menu The MODULAR sub menu provides functions for modular arithmetic with numbers and polynomials These functions are presented in Chapter 5 ...

Страница 848: ...es relational operators e g etc logical operators e g AND OR etc the IFTE function and the ASSUME and UNASSUME commands Relational and logical operators are presented in Chapter 21 in the context of programming the calculator in User RPL language The IFTE function is introduced in Chapter 3 Functions ASSUME and UNASSUME are presented next using their CAS help facility entries see Appendix C ASSUME...

Страница 849: ...enu This command configures the CAS For CAS configuration information see Appendix C The ALGB sub menu The ALGB sub menu includes the following commands These functions except for 0 MAIN MENU and 11 UNASSIGN are available in the ALG keyboard menu Detailed explanation of these functions can be found in Chapter 5 Function UNASSIGN is described in the following entry from the CAS menu ...

Страница 850: ...ough the CALC DIFF sub menu start with Ö These functions are described in Chapters 13 14 and 15 except for function TRUNC which is described next using its CAS help facility entry The MATHS sub menu The MATHS menu is described in detail in Appendix J The TRIGO sub menu The TRIGO menu contains the following functions ...

Страница 851: ...lable in the CALC SOLVE menu start with Ö The functions are described in Chapters 6 11 and 16 The CMPLX sub menu The CMPLX menu includes the following functions The CMPLX menu is also available in the keyboard ß Some of the functions in CMPLX are also available in the MTH COMPLEX menu start with Complex number functions are presented in Chapter 4 The ARIT sub menu The ARIT menu includes the follow...

Страница 852: ...u is also accessible through the keyboard by using Ð The functions in this menu are presented in Chapter 5 The MATR sub menu The MATR menu contains the following functions These functions are also available through the MATRICES menu in the keyboard Ø The functions are described in Chapters 10 and 11 The REWRITE sub menu The REWRITE menu contains the following functions ...

Страница 853: ...e available through the CONVERT REWRITE menu start with Ú The functions are presented in Chapter 5 except for functions XNUM and XQ which are described next using the corresponding entries in the CAS help facility IL HELP XNUM XQ ...

Страница 854: ...s characters to beginning of word SKIP Skips characters to end of word DEL Delete characters to beginning of word DEL Delete characters to end of word DEL L Delete characters in line INS When selected inserts characters at cursor location If not selected the cursor replaces characters overwrites instead of inserting characters EDIT Edits selection BEG Move to beginning of word END Mark end of sele...

Страница 855: ...y Clip Size is the number of characters in the clipboard Sel Size is the number of characters in the current selection EXEC Execute command selected HALT Stop command execution The line editor also provide the following sub menus SEARCH Search characters or words in the command line It includes the following functions GOTO Move to a desired location in the command line It includes the following fu...

Страница 856: ...d is Find next Finds the next search pattern as defined in Find Replace Selection Replace selection with replacement pattern defined with Replace command Replace Find Next Replace a pattern and search for another occurrence The pattern is defined in Replace Replace All Replace all occurrence of a certain pattern This command asks for confirmation from the user before replacing pattern Fast Replace...

Страница 857: ...ve to a specified position in the command line The input form provided for this command is Labels move to a specified label in the command line The Style sub menu The Style sub menu includes the following styles BOL Bold ITALI Italics UNDE Underline Inverse The command FONT allow the user to select the font for the command editor Examples of the different styles are shown below ...

Страница 858: ...Page L 5 ...

Страница 859: ...8 Cantilever Slope 1 10 4 Simple Slope 1 10 9 Cantilever Moment 1 8 5 Simple Moment 1 8 10 Cantilever Shear 1 6 2 Electricity 42 56 1 Coulomb s Law 1 5 13 Capacitor Charge 1 3 2 Ohm s Law and Power 4 4 14 DC Inductor Voltage 3 8 3 Voltage Divider 1 4 15 RC transient 1 6 4 Current Divider 1 4 16 RL transient 1 6 5 Wire Resistance 1 4 17 Resonant Frequency 4 7 6 Series and Parallel R 2 4 18 Plate Ca...

Страница 860: ... 2 Ideal Gas State Change 1 6 6 Real Gas Law 2 8 3 Isothermal Expansion 2 7 7 Real Gas State Change 1 8 4 Polytropic Processes 2 7 8 Kinetic Theory 4 9 6 Heat Transfer 17 31 1 Heat Capacity 2 6 5 Conduction and 2 Thermal Expansion 2 6 Convection 4 14 3 Conduction 2 7 6 Black Body Radiation 5 9 4 Convection 2 6 7 Magnetism 4 14 1 Straight Wire 1 5 3 B Field in Solenoid 1 4 2 Force Between Wires 1 6...

Страница 861: ...eometry 31 21 1 Circle 5 7 4 Regular Polygon 6 8 2 Ellipse 5 8 5 Circular Ring 4 7 3 Rectangle 5 8 6 Triangle 6 107 12 Solid Geometry 18 12 1 Cone 5 9 3 Parallelepiped 4 9 2 Cylinder 5 9 4 Sphere 4 7 13 Solid State Devices 33 53 1 PN Step Junctions 8 19 3 Bipolar Transistors 8 14 2 NMOS Transistors 10 23 4 JFETs 7 15 14 Stress Analysis 16 28 1 Normal Stress 3 7 3 Stress on an Element 3 7 2 Shear S...

Страница 862: ...u 25 3 AMORT 6 31 AMORTIZATION 6 10 AND 19 5 Angle between vectors 9 15 Angle Measure 1 23 Angle symbol G 2 Angle units 22 27 22 29 22 33 Angular measure G 2 ANIMATE 22 27 Animating graphics 22 26 Animation 22 26 Anti derivatives 13 14 Approximate CAS mode C 4 Approximate vs Exact CAS mode C 4 APPS menu F 1 ARC 22 21 AREA in plots 12 6 Area units 3 19 ARG 4 6 ARITHMETIC menu 5 9 ASIN 3 6 ASINH 3 9...

Страница 863: ...stants 3 16 CALCULATOR MODES input form C 1 Calculator restart G 3 Calculus 13 1 Cancel next repeating alarm G 3 Cartesian representation 4 1 CAS help facility listing H 1 CAS HELP facility C 10 CAS independent variable C 2 CAS menu F 6 CAS modulus C 3 CAS settings 1 26 C 1 CASDIR 2 35 CASE construct 21 51 CASINFO 2 37 Cauchy equation 16 51 CEIL 3 14 CENTR 22 7 Chain rule 13 6 Change sign 4 6 Char...

Страница 864: ...ber 11 10 Confidence intervals for the variance 18 33 Confidence intervals in linear regres sion 18 52 Confidence intervals 18 22 Conic curves 12 20 CONJ 4 6 CONLIB 3 29 Constants lib F 2 Continuous self test G 3 CONVERT 3 27 CONVERT Menu 5 26 Convolution 16 47 Coordinate System 1 24 Coordinate transformation 14 9 COPY 2 27 Correlation coefficient 18 11 COS 3 7 COSH 3 9 Covariance 18 11 CRDIR 2 41...

Страница 865: ...ial 14 1 Derivatives step by step 13 16 Derivatives with 13 4 DERVX 13 3 DESOLVE 16 7 DET 11 12 De tagging 21 33 Determinants 11 13 11 40 DIAG 10 13 Diagonal matrix 10 13 DIFF menu 16 3 DIFFE sub menu 6 29 Differential equation graph 12 26 Differential equations 16 1 differential equations 12 26 Differential equations Fourier series 16 40 Differential equations graphical solu tions 16 57 Different...

Страница 866: ... 11 46 EGVL 11 46 Eigenvalues 11 45 eigenvalues 11 10 Eigenvectors 11 45 eigenvectors 11 10 Electric units 3 20 END 2 27 ENDSUB 8 11 Energy units 3 20 Engineering format 1 21 ENGL 3 30 Entering vectors 9 2 EPS 2 37 EPSX0 5 22 EQ 6 26 Equation Library F 6 M 1 Equation Library 27 1 Equation Writer EQW 2 10 Equation writer properties 1 29 Equation Writer Selection Tree E 1 Equations linear systems 11...

Страница 867: ...ier transform 16 47 Fast Replace All L 3 FCOEF 5 11 FDISTRIB 5 28 FFT 16 47 Fields 15 1 File manager menu F 4 Financial calculations 6 9 Find next L 3 Finite arithmetic ring 5 13 Finite population 18 3 Fitting data 18 10 Fixed format 1 19 Flags 24 1 FLOOR 3 14 FOR construct 21 59 Force units 3 20 Format SD card 26 10 FOURIER 16 26 Fourier series 16 26 Fourier series and ODEs 16 41 Fourier series f...

Страница 868: ... 22 1 Graphs 12 1 Graphs bar plots 12 29 Graphs conic curves 12 20 Graphs differential equations 12 26 Graphs Fast 3D plots 12 34 Graphs Gridmap plots 12 40 Graphs histograms 12 29 Graphs parametric 12 22 Graphs polar 12 18 Graphs Pr Surface plots 12 41 Graphs saving 12 7 Graphs scatterplots 12 31 Graphs slope fields 12 33 Graphs SYMBOLIC menu 12 49 Graphs truth plots 12 28 Graphs wireframe plots ...

Страница 869: ...ABCUV 5 10 IBERNOULLI 5 10 ICHINREM 5 10 Identity matrix 11 6 identity matrix 10 1 IDIV2 5 10 IDN 10 9 IEGCD 5 10 IF THEN ELSE END 21 48 IF THEN END 21 47 IFERR sub menu 21 65 IFTE 3 36 ILAP 16 11 Illumination units 3 21 IM 4 6 IMAGE 11 55 Imaginary part 4 1 Improper integrals 13 20 Increasing power CAS mode C 9 INDEP 22 6 Independent variable in CAS C 2 Infinite series 13 20 Infinite series 13 22...

Страница 870: ... ISECT in plots 12 6 ISOL 6 1 ISOM 11 55 ISPRIME 5 10 ITALI L 4 J Jacobian 14 9 JORDAN 11 47 K KER 11 56 Key Click 1 25 Keyboard B 1 Keyboard ALPHA characters B 9 Keyboard ALPHA left shift characters B 10 Keyboard ALPHA right shift charac ters B 12 Keyboard alternate key functions B 4 Keyboard left shift functions B 5 Keyboard main key functions B 2 Keyboard right shift functions B 8 Kronecker s d...

Страница 871: ...2 LINSOLVE 11 41 LIST 2 34 LIST menu 8 8 List of CAS help facility H 1 List of command catalog I 1 Lists 8 1 LN 3 6 Ln X graph 12 8 LNCOLLECT 5 5 LNP1 3 9 Local variables 21 2 LOG 3 5 LOGIC menu 19 5 Logical operators 21 43 Lower triangular matrix 11 50 LQ 11 49 11 51 LQ decomposition 11 49 LSQ 11 24 LU 11 49 LU decomposition 11 49 LVARI 7 11 M Maclaurin series 13 23 MAD 11 48 Main diagonal 10 1 M...

Страница 872: ...trix writer 9 3 Matrix Writer 10 2 MATRIX MAKE menu 10 3 Matrix vector multiplication 11 2 MAX 3 13 Maximum 13 12 14 5 MAXR 3 16 Mean 18 3 Measures of central tendency 18 3 Measures of spreading 18 3 Median 18 3 Memory 26 1 to 26 10 MENU 12 46 Menu numbers 20 2 Menus 1 3 Menus not accessible through key board G 3 MES 7 9 Message box programming 21 37 Method of least squares 18 50 MIN 3 13 Minimum ...

Страница 873: ...ric solver menu F 3 Numeric vs symbolic CAS mode C 3 Numerical solution of ODEs 16 57 Numerical solution to stiff ODEs 16 65 Numerical solver 6 5 NUMX 22 10 NUMY 22 10 O OBJ 9 19 Objects 2 1 24 1 OCT 19 2 Octal numbers 3 2 ODEs ordinary differential equa tions 16 1 ODEs Graphical solution 16 57 ODEs Laplace transform applications 16 17 ODEs Numerical solution 16 57 ODETYPE 16 8 OFF 1 2 ON 1 2 OPER...

Страница 874: ...NDOW environment 12 4 PLOT FLAG menu 22 13 PLOT STAT menu 22 11 PLOT STAT DATA menu 22 12 PLOTADD 12 50 Plots program generated 22 17 Poisson distribution 17 5 Polar coordinate plot 12 18 Polar coordinates double integrals 14 9 Polar plot 12 18 Polar representation 4 1 4 3 POLY sub menu 6 29 Polynomial Equations 6 6 Polynomial fitting 18 59 Polynomials 5 17 Population 18 3 POS 8 11 POTENTIAL 15 3 ...

Страница 875: ...essage box 21 37 Programming modular 22 35 Programming output 21 33 Programming plots 22 14 Programming sequential 21 19 Programming tagged output 21 34 Programming using units 21 37 Programming with GROBs 22 33 Programs with drawing functions 22 24 PROOT 5 21 PROPFRAC 5 10 5 23 Pr Surface plots 12 41 Ps Contour plots 12 38 PSI 3 15 PTAYL 5 11 5 21 PTYPE 22 4 Purging from SD card 26 11 PUT 8 10 PU...

Страница 876: ... REVLIST 8 9 REWRITE menu 5 27 Right shift functions B 8 Rigorous CAS mode C 10 RISCH 13 14 RKF 16 67 RKFERR 16 71 RKFSTEP 16 69 RL 19 6 RLB 19 7 RND 3 14 RNRM 11 9 ROOT 6 26 ROOT in plots 12 5 ROOT sub menu 6 26 Row norm 11 9 Row vectors 9 18 ROW 10 23 ROW 10 23 ROW 10 24 RR 19 6 RRB 19 7 RRK 16 68 RSBERR 16 71 RSD 11 44 RSWP 10 24 R Z 3 2 S Saddle point 14 5 Sample correlation coefficient 18 11 ...

Страница 877: ... 6 SLB 19 7 Slope fields 12 33 Slope fields for differential equations 16 3 SLOPE in plots 12 6 SNRM 11 8 SOFT menus 1 4 SOLVE 5 5 6 2 7 1 27 1 SOLVE menu 6 26 SOLVE menu menu 74 G 3 SOLVE DIFF menu 16 67 SOLVEVX 6 3 SOLVR menu 6 26 SORT 2 34 Special characters G 2 Speed units 3 20 SPHERE 9 15 SQ 3 5 Square root 3 5 Square wave Fourier series 16 38 SR 19 6 SRAD 11 10 SRB 19 7 SREPL 23 3 SST 21 35 ...

Страница 878: ...c division 5 25 SYST2MAT 11 43 System flag EXACT APPROX G 1 System flag 117 CHOOSE SOFT 1 5 G 2 System flag 95 ALG RPN G 1 System flags 24 3 System of equations 11 18 System level operation G 3 T Table 12 17 12 25 TABVAL 12 50 13 9 TABVAR 12 50 13 10 Tagged output programming 21 34 TAIL 8 11 TAN 3 7 TANH 3 9 Taylor polynomial 13 23 Taylor series 13 23 TAYLR 13 24 TAYLR0 13 24 TCHEBYCHEFF 5 22 Tche...

Страница 879: ...6 30 TVMROOT 6 31 Two dimensional plot programs 22 14 Two dimensional vector 9 12 TYPE 24 2 U UBASE 3 22 UFACT 3 28 UNASSIGN K 1 UNASSUME J 3 UNDE L 4 UNDO 2 62 UNIT 3 30 Unit prefixes 3 24 Units 3 17 Units in programming 21 37 Upper triangular matrix 11 29 11 33 USB port P 2 User RPL language 21 1 User defined keys 20 6 Using input forms A 1 UTILITY menu menu 113 G 3 UTPC 17 12 UTPF 17 13 UTPN 17...

Страница 880: ...9 VZIN 12 48 W Warm calculator restart G 3 Weber s equation 16 57 Weibull distribution 17 7 Weighted average 8 17 WHILE construct 21 63 Wireframe plots 12 36 Wordsize 19 4 X XCOL 22 13 XNUM K 5 XOR 19 5 XPON 3 14 XQ K 5 XRNG 22 6 XROOT 3 5 XSEND 2 34 XVOL 22 10 XXRNG 22 10 XYZ 3 2 Y YCOL 22 13 YRNG 22 6 Y Slice plots 12 39 YVOL 22 10 YYRNG 22 10 Z ZAUTO 12 48 ZDECI 12 48 ZDFLT 12 48 ZEROS 6 4 ZFAC...

Страница 881: ...2 ARRY 9 6 9 20 BEG L 1 COL 10 18 DATE 25 3 DIAG 10 12 END L 1 GROB 22 31 HMS 25 3 LCD 22 32 LIST 9 20 ROW 10 22 STK 3 30 STR 23 1 TAG 21 33 23 1 TIME 25 3 UNIT 3 28 V2 9 12 V3 9 12 ΣDAT 18 7 ΔDLIST 8 9 ΣPAR 22 13 ΠPLIST 8 9 ΣSLIST 8 9 ...

Страница 882: ...ondition as warranted you will be entitled to a refund of the purchase price upon prompt return of the product with proof of purchase 4 HP products may contain remanufactured parts equivalent to new in performance or may have been subject to incidental use 5 Warranty does not apply to defects resulting from a improper or inadequate maintenance or calibration b software interfacing parts or supplie...

Страница 883: ...ts accompanying such products and services HP shall not be liable for technical or editorial errors or omissions contained herein FOR CONSUMER TRANSACTIONS IN AUSTRALIA AND NEW ZEALAND THE WARRANTY TERMS CONTAINED IN THIS STATEMENT EXCEPT TO THE EXTENT LAWFULLY PERMITTED DO NOT EXCLUDE RESTRICT OR MODIFY AND ARE IN ADDITION TO THE MANDATORY STATUTORY RIGHTS APPLICABLE TO THE SALE OF THIS PRODUCT T...

Страница 884: ...rica Country Telephone numbers Argentina 0 810 555 5520 Brazil Sao Paulo 3747 7799 ROTC 0 800 157751 Mexico Mx City 5258 9922 ROTC 01 800 472 6684 Venezuela 0800 4746 8368 Chile 800 360999 Columbia 9 800 114726 Peru 0 800 10111 Central America Caribbean 1 800 711 2884 Guatemala 1 800 999 5105 Puerto Rico 1 877 232 0589 Costa Rica 0 800 011 0524 N America Country Telephone numbers U S 1800 HP INVEN...

Страница 885: ...n be determined by turning the equipment off and on the user is encouraged to try to correct the interference by one or more of the following measures Reorient or relocate the receiving antenna Increase the separation between the equipment and the receiver Connect the equipment into an outlet on a circuit different from that to which the receiver is connected Consult the dealer or an experienced r...

Страница 886: ...77269 2000 Or call 1 281 514 3333 To identify this product refer to the part series or model number found on the product Canadian Notice This Class B digital apparatus meets all requirements of the Canadian Interference Causing Equipment Regulations Avis Canadien Cet appareil numérique de la classe B respecte toutes les exigences du Règlement sur le matériel brouilleur du Canada European Union Reg...

Страница 887: ...pment by handing it over to a designated collection point for the recycling of waste electrical and electronic equipment The separate collection and recycling of your waste equipment at the time of disposal will help to conserve natural resources and ensure that it is recycled in a manner that protects human health and the environment For more information about where you can drop off your waste eq...

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