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Summary of Contents for Mathematics II

Page 1: ...USER MANUAL Mathematics II ...

Page 2: ...CALCULUS MATHEMATICS II VER1 0 A mathematical program for calculators Odd Bringslid tsv Postbox 1014 3601 Kongsberg NORWAY ...

Page 3: ...ngslid ISV All rights reserved The author should not be liable for any errors or conse quential or incidential damages connecting with the furnis hing performance or use of the application card JS5 First edition August 1992 ...

Page 4: ...inished 10 Moving in the menu 10 STAT and MATR menu 11 Leaving CALCULUS 11 Intermediate results 12 Flag status and CST menu 12 Linear algebra 13 Linear equations Gauss algo rithm 13 Matrix calculations 16 Addition 16 Multiplication 16 Inverting 16 Determinant 16 Rank 17 Trace 17 Orthogonal matrix 17 Transposed matrix 17 Symmetric 17 ...

Page 5: ...3 dimen sions 23 Translation 23 Scaling 23 Rotation 23 Concatinating 24 Eigenvalueproblems 25 Eigenvalues 25 Eigenvectors 26 Diagonalization 27 Diffequations 28 Vector spaces 30 Basis 30 Norm 31 Norming 31 Scalar product 31 Orthogonalization 31 Orthogonal 32 Orthonorming 32 Vector in new basis 32 Transformation matrix in new basis 34 rr L ...

Page 6: ... 44 Hypergeometric distr function 44 With replacement 46 Combinations unordered 46 Combinations ordered 47 Binomial distribution 48 Binomial dsitr function 49 Negative binomial distribution 49 Negative binomial distribution func tion 50 Pascal distribution 51 Pascal distribution function 52 Normal distribution 53 Poisson distribution 54 Poisson distribution function 55 Info 55 Binomial coefficient...

Page 7: ...n stdev median 65 Fitting 66 Normal ditsribution best fit 66 Hypothesis normaldistribution 67 Hypothesis binomial distribution 68 Hypothesis Poisson distribution 69 L_ Class table 69 T Mean stdev 70 Discrete table 70 it Description of samples 71 Diskrete table DAT 71 Classes KSDAT 71 Cummulativ table 72 72 2DAT mean and st deviation 72 gr Histogram KXDAT 73 Frequency polygon KXDAT 74 2 j Linear re...

Page 8: ...Fourier series 75 Fourier series symbolic form 76 Fourier series numeric form 78 Half range expansions 79 Linear programming 81 ...

Page 9: ...l education As in part I a pedagogicalinterface is stressed CALCULUS mathematics is apedagogicaltool in addition to apackagefor getting things calculated Hardware requirements CALCULUS Math II runs under the calculator HP 48SX The program cardmaybe inserted into either ofthe twoports and Math I could be in the other port 1 Generaly ...

Page 10: ...he START key User interface The CALCULUS meny system is easy to use Using the ar row keys allowsyou to move the dark bar and select by pus hing ENTER In the following example you will enter the submenu for LINEAR ALGEBRA and select Matrices Multiply RAD HOME PRG LAPLACE TRANSFORMS FOURIER SERIES LINEAR PROGRAMMING r 1 Generaly ...

Page 11: ...r Under Matrices you will choose Multiply and you may mul tiply two symbolic matrices The matrices are put into the SYMBOLIC MATRIX WRITER RAD HOME PRG Sum Powers Inverting p v 1 2 In the Matrix Writer you may delete add and echo from the stack see manual for 48SX 1 Generaly ...

Page 12: ...n move around byusing the arrowkeys The cursorisplaced right behind PartAns Y N and hereyou enter Y ifyou want intermediate results The arrowkeys are used to get right behind B Bl and here you enter the right side vector of the system If you have done a mistake you mayalter your input by using the delete keys on the calculator keyboard You will not be able to continue before the data are correctly...

Page 13: ...cursor correctly Calculation finished When a calculation is finished CALCULUS will either show up intermediate results byusing the VIEW routine intrinsic MAII or return directly to the menu In the last caseyouwill needto use the STKkeyto seethe result layingonthe stack Moving up and down in the menu You can move downwardsin the menu system by scrolling the dark bar and pressing ENTER If you need t...

Page 14: ...dless of your current menu position STAT MATR NORM Normal distribution INVN Inverse normal distribution USDAT Sample mean st dev median KSDAT Class table SDAT Discrete table twocolumns ADD Addsymbolicmatrices MULT Multiply INV Invert TRN Transpose DET Determinant Leaving CALCULUS Pushing the EXIT keywill leave CALCULUS 1 Generaly 11 ...

Page 15: ... of the results covers the whole answer In E every case this will give the user a good help Different parts of an answermaybe found on different pages and the page number canbee seen use arrowup down When PartAns Y es ischosen allnumbers will showupwith two figures behind comma If a more accurate answer is ne cessary you will have to look on the stack and perhaps use the N FIX option s r Flag stat...

Page 16: ...tions in two and three dimensions and vector spaces Linear equations Gauss method Linear equations with symbolicparameters are handled The equations have to be ordered to reckognize the coefficient matrix andthe right side The equations are given inthe form E sr r Br g A is the coefficient matrix X a column vector for the un I j knowns and B therightside column vector Symbolic coeffi cients arepos...

Page 17: ...onwillbe givenin the idefinite form If the systemis over __ determined too many equations the solution will be given in the indefinite form if the equations are lineary dependent or as Self contradictory if they are lineary independent L The solution algorithm is the Gauss elimination If PartAns Y es is selected the different stages in the process will be given as matrices on the stackwhich may be...

Page 18: ...rix writer will now appear The following matrix is put into it 2 i i 1 1 5 1 2 4 3 1 1 2 1 3 1 1 1 The example solves the system 2xi X2 3x3 2x4 xs 7 xi 2x2 X3 X4 xs 6 xi 4x2 xs 3x4 xs 0 The system is indefinite too few eqautions and the solu tion is given in terms of xs and X4 2 Linearalgebra 15 ...

Page 19: ...on Multiplication Both matrices are put into the matrix writer and multiplied An error message is given for wrong dimensions The first matrix has to have the same number of columns as the sec ond has rows Be aware of the order of the matrices Inverting The mark isput into the matrix writer andinverted An er ror message is given if its not quadratic Determinant T The determinant of a symbolicmatrix...

Page 20: ... if the matrix is not quadratic Orthogonal matrix This routine is testing whether the matrix is orthogonal i e the inverse is equal to the transpose An error message isgi ven for wrong dimension must be quadratic The answer is logic 0 or 1 Maybe used to investigate if rowvectors are ort hogonal i e is an orthogonal basis of a vector space Transpose matrix The transpose of a symbolicmatrix is calcu...

Page 21: ...ve a rectangular coordinate sys tem Mixedtransformations concatinating ispossible The order of the transformationsis important if rotation is one of the them 2 gir TC 2 D transformations two dimensions Rotation The rotationangle mustbe givenin degrees andthe transfor med point is given ascomponents of alist to allowsymbols The rotation is counterclockwise for positive angles about an arbitrary poi...

Page 22: ...G Rotation about xO yO X Y 2 5 XO YO 2 2 45 T5 ra KO J A The point 2 5 is rotated about 2 2 an angle 45 To rotate a triangle allthree points have to be transformed Translation E15 This is a pure translation of apoint the coordinates are given r an addition 2 Linearalgebra 19 ...

Page 23: ...rir cTi i The example moves the point 2 5 to 2 5 3 6 5 11 Scaling The coordinates are multiplied by a factor For a geometric figure where the points are scaled this will give a smaller or bigger figure If the X and Y coordinates are scaled different ly this will alter the shape of the geometric figure 3 ff 2 Linear algebra 20 ...

Page 24: ...riangle Interface RAD HOME PRO XY Sx Sy X Sx Yt 2 5 3 6 _ 1 The example multiplies 2 with 3 and 5 with 6 and the point 2 5 is moved Concatinating mixed transformations Concatinating means a mixture of several transformations 2 Linear algebra 21 ...

Page 25: ... 25 Sx Sy l 1 TxTy 2 6 0XOYO STR 4522 RTS In the example the point 2 5 is rotated clockwise 45 about the point 2 2 first then a translation of 2 in the x direction and 6in the y direction There is no scaling indicated by 11 for the scaling factors Rent No translation gives Tx Ty 0 and no rotation gives 0 0 Sa 2 Linear algebra 22 ...

Page 26: ...aling The interface is the same as 2D scaling with one extra coor dinate and one extra scaling Rotation 3D rotation is somewhat more complicated than in two di mensions The rotation axeshasto be specified i e angles re lative the coordinate axes and a point Interface RAD HOME XYZ XO YO ZO ap y 0 v w PRG 2 54 221 45 45 60 45 i S 2 Linear algebra 23 ...

Page 27: ...lative the x y and z axes equal to 45 45 and 60 Concatinating mixed The same interface as in the 2D case but the rotation axes now has to be specified Interface e s gr fe RAD HOME XYZ XO YO ZO Sx Sy Sz TxTyTz PRG 2 54 221 346 252 RAD HOME a p7 45 45 60 0 STR 45 RTS PRG 3 rt 2 Linear algebra 24 ...

Page 28: ...eigenvalues and the eigenvectors of a matrix diagonalize a matrix and solve a system of linear dif ferential equations Eigenvalues The eigenvalues of a matrix are determined by the equation A X x X where Ais the matrix and X a column vector ei genvector x is called the eigenvalue Interface RAD HOME Det A x I 0 rPartAns Y N Y PRG WJ J f SS yS l i 2 Linear algebra 25 ...

Page 29: ...igenvalue with multiplicity 2 x 2 Eigenvectors A matrix has infinite many eigenvectors because the system of equations that determines the vectors are indefinite The eigenvectors are givenin terms ofarbitrary parameters Aset of eigenvectors will normaly be linear independent even if the eigenvalues have multiplicity greater than 1 But this is not always the case fcr 2 Linear algebra 26 n ...

Page 30: ... system of equations that determines the eigenvectors will be given We see that only two ofthe eigenvectors are lineary independent onlyone ar bitrary parameter c Diagonalization For a matrix A we can write Here K is a matrix composed of the eigenvectors ofAwhich have to be lineary independent D is a diagonal matrix with the eigenvalues on the diagonal If the eigenvectors are 2 Linear algebra 27 ...

Page 31: ... output is the matrices K and D The matrix K may be found by inverting K Rera If intermediate results are wanted you may look at the problems of finding eigenva lues and eigenvectors separately r g System of differential equations Here a set of linear homogenous differential equations with constant coefficients are solved byusing the method of dia gonalization L 2 Linear algebra 28 ...

Page 32: ... HOME PRG dX dt A X X x y XYZ t t r jfv ifTx 4 5 5 5 6 5 5 5 6 The following system is solved dx dt dy dt 5x 6y 5 The output contains the constants Cl C2 and C3 and the in dependent variable is t 2 Linear algebra 29 ...

Page 33: ... collection ofvectors relative a basiswhe re certain operations on them are defined A basis is a setof linear independent vectors from the space In anorthogonal basis the vectors are mutualy orthogonal inner product equals zero Basis This routine examines whether a set ofvectors in the space islinear independent The vectors areput into the matrixwri ter as rows and the output is logic 0 or 1 2 Lin...

Page 34: ...alculated The input vector is viV2vs and the output is anumber or anexpres sion if the vector issymbolic Norming A vector is transformed into an unit vector e V NORM V V Scalar product inner product The scalarproductoftwovectorsis calculated Symbolicvec tors are possible Orthogonalization An orthogonal basis is calculated with an arbitrary basis as a starting point using the Gram Schmidt process _...

Page 35: ...s are not possible Orthogonal The routine examines whether a matrix is orthogonal If the row vectors building up the matrix form an orthogonalbasis then the matrix is orthogonal Orthonorming The routine is norming an orthogonal basis Vector in new basis U Given a vector VBI i e relative a basis Bl A new vector re I lative a basis B2 is calculated 2 Linear algebra 32 ...

Page 36: ...Interface RAD HOME Xbl Xb2 X xl 142 PRG 1 2 3 3 1 2 2 2 5 1 1 3 4 1 2 2 6 5 Thevector 142 relative the basis 12 3 3 1 2 225 is transformed to the new basis 2 Linear algebra 33 ...

Page 37: ...avector spacere tp lative the natural basis This routine calculates a new trans formation matrix relative a new basis The natural basis is 100 010 001 inR3 1 1 1 O i l 0 0 1 f r r The transformation matrix 110 011 101 in natu ral basis defines the transformation E L X1 X2 X3 XI X2 X2 X3 X3 Xl ...

Page 38: ...The example calculates the new transformation matrix rela tive basis 111 0 11 0 0 1 Symbolic elements are pos sible in the matrices 5 53 S H n 1 3 r a e 2 Linear algebra 35 ...

Page 39: ...to other methods deal withfunctions f t that are discontinous in the equation a y b y c y f t e Discontinous f t may be composed by using the Unit Step function u t a defined as This function is not implemented in CALCULUS in other ways than as a symbol and the user has to make a program to define itfor evaluation I 3 3 3 Laplace transforms_ 36 ...

Page 40: ... arbitrary f t Cos a t a arbitrary g t f t ea a arbitrary g t f t u t a a 0 g t f t u t a ebt a 0 b arbitrary h t g t t Linear combinations oftheese functions Interface RAD HOME PRG F s L f t t s t s f t t 2 u t l The example calculates the Laplace transform of t 2 u t l 3 Laplacetransforms 37 ...

Page 41: ...or its not implemen ted Inverse Laplace transform The inverse transform is calculated The types of functions which can be inverted are the transforms of the functions lis ted on page 37 Interface n RAD HOME PRG f t InvL F s s t s t The example calculates the inverse transform of 3 Laplace transforms 38 ...

Page 42: ...denominator has to be split into partial fractions Intermediate results PartialAnswers ispossible to showthe splitting into partial fractions Rem If the transformation does not exist the error message does not exist is given If the expressionis too complicatedthemessage not rational may appear The expression may then be split up 3 Laplace transforms 39 ...

Page 43: ...J I PRG RAD HOME Input cont Numerator P Exp ir s Denominator Q s 2 1 PRG v In the example F s s 1 is split into partial fractions and then transformed The shift e 778 willbe taken care of be fore the splitting into partial fractions 3 Laplace transforms 40 5 S c S Jjjp C I B e ...

Page 44: ...is discontinous The answer is given in the form Y s P s Q s R s which has to be transformed into P s Q s R s before the ro utine for partial fractions is used to solve the problem Interface RAD HOME PRG ay by cy f t y 0 yOy 0 DyO abcyODyO 13201 f t t SIN t t The equation y 3y 4 2y Sin t with initial conditions y 0 0andy 0 1is transformed 3 Laplacetransforms 41 ...

Page 45: ...r the discrete distributions both the cummulative probability and the point probability may be calculated For the discrete distributions and in connection with pure combinatorial calculations we have distinquished between with and without replacement e B 5 3 Rem Probabilities must be less then or equal to 1 and greater than or equal to 0 UT L 4 Probability 42 WBT t_ ...

Page 46: ...his routine calculates the number ofpossibilities to draw k elements of total n without replacement and without regard to order Interface RAD HOME N n k PRG Draw k of n F n n k k 15 3 _ _ The example calculates the number og possibilities to draw 3 elements form total 15elements without regard to order 4 Probability 43 ...

Page 47: ... n k 15 3 J i JJ X tWw J S W 1 if i1 l K si PRG i X I e e The example calculates the number of combinations when drawing 3 elements from 15with regard to order Hypergeometric distribution Here the probability of drawing exactly k X es from a popu lation of n when a elements are drawn at a time without re placementiscalculated The probability for the Xtobedrawn is p 3 Ew c gw E 4 Probability 44 ...

Page 48: ...s 0 6 If k a or p 1the probability is0 The example maybe drawing individuals from apopulation of 20 where 12 is women p 12 20 0 6 The probability that of 8 drawn individuals 3 is women is calculated Hypergeometric distributionfunction The cummulativeprobability iscalculated i e theprobability that maximum k elements are drawn This is the sum of the probabilities ofk 0 k l k 2 andk 3 4 Probability ...

Page 49: ... of20bydrawing 4 at a time or 1by 1without replacement With replacement Here the elements are replaced by drawing so that the prob ability is the same every time an element is drawn uncondi tional drawing Combinations unordered This routine calculates the number of combinations of dra wingk elements from n without replacement without regard to order 4 Probability 46 ...

Page 50: ...wing 3 elements of total 15is cal culated Order isindifferent Combinations ordered If the order is critical this routine has to be used The same elements in different orders are two separate events Interface RAD HOME PRG n k Draw k of n ordered N n k 15 3 4 Probability 47 ...

Page 51: ...al n where the probabilityof X itself is p Independent trials with replacement Interface RAD HOME np k ofrrsf PRG kX es of n P X p 10 0 6 3 The probability of drawing3 elements with the mark Xwhen X has the probability of 0 6 is calculated The number of in dependent trials is 10 p 1gives an error message The example may be the production of glasses where the probability of first assortment is 0 6 ...

Page 52: ... k 0 k l k 2 andk 3 if k 3 Interface RAD HOME PRG maxkX es of n P X p 10 0 6 3 The example calculates the probability that maximum 3 ele ments have the mark X p 0 6 in 10independent trials Negative binomial distribution This distributiongivesthe probability of kfailures before the rth successin a series of independent trials each ofwhich the probability of successis p 4 Probability 49 ...

Page 53: ...ess when the probability of success is 0 25 is calculated The example maybe the drawing ofcards and the calculation of the probability of drawing 15cards that are not clubs be fore the 10th club S Negative binomial distribution function This routine calculates probability ofmaximum k failures be fore the rth success 4 Probability 50 ...

Page 54: ...culated Probability of success is 0 4 The example maybe the drawing ofballs from ahat that con tains 40 white balls The probability of finding 5 not white balls before drawingmaximum 4 white balls is calculated Pascal distribution The probability ofthe rth success in kth trial in a series of in dependent trials is calculated 4 Probability 51 ...

Page 55: ...ty offinding the mark X 5th time in the 8th tri al is calculated _ L Rem The geometric distribution is a special case with r l a fee Pascal distribution function The probability of the rth success in maximum k trials is cal culated The probability of success is p 4 Probability 52 ...

Page 56: ...culates the probability offindingthe mark X the 5th time in maximum 8trials Tossing a fair coinwe find the probability of finding the 5th head in maximum 8 trials Normal distribution This is a continous distribution and only cummulative prob abilities are calculated 4 Probability 53 ...

Page 57: ...ndom variable is less than or equal to 1is calculated The mean and the standard deviation is 0 and 1 Rem P a x b P x b P x a and P x a l P x a Poisson distribution The Poisson distribution is used as a model when we are in terested in events within intervals of time or other variables 3 T 4 Probability 54 ...

Page 58: ... distr mean p and P X k ak 4 5 The example calculates the probability that a random varia ble X is exactly 5 when the mean is4 Poisson distribution function Interface RAD HOME PRG Poisson distr meanjx and P X k 4 5 4 Probability 55 ...

Page 59: ...Here information about probability and some distributions is given Binomial coefficients Binomial coefficients Bnk n n k k are calculated from k 0to k n andput in a list Interface RAD HOME PRO Binomial coeff n n k k k 0 n The example calculates BnO Bnl Bn2 Bn3 Bn4 Bn5 4 Probability 56 ...

Page 60: ...ics by using the mean value of the intervals asthe discrete value Statistical methods are represented by confidence intervals and hypothesistesting for distributions The best fit for the normal distribution uses the method of least squares The normal distribution kjisquare distribution and student t distribution are included and its possible to find both the probabilityand the value ofthe randomva...

Page 61: ...andard deviation o have to be known Interface RAD HOME PRO Normal distribution pararn and o gives P X x cr 0 1 x 1 3 1 V f 1 1 1 A j i P X 1 for n 0 and o 1is calculated Inverse normal distribution The routine finds the value of the random variable xwithgi ven probability p and o are known 5 Statistics 58 ...

Page 62: ... The value of x with P X x 0 6 jj 0 and a 1 is calcula ted Kjisquaredistribution The kjisquare distribution isused to find confidence intervals and in connection with fitting a distribution to a sample Interface RAD HOME PRG Kjisquare distr degrees of freed KP X x K x 3 5 6 5 Statistics 59 ...

Page 63: ... value ofx for given probability is calculated Interface f RAD HOME PRG Kjisquare distr degrees offreed KP X x p K p 3 0 85 3 The example calculates the value ofX sothat P X x 0 85 with 3 degrees of freedom Studen t distribution This distribution isused to find confidenceintervals in CAL CULUS T 5 Statistics 60 ...

Page 64: ... with 3 degrees offreedom is calculated Inverse student t This routine calculates the value of the random variable x Interface RAD HOME PRG Student t distr degrees offreed KP X x p K p 3 0 85 The value of x is calculated so that P X x 0 85 with 3 de grees of freedom 5 Statistics 61 ...

Page 65: ...ues may be used aspoint estimates for theparameters in the distribution function Confidence interval for the mean x givenvalue of a For known a wemayuse the normal dsirribution to find the value c so that F c P x c l 2 T with confidence le vel The interval is given in the form a x b The interval is calculated from a sample xi and the mean value has to calculated in advance byusing the menu option ...

Page 66: ...values in the sample is20and the normal distribution has the standard deviation 1 12 Confidence intervalfor the mean ji unknowno If a is not known the estimate s for standard deviation from the sample is used To find the value of c so that F c 1 2 7 1 the student t distribution is used The interval is calculated from a sample xi with n values and the mean and the standard deviation of the sample h...

Page 67: ...e mean is calculated based on a sample with 20values standard deviation 1 2 mean 3 and with confidencelevel 90 Confidence intervalforvariance isunknown The standarddeviationofthe samplehastobe calculated first separate menu option The calculation is based on the kji square distribution L fe 5 Statistics 64 fc ...

Page 68: ...e example calculates the confidence interval for a based on a sample with standard deviation 1 2 confidence level 90 and 20 values in the sample Sample x s n andmedian The mean value standard deviation number of values and the median are calculated The table is stored as UsDAT Interface RAD HOME PRG St dev s and mean x 237 5 Statistics 65 ...

Page 69: ...f Fit The sample has to be given as a class statistic with given fre quences se description of samples In order to calculate estimates for the the distribution para meters a discrete statistic has to be stored as SDAT see se parate menu option Normal distribution A best fit based on the least squares is calculated The sam ple has to be stored as a class statistic KSDAT in advance When KsDATisstore...

Page 70: ...it for the class intervals For the calculation of mean and standard deviation as esti mates for the parameters the discrete statistic has to be sto red with known frequencies SDAT Interface RAD HOME IJL a anr i Normald n values Level a Num est r 360 26 0 05 1002 PRG 71 The example is testing whether the sample KsDAT may be fitted to a normal distribution with significance level 5 and a are estimat...

Page 71: ...sis binomialdistribution The class statistic is stored and the parameter p is if neces sary estimated asp i n Interface RAD HOME pa nr F r PRG Binomiald nvalues Level a Num est r 0 5 0 05 501 _ r _ fl 3 The example istesting whether the samplemaybe fitted to a binomial distribution with significancelevel 5 p is estima ted 0 5 and the value of r is 1 Number of values is 50 rt 5 Statistics 68 ...

Page 72: ...05 100 1 PRG The example istesting whether the sample maybe fitted to a Poisson distribution with significance level 5 L is estima ted and r 1 Number of values in the sample is 100 Class tabel class statistic A class table is a double list Ik fk where the first list is the limits of the class intervals and the other list is the num ber ofvalues in the separate intervals 5 Statistics 69 ...

Page 73: ...ncy ta ble If the data is stored as a frequency table we cannot use the ordinary sample routine to find the mean etc The data is sto red in SDAT Use F for frequency table Storing discrete table The sample is stored as 2DAT The data is put directly into the matrix writer where the first column contains the values and the second column the frequencies T T T 5 Statistics 70 ...

Page 74: ... polygons Discrete table SDAT Here the values are stored in the first column and the fre quencies in the second In two variable statistics the second column will be the data for the second variable Classes KsDAT A class table is a double list Ik fk where the first list is the limits of the intervals and the other the numbers in each interval Interface RAD HOME PRG Class table 3467 254 5 Statistics...

Page 75: ...es co lumn 4 cummulative frequencies column3 and cummula tive relative frequencies column 5 This routine transforms a class statistics to a discrete statis tics by using the mean value of each interval as the repre sentative value The table is stored as DAT Input data is KsDATwhich is storedinadvance The meanvalue and the standarddeviation is calculated ba sed on a frequency table F or a twovariab...

Page 76: ... the mean and standard deviation for each variable in a two variable statistics Rem Under menu confidence intervals and STAT menu themean andstandard deviation for simple samples arecalculated one dimen sional tables Histogram KxDAT Creates a histogrambased on the class table KsDATwhich has to be stored in advance 5 Statistics 73 ...

Page 77: ...quency polygonfrom KXDAT Linear regression and correlation A straight line isfitted bythe use ofleast squares from the ta ble IDAT The line will be given as Y aX b where X is the data in the second column in 2DAT The correlation coefficient is a measure of the goodnessof the fit and a value between 0 7 and 0 7is a good fit E E E 5 Statistics 74 ...

Page 78: ...number of terms may also be found The input function maybe bifurca ted with different expressions in two different intervals The Fourier series are generaly given as f x aO an Cos 2 Tr x n T 2 bn Sin 2 Tr x n T ri l n l T is the period and the coefficients are given T 2 aO l rjf x dx T 2 T 2 r an 2 TJf x Cos 2 ir x n r dx T 2 T 2 bn 2 T f x Sin 2 Tr x n T dx T 2 6 Fourier series 75 ...

Page 79: ...ound for given f x The series itself must be set up bythe user The functionf x may be given as two expressions in two intervals 1 a x b 2 c x d The functions are given as f a b where a and b are defining the interval or bifurcated as f1 a b 2c d Rem If the function issplit in more than two intervals CALCULUS may be used on two and two or twoplus one intervals 3 3 E 6 Fourier series 76 5 ...

Page 80: ...N t 00 2 11 1 t T t 2V PartAns Y N Y Ir x fe w RAD HOME PRG Input cont fab W010V V p y The example calculates the Fourier coefficients of f t _ 1 i r t T The answer is given with intermediate results indefinite in tegrals are given 6 Fourierseries 77 ...

Page 81: ...ame asinsymbo licform but the number ofterms has to be included and also the start and stop ofthe summation index The integration is numeric and intermediate results are not given Interface RAD HOME PRG f t aO S m n an COS co t 2 m n bn SIN w t co 2 or n T t T t 2V 3 ru RAD HOME PRG Input cont m n 0 2 fab 1 W OIOV 6 Fourier series 78 n ...

Page 82: ...n manysituations there is apractical need to use Fourier se ries in connection with functions that are given merely on some definite interval They maybe done periodic by an ex tension withthe period asthe double interval The extension may be even or odd by choice E O Interface RAD HOME PRG f t aO Xan COS w t Sbn SIN w t co 2 Tr t T t 2 PartAnsY N Y SHTjir 6 Fourier series 79 ...

Page 83: ...RAD HOME PRG Input cont fab 1 WO O E O Heref t 1 ir t 0isgiven An odd extension ismarked by O 3 6 Fourier series 80 g ...

Page 84: ...s Rem If such constraints aregiven and a deci mal number is the answer one cannot simply round off to nearest naturalnumber This will not always give the optimal solution The algorithm used is the simplex method and it finds only the minimum ofan objectfunction Anyproblem canbe writ ten as a minimum problem If a maximum value of f x is going to be found one may simply change the sign and find the ...

Page 85: ...ality sign Ifwe are goingto maximize f x xl 2x2and one ofthe con straints are xl 2x2 2 then minimize xl 2x2 with the con straint xl 2x2 2 Allindependent variables are assumed to be positive or zero Rem The independent variables have the symbol xi regardless of the symbols used in a givenproblem Ed Interface RAD HOME PRG minf X OX A X B A is the last input B 8 5 C 25 7 24 7 Linear programming 82 ...

Page 86: ...5x1 7x2 24x3 under the constraints 3xl x2 5x3 8 5xl x2 3x3 5 0 xl x2 x3 C vector is the object function f B vector is the right side of the constraints and A is the constraint matrix left side The solution is given as 39 5 which means that the maximum is 39 5 7 Linear programming 83 ...

Page 87: ...eigenvectors o Linear Programming Laplace Transforms Fourier Series Statistics With the symbolic MATRIX WRITER Odd Brincs ic fntern tional Software Vendor Box1014KNP 3601 Kongsberg NORWAY Tel 47 3 73 88 40 Fax 47 3 73 86 33 Distributor BB Marketing ANS Ensjeweien 12 B 0655 Oslo Norway Tel 47 2 67 11 57 Fax 47 2 67 11 37 ...

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