Texas Instruments Titanium TI-89, User Manual

Page: 856 / 977

Appendix A: Functions and Instructions 853
If all of the equations are polynomials and if you do
NOT specify any initial guesses,
solve()
uses the
lexical Gröbner/Buchberger elimination method to
attempt to determine
all real solutions.
For example, suppose you have a circle of radius r
at the origin and another circle of radius r centered
where the first circle crosses the positive x-axis. Use
solve()
to find the intersections.
As illustrated by r in the example to the right,
simultaneous
polynomial equations can have extra
variables that have no values, but represent given
numeric values that could be substituted later.
solve(x^2+y^2=r^2 and
(x
ì r)^2+y^2=r^2,{x,y}) ¸
x=
r
2
and y=
3
ø
r
2
or x=
r
2
and y=
ë
3
ø
r
2
You can also (or instead) include solution variables
that do not appear in the equations. For example,
you can include z as a solution variable to extend
the previous example to two parallel intersecting
cylinders of radius r.
The cylinder solutions illustrate how families of
solutions might contain arbitrary constants of the
form @
k, where k is an integer suffix from 1 through
255. The suffix resets to 1 when you use
ClrHome
or
ƒ
8:Clear Home
.
For polynomial systems, computation time or
memory exhaustion may depend strongly on the
order in which you list solution variables. If your
initial choice exhausts memory or your patience, try
rearranging the variables in the equations and/or
varOrGuess
list.
solve(x^2+y^2=r^2 and
(x
ì r)^2+y^2=r^2,{x,y,z}) ¸
x=
r
2
and y=
3
ø
r
2
and z=@1
or x=
r
2
and y=
ë
3
ø
r
2
and z=@1
If you do not include any guesses and if any
equation is non-polynomial in any variable but all
equations are linear in the solution variables,
solve()
uses Gaussian elimination to attempt to
determine all real solutions.
solve(x+
e
^(z)
ù
y=1 and
x ì y=sin(z),{x,y}) ¸
x=
e z
ø
sin(z)+1
e
z +1
and y=
ë
(sin(z)
ì
1)
e
z +1
If a system is neither polynomial in all of its
variables nor linear in its solution variables,
solve()
determines at most one solution using an
approximate iterative method. To do so, the number
of solution variables must equal the number of
equations, and all other variables in the equations
must simplify to numbers.
solve( e ^(z) ù y=1 and
ë
y=sin(z),{y,z})
¸
y=.041… and z=3.183…