restart:
# > This worksheet describes using solve and fsolve to solve equations.
# Consider the solution of a single equation for a single variable.
> solve(sin(x)+y^2-2,x);
/ 2 \
-arcsin\y - 2/
# Solve for y; you get a sequence of 2 solutions
> sy:=solve(sin(x)+y^2-2,y);
(1/2) (1/2)
sy := (2 - sin(x)) , -(2 - sin(x))
# Refer to the 2nd solution in the sequence using [2]
> sy[2];
(1/2)
-(2 - sin(x))
# Note that the solution implicitly assumes the expression for the
# equation is set equal to zero. If we add a second independent
# equation, we have
> solve({sin(x)+y^2-2,y-ln(x)},{x,y});
# We don't get a solution, because there is no analytic answer to this
# set of coupled equations; we can only determine a numerical answer,
# e.g. using fsolve.
> fsolve({sin(x)+y^2-2,y-ln(x)},{x,y});
{x = 0.2678513736, y = -1.317323028}
# Is this the only solution? It would be interesting to plot the
# functions to determine if that is the case. To do this we need the
# plots package.
> with(plots):
# Warning, the name changecoords has been redefined
# The plot command takes an equation as its first argument, the
# variable range as its second (and optional third), and additional
# plot options as the final argument.
> implicitplot([sin(x)+y^2-2,y-ln(x)],x=0..10,y=-3..5,color=[red,blue],n
> umpoints=2500);
# So it looks like there's another solution around x=5; let's try to
# find it.
> fsolve({sin(x)+y^2-2,y-ln(x)},{x,y},x=4..6);
{x = 5.342510434, y = 1.675695661}
> fsolve({sin(x)+y^2-2,y-ln(x)},{x,y},x=0..1);
{x = 0.2678513736, y = -1.317323028}
# You can also predefine the equations used by solve or fsolve.
> f:=sin(x)+y^2-2:
> g:=y-ln(x);
g := y - ln(x)
> fsolve({f,g},{x,y},x=4..6);
{y = 1.675695661, x = 5.342510434}
# Notice that the order of x and y in the return set is different than
# above; this underlines the fact that order is not preserved in sets.
# The above can also use the functional operator specification of
# functions. For example
> x:='x':y:='y':
> f:=(x,y)->sin(x)+y^2-2:
> g:=(x,y)->y-ln(x);
g := (x, y) -> y - ln(x)
> fsolve({f(w,z),g(w,z)},{w,z},w=4..6);
{w = 5.342510434, z = 1.675695661}
# You cannot, however, omit the arguments of f and g:
> fsolve({f,g},{x,y},x=4..6);
# Error, (in fsolve) Case of systems of equations involving procedures
# is not implemented yet
# On the other hand, you can use a richer set of arguments
> fsolve({f(x^2,y^2),g(x^2,y^2)},{x,y},x=2..3,y=1..2);
{x = 2.311387123, y = 1.294486640}
> fsolve({f(x^2,y^2),g(x^2,y^2)},{x,y},x=2..3,y=-2..1);
{y = -1.294486640, x = 2.311387123}
# Switching to a polynomial example, we encounter RootOf and allvalues:
> f:=x->3*x^5-4*x-1;
5
f := x -> 3 x - 4 x - 1
> solve(f(x));
/ 4 3 2 \
-1, RootOf\3 _Z - 3 _Z + 3 _Z - 3 _Z - 1, index = 1/,
/ 4 3 2 \
RootOf\3 _Z - 3 _Z + 3 _Z - 3 _Z - 1, index = 2/,
/ 4 3 2 \
RootOf\3 _Z - 3 _Z + 3 _Z - 3 _Z - 1, index = 3/,
/ 4 3 2 \
RootOf\3 _Z - 3 _Z + 3 _Z - 3 _Z - 1, index = 4/
> RootOf(f(x));
/ 5 \
RootOf\3 _Z - 4 _Z - 1/
> allvalues(%);
/
|
1 1 (1/2) |
-1, - + -- 3 |
4 12 |
|
\
(1/3) (2/3) \ //
/ (1/2)\ / (1/2)\ | ||
-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| 1 ||
-------------------------------------------------------------|^(1/2) + -- ||
(1/3) | 12 ||
/ (1/2)\ | ||
\140 + 4 2089 / / \\
/
(1/3) |
/ (1/2)\ |
-30 \140 + 4 2089 / |
|
|
\
(1/3) (2/3) \ /
/ (1/2)\ / (1/2)\ | |
-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| |
-------------------------------------------------------------|^(1/2) - 6 |
(1/3) | |
/ (1/2)\ | |
\140 + 4 2089 / / \
(1/3) (2/3) \
/ (1/2)\ / (1/2)\ |
-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48|
-------------------------------------------------------------|^(1/2)
(1/3) |
/ (1/2)\ |
\140 + 4 2089 / /
(2/3)
/ (1/2)\
\140 + 4 2089 /
/ (1/3) (2/3) \
| / (1/2)\ / (1/2)\ |
|-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48|
+ 144 |-------------------------------------------------------------|^(1/2)
| (1/3) |
| / (1/2)\ |
\ \140 + 4 2089 / /
\// /
(1/3)| | (1/3) |
(1/2) / (1/2)\ | |/ (1/2)\ |
+ 90 3 \140 + 4 2089 / | |\140 + 4 2089 / |
| | |
| | |
/ \ \
(1/3) (2/3) \ \\
/ (1/2)\ / (1/2)\ | ||
-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| ||
-------------------------------------------------------------|^(1/2)||^(1/2),
(1/3) | ||
/ (1/2)\ | ||
\140 + 4 2089 / / //
/ (1/3) (2/3)
| / (1/2)\ / (1/2)\
1 1 (1/2) |-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48
- + -- 3 |-------------------------------------------------------------
4 12 | (1/3)
| / (1/2)\
\ \140 + 4 2089 /
\ //
| ||
| 1 ||
|^(1/2) - -- ||
| 12 ||
| ||
/ \\
/
(1/3) |
/ (1/2)\ |
-30 \140 + 4 2089 / |
|
|
\
(1/3) (2/3) \ /
/ (1/2)\ / (1/2)\ | |
-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| |
-------------------------------------------------------------|^(1/2) - 6 |
(1/3) | |
/ (1/2)\ | |
\140 + 4 2089 / / \
(1/3) (2/3) \
/ (1/2)\ / (1/2)\ |
-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48|
-------------------------------------------------------------|^(1/2)
(1/3)
| / (1/2)\ |
\140 + 4 2089 / /
(2/3)
/ (1/2)\
\140 + 4 2089 /
/ (1/3) (2/3) \
| / (1/2)\ / (1/2)\ |
|-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48|
+ 144 |-------------------------------------------------------------|^(1/2)
| (1/3) |
| / (1/2)\ |
\ \140 + 4 2089 / /
\// /
(1/3)| | (1/3) |
(1/2) / (1/2)\ | |/ (1/2)\ |
+ 90 3 \140 + 4 2089 / | |\140 + 4 2089 / |
| | |
| | |
/ \ \
(1/3) (2/3) \ \\
/ (1/2)\ / (1/2)\ | ||
-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| ||
-------------------------------------------------------------|^(1/2)||^(1/2),
(1/3) | ||
/ (1/2)\ | ||
\140 + 4 2089 / / //
/ (1/3) (2/3)
| / (1/2)\ / (1/2)\
1 1 (1/2) |-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48
- - -- 3 |-------------------------------------------------------------
4 12 | (1/3)
| / (1/2)\
\ \140 + 4 2089 /
\ // /
| || (1/3) |
| 1 || / (1/2)\ |
|^(1/2) + -- I ||30 \140 + 4 2089 / |
| 12 || |
| || |
/ \\ \
(1/3) (2/3) \ /
/ (1/2)\ / (1/2)\ | |
-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| |
-------------------------------------------------------------|^(1/2) + 6 |
(1/3) | |
/ (1/2)\ | |
\140 + 4 2089 / / \
(1/3) (2/3) \
/ (1/2)\ / (1/2)\ |
-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48|
-------------------------------------------------------------|^(1/2)
(1/3) |
/ (1/2)\ |
\140 + 4 2089 / /
(2/3)
/ (1/2)\
\140 + 4 2089 /
/ (1/3) (2/3) \
| / (1/2)\ / (1/2)\ |
|-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48|
- 144 |-------------------------------------------------------------|^(1/2)
| (1/3) |
| / (1/2)\ |
\ \140 + 4 2089 / /
\// /
(1/3)| | (1/3) |
(1/2) / (1/2)\ | |/ (1/2)\ |
+ 90 3 \140 + 4 2089 / | |\140 + 4 2089 / |
| | |
| | |
/ \ \
(1/3) (2/3) \ \\
/ (1/2)\ / (1/2)\ | ||
-5 \140 + 4 2089 / + 2 \140 + 4 2089 / - 48| ||
-------------------------------------------------------------|^(1/2)||^(1/2),
(1/3) | ||
/ (1/2)\ | ||
\140 + 4 2089 / / //
/ ...