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@node Nonlinear Equations
@chapter Nonlinear Equations
@cindex nonlinear equations
@cindex equations, nonlinear

Octave can solve sets of nonlinear equations of the form
@tex
$$
 f (x) = 0
$$
@end tex
@ifnottex

@example
F (x) = 0
@end example

@end ifnottex

@noindent
using the function @code{fsolve}, which is based on the @sc{Minpack}
subroutine @code{hybrd}.  This is an iterative technique so a starting
point will have to be provided.  This also has the consequence that
convergence is not guaranteed even if a solution exists.

@DOCSTRING(fsolve)

Here is a complete example.  To solve the set of equations
@tex
$$
 \eqalign{-2x^2 + 3xy + 4\sin(y) - 6 &= 0\cr
           3x^2 - 2xy^2 + 3\cos(x) + 4 &= 0}
$$
@end tex
@ifnottex

@example
@group
-2x^2 + 3xy   + 4 sin(y) = 6
 3x^2 - 2xy^2 + 3 cos(x) = -4
@end group
@end example

@end ifnottex

@noindent
you first need to write a function to compute the value of the given
function.  For example:

@example
@group
function y = f (x)
  y = zeros (2, 1);
  y(1) = -2*x(1)^2 + 3*x(1)*x(2)   + 4*sin(x(2)) - 6;
  y(2) =  3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4;
endfunction
@end group
@end example

Then, call @code{fsolve} with a specified initial condition to find the
roots of the system of equations.  For example, given the function
@code{f} defined above,

@example
[x, fval, info] = fsolve (@@f, [1; 2])
@end example

@noindent
results in the solution

@example
@group
x =

  0.57983
  2.54621

fval =

  -5.7184e-10
   5.5460e-10

info = 1
@end group
@end example

@noindent
A value of @code{info = 1} indicates that the solution has converged.

The function @code{perror} may be used to print English messages
corresponding to the numeric error codes.  For example:

@example
@group
perror ("fsolve", 1)
     @print{} solution converged to requested tolerance
@end group
@end example

When no Jacobian is supplied (as in the example above) it is approximated
numerically.  This requires more function evaluations, and hence is
less efficient.  In the example above we could compute the Jacobian 
analytically as

@iftex
@tex
$$
\left[\matrix{ {\partial f_1 \over \partial x_1} &
               {\partial f_1 \over \partial x_2} \cr
               {\partial f_2 \over \partial x_1} &
               {\partial f_2 \over \partial x_2} \cr}\right] =
\left[\matrix{ 3 x_2 - 4 x_1                  &
               4 \cos(x_2) + 3 x_1            \cr
               -2 x_2^2 - 3 \sin(x_1) + 6 x_1 &
               -4 x_1 x_2                     \cr }\right]
$$
@end tex
and compute it with the following Octave function
@end iftex

@example
@group
function [y, jac] = f (x)
  y = zeros (2, 1);
  y(1) = -2*x(1)^2 + 3*x(1)*x(2)   + 4*sin(x(2)) - 6;
  y(2) =  3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4;
  if (nargout == 2)
    jac = zeros (2, 2);
    jac(1,1) =  3*x(2) - 4*x(1);
    jac(1,2) =  4*cos(x(2)) + 3*x(1);
    jac(2,1) = -2*x(2)^2 - 3*sin(x(1)) + 6*x(1);
    jac(2,2) = -4*x(1)*x(2);
  endif
endfunction
@end group
@end example

@noindent
The Jacobian can then be used with the following call to @code{fsolve}:

@example
[x, fval, info] = fsolve (@@f, [1; 2], optimset ("jacobian", "on"));
@end example

@noindent
which gives the same solution as before.

@DOCSTRING(fzero)
