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## 20 Nonlinear Equations

Octave can solve sets of nonlinear equations of the form

         F (x) = 0

using the function `fsolve`, which is based on the Minpack subroutine `hybrd`.

— Loadable Function: [x, info, msg] = fsolve (fcn, x0)

> Given fcn, the name of a function of the form `f (`x`)` and an initial starting point x0, `fsolve` solves the set of equations such that `f(`x`) == 0`.

> If fcn is a two-element string array, or a two element cell array containing either the function name or inline or function handle. The first element names the function f described above, and the second element names a function of the form `j (`x`)` to compute the Jacobian matrix with elements

>                          df_i
>               jac(i,j) = ----
>                          dx_j

> You can use the function `fsolve_options` to set optional parameters for `fsolve`.

— Loadable Function: fsolve_options (opt, val)

> When called with two arguments, this function allows you set options parameters for the function `fsolve`. Given one argument, `fsolve_options` returns the value of the corresponding option. If no arguments are supplied, the names of all the available options and their current values are displayed.

> Options include

> `"tolerance"`
>   Nonnegative relative tolerance.

Here is a complete example. To solve the set of equations

         -2x^2 + 3xy   + 4 sin(y) = 6
          3x^2 - 2xy^2 + 3 cos(x) = -4

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

         function y = f (x)
           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

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

         [x, info] = fsolve ("f", [1; 2])

results in the solution

         x =

           0.57983
           2.54621

         info = 1

A value of `info = 1` indicates that the solution has converged.

The function `perror` may be used to print English messages corresponding to the numeric error codes. For example,

         perror ("fsolve", 1)
              -| solution converged to requested tolerance