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

@menu
* Solvers::
* Minimizers::
@end menu

@node Solvers
@section Solvers

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 must be provided.  This also has the consequence that
convergence is not guaranteed even if a solution exists.

@c fsolve scripts/optimization/fsolve.m
@anchor{XREFfsolve}
@deftypefn  {} {} fsolve (@var{fcn}, @var{x0}, @var{options})
@deftypefnx {} {[@var{x}, @var{fvec}, @var{info}, @var{output}, @var{fjac}] =} fsolve (@var{fcn}, @dots{})
Solve a system of nonlinear equations defined by the function @var{fcn}.

@var{fcn} should accept a vector (array) defining the unknown variables,
and return a vector of left-hand sides of the equations.  Right-hand sides
are defined to be zeros.  In other words, this function attempts to
determine a vector @var{x} such that @code{@var{fcn} (@var{x})} gives
(approximately) all zeros.

@var{x0} determines a starting guess.  The shape of @var{x0} is preserved
in all calls to @var{fcn}, but otherwise it is treated as a column vector.

@var{options} is a structure specifying additional options.  Currently,
@code{fsolve} recognizes these options:
@qcode{"FunValCheck"}, @qcode{"OutputFcn"}, @qcode{"TolX"},
@qcode{"TolFun"}, @qcode{"MaxIter"}, @qcode{"MaxFunEvals"},
@qcode{"Jacobian"}, @qcode{"Updating"}, @qcode{"ComplexEqn"}
@qcode{"TypicalX"}, @qcode{"AutoScaling"} and @qcode{"FinDiffType"}.

If @qcode{"Jacobian"} is @qcode{"on"}, it specifies that @var{fcn}, called
with 2 output arguments also returns the Jacobian matrix of right-hand sides
at the requested point.  @qcode{"TolX"} specifies the termination tolerance
in the unknown variables, while @qcode{"TolFun"} is a tolerance for
equations.  Default is @code{1e-7} for both @qcode{"TolX"} and
@qcode{"TolFun"}.

If @qcode{"AutoScaling"} is on, the variables will be automatically scaled
according to the column norms of the (estimated) Jacobian.  As a result,
TolF becomes scaling-independent.  By default, this option is off because
it may sometimes deliver unexpected (though mathematically correct) results.

If @qcode{"Updating"} is @qcode{"on"}, the function will attempt to use
@nospell{Broyden} updates to update the Jacobian, in order to reduce the
amount of Jacobian calculations.  If your user function always calculates
the Jacobian (regardless of number of output arguments) then this option
provides no advantage and should be set to false.

@qcode{"ComplexEqn"} is @qcode{"on"}, @code{fsolve} will attempt to solve
complex equations in complex variables, assuming that the equations possess
a complex derivative (i.e., are holomorphic).  If this is not what you want,
you should unpack the real and imaginary parts of the system to get a real
system.

For description of the other options, see @code{optimset}.

On return, @var{fval} contains the value of the function @var{fcn}
evaluated at @var{x}.

@var{info} may be one of the following values:

@table @asis
@item 1
Converged to a solution point.  Relative residual error is less than
specified by TolFun.

@item 2
Last relative step size was less that TolX.

@item 3
Last relative decrease in residual was less than TolF.

@item 0
Iteration limit exceeded.

@item -3
The trust region radius became excessively small.
@end table

Note: If you only have a single nonlinear equation of one variable, using
@code{fzero} is usually a much better idea.

Note about user-supplied Jacobians:
As an inherent property of the algorithm, a Jacobian is always requested for
a solution vector whose residual vector is already known, and it is the last
accepted successful step.  Often this will be one of the last two calls, but
not always.  If the savings by reusing intermediate results from residual
calculation in Jacobian calculation are significant, the best strategy is to
employ OutputFcn: After a vector is evaluated for residuals, if OutputFcn is
called with that vector, then the intermediate results should be saved for
future Jacobian evaluation, and should be kept until a Jacobian evaluation
is requested or until OutputFcn is called with a different vector, in which
case they should be dropped in favor of this most recent vector.  A short
example how this can be achieved follows:

@example
function [fvec, fjac] = user_func (x, optimvalues, state)
persistent sav = [], sav0 = [];
if (nargin == 1)
  ## evaluation call
  if (nargout == 1)
    sav0.x = x; # mark saved vector
    ## calculate fvec, save results to sav0.
  elseif (nargout == 2)
    ## calculate fjac using sav.
  endif
else
  ## outputfcn call.
  if (all (x == sav0.x))
    sav = sav0;
  endif
  ## maybe output iteration status, etc.
endif
endfunction

## @dots{}

fsolve (@@user_func, x0, optimset ("OutputFcn", @@user_func, @dots{}))
@end example
@seealso{@ref{XREFfzero,,fzero}, @ref{XREFoptimset,,optimset}}
@end deftypefn


The following 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.

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.

@c fzero scripts/optimization/fzero.m
@anchor{XREFfzero}
@deftypefn  {} {} fzero (@var{fun}, @var{x0})
@deftypefnx {} {} fzero (@var{fun}, @var{x0}, @var{options})
@deftypefnx {} {[@var{x}, @var{fval}, @var{info}, @var{output}] =} fzero (@dots{})
Find a zero of a univariate function.

@var{fun} is a function handle, inline function, or string containing the
name of the function to evaluate.

@var{x0} should be a two-element vector specifying two points which
bracket a zero.  In other words, there must be a change in sign of the
function between @var{x0}(1) and @var{x0}(2).  More mathematically, the
following must hold

@example
sign (@var{fun}(@var{x0}(1))) * sign (@var{fun}(@var{x0}(2))) <= 0
@end example

If @var{x0} is a single scalar then several nearby and distant values are
probed in an attempt to obtain a valid bracketing.  If this is not
successful, the function fails.

@var{options} is a structure specifying additional options.  Currently,
@code{fzero} recognizes these options:
@qcode{"FunValCheck"}, @qcode{"OutputFcn"}, @qcode{"TolX"},
@qcode{"MaxIter"}, @qcode{"MaxFunEvals"}.
For a description of these options, see @ref{XREFoptimset,,optimset}.

On exit, the function returns @var{x}, the approximate zero point and
@var{fval}, the function value thereof.

@var{info} is an exit flag that can have these values:

@itemize
@item 1
 The algorithm converged to a solution.

@item 0
 Maximum number of iterations or function evaluations has been reached.

@item -1
The algorithm has been terminated from user output function.

@item -5
The algorithm may have converged to a singular point.
@end itemize

@var{output} is a structure containing runtime information about the
@code{fzero} algorithm.  Fields in the structure are:

@itemize
@item iterations
 Number of iterations through loop.

@item @nospell{nfev}
 Number of function evaluations.

@item bracketx
 A two-element vector with the final bracketing of the zero along the
x-axis.

@item brackety
 A two-element vector with the final bracketing of the zero along the
y-axis.
@end itemize
@seealso{@ref{XREFoptimset,,optimset}, @ref{XREFfsolve,,fsolve}}
@end deftypefn


@node Minimizers
@section Minimizers
@cindex local minimum
@cindex finding minimums

Often it is useful to find the minimum value of a function rather than just
the zeroes where it crosses the x-axis.  @code{fminbnd} is designed for the
simpler, but very common, case of a univariate function where the interval
to search is bounded.  For unbounded minimization of a function with
potentially many variables use @code{fminunc} or @code{fminsearch}.  The two
functions use different internal algorithms and some knowledge of the objective
function is required.  For functions which can be differentiated,
@code{fminunc} is appropriate.  For functions with discontinuities, or for
which a gradient search would fail, use @code{fminsearch}.
@xref{Optimization}, for minimization with the presence of constraint
functions.  Note that searches can be made for maxima by simply inverting the
objective function
@tex
($F_{max} = -F_{min}$).
@end tex
@ifnottex
(@code{Fto_max = -Fto_min}).
@end ifnottex

@c fminbnd scripts/optimization/fminbnd.m
@anchor{XREFfminbnd}
@deftypefn {} {[@var{x}, @var{fval}, @var{info}, @var{output}] =} fminbnd (@var{fun}, @var{a}, @var{b}, @var{options})
Find a minimum point of a univariate function.

@var{fun} should be a function handle or name.  @var{a}, @var{b} specify a
starting interval.  @var{options} is a structure specifying additional
options.  Currently, @code{fminbnd} recognizes these options:
@qcode{"FunValCheck"}, @qcode{"OutputFcn"}, @qcode{"TolX"},
@qcode{"MaxIter"}, @qcode{"MaxFunEvals"}.  For a description of these
options, see @ref{XREFoptimset,,optimset}.

On exit, the function returns @var{x}, the approximate minimum point and
@var{fval}, the function value thereof.

@var{info} is an exit flag that can have these values:

@itemize
@item 1
The algorithm converged to a solution.

@item 0
Maximum number of iterations or function evaluations has been exhausted.

@item -1
The algorithm has been terminated from user output function.
@end itemize

Notes: The search for a minimum is restricted to be in the interval bound by
@var{a} and @var{b}.  If you only have an initial point to begin searching
from you will need to use an unconstrained minimization algorithm such as
@code{fminunc} or @code{fminsearch}.  @code{fminbnd} internally uses a
Golden Section search strategy.
@seealso{@ref{XREFfzero,,fzero}, @ref{XREFfminunc,,fminunc}, @ref{XREFfminsearch,,fminsearch}, @ref{XREFoptimset,,optimset}}
@end deftypefn


@c fminunc scripts/optimization/fminunc.m
@anchor{XREFfminunc}
@deftypefn  {} {} fminunc (@var{fcn}, @var{x0})
@deftypefnx {} {} fminunc (@var{fcn}, @var{x0}, @var{options})
@deftypefnx {} {[@var{x}, @var{fval}, @var{info}, @var{output}, @var{grad}, @var{hess}] =} fminunc (@var{fcn}, @dots{})
Solve an unconstrained optimization problem defined by the function
@var{fcn}.

@var{fcn} should accept a vector (array) defining the unknown variables, and
return the objective function value, optionally with gradient.
@code{fminunc} attempts to determine a vector @var{x} such that
@code{@var{fcn} (@var{x})} is a local minimum.

@var{x0} determines a starting guess.  The shape of @var{x0} is preserved in
all calls to @var{fcn}, but otherwise is treated as a column vector.

@var{options} is a structure specifying additional options.  Currently,
@code{fminunc} recognizes these options:
@qcode{"FunValCheck"}, @qcode{"OutputFcn"}, @qcode{"TolX"},
@qcode{"TolFun"}, @qcode{"MaxIter"}, @qcode{"MaxFunEvals"},
@qcode{"GradObj"}, @qcode{"FinDiffType"}, @qcode{"TypicalX"},
@qcode{"AutoScaling"}.

If @qcode{"GradObj"} is @qcode{"on"}, it specifies that @var{fcn}, when
called with two output arguments, also returns the Jacobian matrix of
partial first derivatives at the requested point.  @code{TolX} specifies
the termination tolerance for the unknown variables @var{x}, while
@code{TolFun} is a tolerance for the objective function value @var{fval}.
 The default is @code{1e-7} for both options.

For a description of the other options, see @code{optimset}.

On return, @var{x} is the location of the minimum and @var{fval} contains
the value of the objective function at @var{x}.

@var{info} may be one of the following values:

@table @asis
@item 1
Converged to a solution point.  Relative gradient error is less than
specified by @code{TolFun}.

@item 2
Last relative step size was less than @code{TolX}.

@item 3
Last relative change in function value was less than @code{TolFun}.

@item 0
Iteration limit exceeded---either maximum number of algorithm iterations
@code{MaxIter} or maximum number of function evaluations @code{MaxFunEvals}.

@item -1
Algorithm terminated by @code{OutputFcn}.

@item -3
The trust region radius became excessively small.
@end table

Optionally, @code{fminunc} can return a structure with convergence
statistics (@var{output}), the output gradient (@var{grad}) at the
solution @var{x}, and approximate Hessian (@var{hess}) at the solution
@var{x}.

Application Notes: If the objective function is a single nonlinear equation
of one variable then using @code{fminbnd} is usually a better choice.

The algorithm used by @code{fminunc} is a gradient search which depends
on the objective function being differentiable.  If the function has
discontinuities it may be better to use a derivative-free algorithm such as
@code{fminsearch}.
@seealso{@ref{XREFfminbnd,,fminbnd}, @ref{XREFfminsearch,,fminsearch}, @ref{XREFoptimset,,optimset}}
@end deftypefn


@c fminsearch scripts/optimization/fminsearch.m
@anchor{XREFfminsearch}
@deftypefn  {} {@var{x} =} fminsearch (@var{fun}, @var{x0})
@deftypefnx {} {@var{x} =} fminsearch (@var{fun}, @var{x0}, @var{options})
@deftypefnx {} {@var{x} =} fminsearch (@var{fun}, @var{x0}, @var{options}, @var{fun_arg1}, @var{fun_arg2}, @dots{})
@deftypefnx {} {[@var{x}, @var{fval}, @var{exitflag}, @var{output}] =} fminsearch (@dots{})

Find a value of @var{x} which minimizes the function @var{fun}.

The search begins at the point @var{x0} and iterates using the
@nospell{Nelder & Mead} Simplex algorithm (a derivative-free method).  This
algorithm is better-suited to functions which have discontinuities or for
which a gradient-based search such as @code{fminunc} fails.

Options for the search are provided in the parameter @var{options} using the
function @code{optimset}.  Currently, @code{fminsearch} accepts the options:
@qcode{"TolX"}, @qcode{"TolFun"}, @qcode{"MaxFunEvals"}, @qcode{"MaxIter"},
@qcode{"Display"}, @qcode{"FunValCheck"}, and @qcode{"OutputFcn"}.
For a description of these options, see @code{optimset}.

Additional inputs for the function @var{fun} can be passed as the fourth
and higher arguments.  To pass function arguments while using the default
@var{options} values, use @code{[]} for @var{options}.

On exit, the function returns @var{x}, the minimum point, and @var{fval},
the function value at the minimum.

The third return value @var{exitflag} is

@table @asis
@item 1
if the algorithm converged
(size of the simplex is smaller than @code{@var{options}.TolX} @strong{AND}
the step in the function value between iterations is smaller than
@code{@var{options}.TolFun}).

@item 0
if the maximum number of iterations or the maximum number of function
evaluations are exceeded.

@item -1
if the iteration is stopped by the @qcode{"OutputFcn"}.
@end table

The fourth return value is a structure @var{output} with the fields,
@code{funcCount} containing the number of function calls to @var{fun},
@code{iterations} containing the number of iteration steps,
@code{algorithm} with the name of the search algorithm (always:
@nospell{@qcode{"Nelder-Mead simplex direct search"}}), and @code{message}
with the exit message.

Example:

@example
fminsearch (@@(x) (x(1)-5).^2+(x(2)-8).^4, [0;0])
@end example
@seealso{@ref{XREFfminbnd,,fminbnd}, @ref{XREFfminunc,,fminunc}, @ref{XREFoptimset,,optimset}}
@end deftypefn


The function @code{humps} is a useful function for testing zero and
extrema finding functions.

@c humps scripts/optimization/humps.m
@anchor{XREFhumps}
@deftypefn  {} {@var{y} =} humps (@var{x})
@deftypefnx {} {[@var{x}, @var{y}] =} humps (@var{x})
Evaluate a function with multiple minima, maxima, and zero crossings.

The output @var{y} is the evaluation of the rational function:

@tex
$$y = -{ {1200x^4 - 2880x^3 + 2036x^2 - 340x - 88} \over {200x^4 - 480x^3 + 406x^2 - 138x - 17} }$$
@end tex
@ifnottex

@example
@group
        1200*@var{x}^4 - 2880*@var{x}^3 + 2036*@var{x}^2 - 348*@var{x} - 88
 @var{y} = - ---------------------------------------------
         200*@var{x}^4 - 480*@var{x}^3 + 406*@var{x}^2 - 138*@var{x} + 17
@end group
@end example

@end ifnottex

@var{x} may be a scalar, vector or array.  If @var{x} is omitted, the
default range [0:0.05:1] is used.

When called with two output arguments, [@var{x}, @var{y}], @var{x} will
contain the input values, and @var{y} will contain the output from
@code{humps}.

Programming Notes: @code{humps} has two local maxima located near @var{x} =
0.300 and 0.893, a local minimum near @var{x} = 0.637, and zeros near
@var{x} = -0.132 and 1.300.  @code{humps} is a useful function for testing
algorithms which find zeros or local minima and maxima.

Try @code{demo humps} to see a plot of the @code{humps} function.
@seealso{@ref{XREFfzero,,fzero}, @ref{XREFfminbnd,,fminbnd}, @ref{XREFfminunc,,fminunc}, @ref{XREFfminsearch,,fminsearch}}
@end deftypefn