@c This file is generated automatically by the packages munge-texi.pl.

\input texinfo
@c %**start of header
@setfilename optim.info
@settitle optim_doc
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@c Nowadays the predined function index has entries for each @deftypefn
@c in additiont to each @findex. This would hopelessly clutter the index
@c here.
@defcodeindex mfn

@copying
Additional documentation for the optim package for Octave.

Copyright @copyright{} @email{Olaf Till <i7tiol@@t-online.de>}

You can redistribute this documentation and/or modify it under the terms
of the GNU General Public License as published by the Free Software
Foundation; either version 3 of the License, or (at your option) any
later version.

This documentation is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
Public License for more details.

You should have received a copy of the GNU General Public License along
with this documentation; if not, see <http://www.gnu.org/licenses/>.
@end copying

@include macros.texi

@macro mysee
@ifhtml
see
@end ifhtml
@end macro

@titlepage
@title Additional documentation for the optim package for Octave
@page
@vskip 0pt plus 1filll
@insertcopying
@end titlepage

@c No table of contents. The table would occupy most of the top node in
@c html and IMHO misleads the user to use the table instead of the menu
@c structure of the nodes, which would let some information unused.
@c
@c @contents

@c ------------------------------------------------------------------

@node Top
@top Additional documentation for the optim package for Octave

@ifhtml
The info version of this document is accessible, after package
installation, from the Octave commandline with @code{optim_doc()}.
@end ifhtml

This documentation applies to version 1.6.0 of the optim
package.

The optim package is a collection of additional functions related to
numerical optimization.  For Octaves core optimization functions (not
contained in this package)
@mysee
@ref{Optimization,,,octave}.

@menu
Types of functions in the optim package
* Scalar optimization::         Functions for optimization of a scalar
                                  objective function.
* Residual optimization::       Functions for optimization of a model
                                  function returning an array.
* Zero finders::                Functions for finding the zero of a
                                  scalar or array valued nonlinear user
                                  function.
* Gradient functions::          Functions for numerical approximation of
                                  gradients and Hessians.
* Helper functions::            Functions for algebraic tasks common to
                                  optimization problems.
* Documentation::               Function optim_doc to view documentation.

* Compatibility functions::     Functions with traditional names and
                                  arguments which provide a subset of
                                  the functionality of other functions.

Configuration
* Common frontend options::     Options common to all frontends.
* Common optimization options:: Options common to all optimization
                                  frontends.

* Parameter structures::        Handling of structures of optimized
                                  parameters.
* Additional parameters::       Passing additional parameters to user
                                  functions.

Indices
* Function index::              Index of functions in optim.
* Concept index::               Concept index.
@end menu

@c ------------------------------------------------------------------

@node Scalar optimization
@chapter Functions for optimization of a scalar objective function
@cindex scalar optimization

@menu
Frontend
* nonlin_min::                 Interface for scalar non-linear
                                 optimization.

Backends
* lm_feasible::                L/M-like optimizer, constraints met
                                 throughout optimization.
* octave_sqp::                 A wrapper to core Octaves sqp function.
* siman::                      Simulated annealing with constraints.
* samin::                      Simulated annealing covering a range.
* d2_min::                     Newton-like optimizer, no constraints.

Alternative frontend
* fmins::                      Frontend for direct search
                               (gradient-less) algorithms.

Standalone functions
* nmsmax::                     A Nelder-Mead simplex algorithm.
* mdsmax::                     A multidirectional search algorithm.
* adsmax::                     An alternating directions algorithm.
* nelder_mead_min::            Another Nelder-Mead algorithm.
* powell::                     Direction-set (Powell's) method.
* bfgsmin::                    Unconstrained BFGS algorithm.
* nrm::                        Newton-Raphson algorithm.
* cg_min::                     A conjugate gradient method.
* brent_line_min::             Linesearch, Brent method.
* line_min::                   Linesearch (minimize a function along dx).
* de_min::                     A differential evolution algorithm.
* battery::                    Repeatedly call bfgsmin.
@end menu

@c ------------------------------------------------------------------

@node nonlin_min
@section Frontend nonlin_min for scalar non-linear optimization
@mfnindex nonlin_min

@c nonlin_min ../inst/nonlin_min.m
@anchor{XREFnonlin_min}
@deftypefn {Function File} {[@var{p}, @var{objf}, @var{cvg}, @var{outp}] =} nonlin_min (@var{f}, @var{pin})
@deftypefnx {Function File} {[@var{p}, @var{objf}, @var{cvg}, @var{outp}] =} nonlin_min (@var{f}, @var{pin}, @var{settings})
Frontend for nonlinear minimization of a scalar objective function.

The functions supplied by the user have a minimal interface; any
additionally needed constants can be supplied by wrapping the user
functions into anonymous functions.

The following description applies to usage with vector-based
parameter handling. Differences in usage for structure-based
parameter handling will be explained separately.

@var{f}: objective function. It gets a column vector of real
parameters as argument. In gradient determination, this function may
be called with an informational second argument, whose content
depends on the function for gradient determination.

@var{pin}: real column vector of initial parameters.

@var{settings}: structure whose fields stand for optional settings
referred to below. The fields can be set by @code{optimset()}.

The returned values are the column vector of final parameters
@var{p}, the final value of the objective function @var{objf}, an
integer @var{cvg} indicating if and how optimization succeeded or
failed, and a structure @var{outp} with additional information,
curently with possible fields: @code{niter}, the number of
iterations, @code{nobjf}, the number of objective function calls
(indirect calls by gradient function not counted), @code{lambda}, the
lambda of constraints at the result, and @code{user_interaction},
information on user stops (see settings). The backend may define
additional fields. @var{cvg} is greater than zero for success and
less than or equal to zero for failure; its possible values depend on
the used backend and currently can be @code{0} (maximum number of
iterations exceeded), @code{1} (success without further specification
of criteria), @code{2} (parameter change less than specified
precision in two consecutive iterations), @code{3} (improvement in
objective function less than specified), @code{-1} (algorithm aborted
by a user function), or @code{-4} (algorithm got stuck).

@c The following block will be cut out in the package info file.

@end deftypefn


@subheading Settings

The fields of the @var{settings} structure can be set with
@ref{XREFoptimset,,optimset,octave}.

For settings common to all frontends (including these for statistics)
@mysee
@ref{Common frontend options}.

For additional settings common to all optimization frontends
@mysee
@ref{Common optimization options}.

@subsubheading Specific defaults:

@multitable {@code{Algorithm}} {"lm_feasible"}
@item @code{Algorithm}:
@tab "lm_feasible"
@end multitable

@subsubheading Additional settings:

@table @code
@item objf_grad
Function computing the gradient of the objective function with respect
to the parameters.  Default: real finite differences.  Will be called
with the column vector of parameters and an informational structure as
arguments.  If @code{dfdp} was specified by the user, the informational
structure has the fields @code{f}: value of objective function for
current parameters, @code{fixed}: logical vector indicating which
parameters are not optimized, so these partial derivatives need not be
computed and can be set to zero, @code{diffp}, @code{diff_onesided},
@code{lbound}, @code{ubound}: identical to the user settings of this
name, @code{plabels}: 1-dimensional cell-array of column-cell-arrays,
each column with labels for all parameters; the first column contains
the numerical indices of the parameters; the second and third columns,
present for structure based parameter handling,
@mysee
@ref{Parameter structures},
contain the names of the parameters and the subindices of
the parameters,
@mysee
@ref{Non-scalar parameters}, respectively.  The default
gradient function will call the objective function with the second
argument set with fields @code{f}: as the @code{f} passed to the
gradient function, @code{plabels}: cell-array of 1x1 cell-arrays with
the entries of the column-cell-arrays of @code{plabels} as passed to the
jacobian function corresponding to current parameter, @code{side}:
@code{0} for one-sided interval, @code{1} or @code{2}, respectively, for
the sides of a two-sided interval, and @code{parallel}: logical scalar
indicating parallel computation of partial derivatives.  This
information can be useful if the model function can omit some
computations depending on the currently computed partial derivative.
@item objf_hessian
Function computing the Hessian of the objective function with respect to
the parameters.  The default is backend specific.  Will be called with
the column vector of parameters as argument.
@item inverse_hessian
Logical scalar, indicating whether the Hessian function passed by the
user actually returns the inverse of the Hessian.
@item complex_step_derivative_objf
Logical scalar, default: @code{false}.  Estimate gradient of objective
function with complex step derivative approximation.  Use only if you
know that your objective function is suitable for this.  No user
function for the gradient (@code{objf_grad}) must be specified.
@item save_state
String with path to a file which will be created for periodical saving
of the state of optimization.  Useful for very long optimizations which
might get interrupted.  The format of the saved state will be
backend-specific.  Currently, only the @qcode{"siman"} backend honours
this option.  Default: empty string, meaning no saving of state.
@item recover_state
String with path to a file created due to option @code{save_state},
which is used to recover a saved state before starting optimization.
Default: empty string, meaning no recovering of state.
@end table

@subheading Structure based parameter handling

Please
@mysee
@ref{Parameter structures}.

@subheading Backend information

Please
@mysee
@ref{Scalar optimization} and choose backend from menu.

@c ------------------------------------------------------------------

@node lm_feasible
@section Default backend lm_feasible of scalar optimization
@cindex lm_feasible

A Levenberg/Marquardt-like algorithm, attempting to honour constraints
throughout the course of optimization.  This means that the initial
parameters must not violate constraints (to find an initial feasible set
of parameters, e.g. core Octaves @code{sqp} can be used
(
@mysee
@ref{octave_sqp}), by specifying an objective function which is
constant or which returns a norm of the distances to the initial
values).  The Hessian is either supplied by the user or is approximated
by the BFGS algorithm.  Core Octaves @code{sqp} performed better in some
tests with @emph{unconstrained} problems.

Returned value @var{cvg} will be @code{2} or @code{3} for success and
@code{0} or @code{-4} for failure (
@mysee
@ref{nonlin_min} for
meaning). Returned structure @var{outp} will have the fields
@code{niter}, @code{nobjf}, and @code{user_interaction}.

Backend-specific defaults are: @code{MaxIter}: 20, @code{fract_prec}:
@code{zeros (size (parameters))}, @code{max_fract_change}: @code{Inf}
for all parameters. The setting @code{TolX} is not honoured.

Interpretation of @code{Display}: if set to @qcode{"iter"}, currently
only information on applying @code{max_fract_change} is printed.

@c ------------------------------------------------------------------

@node octave_sqp
@section Backend wrapping core Octaves sqp function
@cindex octave_sqp

This backend calls the @code{sqp} function of core Octave, a sequential
quadratic programming algorithm with BFGS, so that it is usable with the
@code{nonlin_min} frontend of the optim package. @code{sqp} honours
constraints for the returned result, but not necessarily during the
course of optimization.
 
Compared to calling @code{sqp} directly (
@mysee
@ref{XREFsqp,,sqp,octave}),

@itemize
@item a different default (numerical) gradient function is used (that of
the frontend),
@item bounds are set to @code{+-Inf} by default, not to @code{+-realmax}.
@end itemize

The value of the additional setting @code{octave_sqp_tolerance}, a
tolerance to several termination criteria, is passed as the respective
argument to Octaves @code{sqp}, which provides the default @code{sqrt
(eps)}.  The settings @code{TolFun}, @code{TolX}, @code{fract_prec},
@code{max_fract_change}, @code{Display}, and @code{user_interaction} are
not honoured.

Returned value @var{cvg} will be @code{1} for success and @code{0} or
@code{-4} for failure (
@mysee
@ref{nonlin_min} for meaning).  Returned
structure @var{outp} will have the fields @code{niter}, @code{nobjf},
and @code{lambda}.

The default of @code{MaxIter} is that of Octaves @code{sqp} (100).

@c ------------------------------------------------------------------

@node siman
@section Simulated annealing backend siman of scalar optimization
@cindex siman

A simulated annealing (stochastic) optimizer, changing all parameters at
once in a single step, so being suitable for non-bound constraints.

No gradient or hessian of the objective function is used. The settings
@code{MaxIter}, @code{fract_prec}, @code{TolFun}, @code{TolX}, and
@code{max_fract_change} are not honoured.

Accepts the additional settings @code{T_init} (initial temperature,
default 0.01), @code{T_min} (final temperature, default 1.0e-5),
@code{mu_T} (factor of temperature decrease, default 1.005),
@code{iters_fixed_T} (iterations within one temperature step, default
10), @code{max_rand_step} (column vector or structure-based
configuration of maximum random steps for each parameter, default 0.005
* @var{pin}), @code{stoch_regain_constr} (if @code{true}, regain
constraints after a random step, otherwise take new random value until
constraints are met, default @code{false}), @code{trace_steps} (set
field @code{trace} of @var{outp} with a matrix with a row for each step,
first column iteration number, second column repeat number within
iteration, third column value of objective function, rest columns
parameter values, default @code{false}), and @code{siman_log} (set field
@code{log} of @var{outp} with a matrix with a row for each iteration,
first column temperature, second column value of objective function,
rest columns numbers of tries with decrease, no decrease but accepted,
and no decrease and rejected.

Steps with increase @code{diff} of objective function are accepted if
@code{rand (1) < exp (- diff / T)}, where @code{T} is the temperature of
the current iteration.

If regaining of constraints failed, optimization will be aborted and
returned value of @var{cvg} will be @code{0}. Otherwise, @var{cvg} will
be @code{1}. Returned structure @var{outp}, additionally to the possible
fields @code{trace} and @code{log} described above, will have the fields
@code{niter} and @code{user_interaction}.

Interpretation of @code{Display}: if set to @qcode{"iter"}, an
informational line is printed after each iteration.

If @code{parallel_local} is equivalent to @code{true}, the objective
function is evaluated for several parameter combinations in parallel. If
@code{parallel_local} is set to an integer @code{> 1}, this is the
maximal number of parallel processes; if it is @code{<= 1}, the maximal
number will be the number of available processor cores.  The course of
optimization won't be changed by parallelization, provided the random
number generator starts with the same state.  To achieve this, some of
the parallel results are discarded, causing the speedup to be smaller if
the rate of acceptance of results is high.  Also, due to overhead, there
won't be any speedup, but even a slowdown, if the objective function is
not computationally extensive enough.

Honours options @code{save_state} and @code{recover_state}, described
for the frontend.

@c ------------------------------------------------------------------

@node samin
@section Simulated annealing backend samin of scalar optimization
@cindex samin

A simulated annealing (stochastic) optimizer, changing each parameter
separately, so not suitable for non-bound constraints. Requires
specification of lower and upper bounds for each parameter and attempts
to scan the whole region so defined.

No gradient or hessian of the objective function is used. The settings
@code{fract_prec} and @code{max_fract_change} are not honoured.

Maximal stepwidth is continuously adjusted according to the ratio of
accepted steps. Temperature is increased at the start until accepted
steps cover the whole parameter range, if this is not the case at the
initial temperature.

Accepts the additional settings @code{T_init} (initial temperature,
default 0.1), @code{mu_T} (factor of temperature decrease, default 1.2),
@code{iters_fixed_T} (subiterations within one temperature step, default
100), @code{iters_adjust_step} (number of subiterations after which
maximal stepwidth is adjusted, default 5), @code{niter_check_tolfun}
(number of iterations to compare with if checking the value of the
objective function for convergence, default 5), @code{trace_steps} (set
field @code{trace} of @var{outp} with a matrix with a row for each step,
first column iteration number, second column repeat number within
iteration, third column value of objective function, fourth column
temperature, rest columns parameter values, default @code{false}), and
@code{siman_log} (set field @code{log} of @var{outp} with a matrix with
a row for each iteration, first column temperature, second column value
of objective function, rest columns numbers of tries with decrease, no
decrease but accepted, no decrease and rejected, steps corrected back
into bounds, and steps yielding a new best value of the objective
function).

Steps with increase @code{diff} of objective function are accepted if
@code{rand (1) < exp (- diff / T)}, where @code{T} is the temperature of
the current iteration.

If converged, the returned value of @var{cvg} will be @code{1}, if
@code{MaxIters} is exceeded before convergence, @var{cvg} will be
@code{0}. The returned structure @var{outp}, additionally to the
possible fields @code{trace} and @code{log} described above, will have
the fields @code{niter} and @code{user_interaction}.

Interpretation of @code{Display}: if set to @qcode{"iter"}, 
informational text is printed after each iteration and at the end of
optimization. If set to @qcode{"final"}, informational text is printed
only at the end of optimization.

No parallel execution is performed in this backend.

Options @code{save_state} and @code{recover_state} are not honoured.

@c ------------------------------------------------------------------

@node d2_min
@section Unconstrained Newton-like optimization
@cindex d2_min

This backend features a Newton-like algorithm. The user has to supply a
Hessian function. No constraints are honoured. If the supplied Hessian
function actually returns the inverse of the Hessian, set
@code{inverse_hessian} to @code{true}. Supplying the inverse Hessian is
preferable, if possible.

Returned value @var{cvg} will be @code{2} or @code{3} for success and
@code{0} or @code{-1} for failure (
@mysee
@ref{nonlin_min} for
meaning). Returned structure @var{outp} will have the fields
@code{niter}, @code{nobjf}, and @code{user_interaction}.

Interpretation of @code{Display}: if set to @qcode{"iter"}, some
diagnostics are printed.

@c ------------------------------------------------------------------

@node fmins
@section Alternative frontend for gradient-less algorithms
@mfnindex fmins

@c fmins ../inst/fmins.m
@anchor{XREFfmins}
@deftypefn {Function File} {[@var{x}] =} fmins (@var{f},@var{X0},@var{options},@var{grad},@var{P1},@var{P2}, ...)

Find the minimum of a funtion of several variables.
By default the method used is the Nelder&Mead Simplex algorithm

Example usage:
  fmins(inline('(x(1)-5).^2+(x(2)-8).^4'),[0;0])

@strong{Inputs}
@c use @asis and explicite @var in @table to avoid makeinfo warning
@c `unlikely character , in @var' for `P1, P2, ...'.
@table @asis 
@item @var{f} 
A string containing the name of the function to minimize
@item @var{X0}
A vector of initial parameters fo the function @var{f}.
@item @var{options}
Vector with control parameters (not all parameters are used)
@verbatim
options(1) - Show progress (if 1, default is 0, no progress)
options(2) - Relative size of simplex (default 1e-3)
options(6) - Optimization algorithm
   if options(6)==0 - Nelder & Mead simplex (default)
   if options(6)==1 - Multidirectional search Method
   if options(6)==2 - Alternating Directions search
options(5)
   if options(6)==0 && options(5)==0 - regular simplex
   if options(6)==0 && options(5)==1 - right-angled simplex
      Comment: the default is set to "right-angled simplex".
        this works better for me on a broad range of problems,
        although the default in nmsmax is "regular simplex"
options(10) - Maximum number of function evaluations
@end verbatim
@item @var{grad}
Unused (For compatibility with Matlab)
@item @var{P1}, @var{P2}, ...
Optional parameters for function @var{f} 

@end table
@end deftypefn


@c ------------------------------------------------------------------

@node nmsmax
@section A Nelder-Mead simplex algorithm
@mfnindex nmsmax

This function is deprecated. It is available with a slightly different
interface in core Octave as `fminsearch'.

@subheading Helptext:

@anchor{XREFnmsmax}
@verbatim
NMSMAX  Nelder-Mead simplex method for direct search optimization.
       [x, fmax, nf] = NMSMAX(FUN, x0, STOPIT, SAVIT) attempts to
       maximize the function FUN, using the starting vector x0.
       The Nelder-Mead direct search method is used.
       Output arguments:
              x    = vector yielding largest function value found,
              fmax = function value at x,
              nf   = number of function evaluations.
       The iteration is terminated when either
              - the relative size of the simplex is <= STOPIT(1)
                (default 1e-3),
              - STOPIT(2) function evaluations have been performed
                (default inf, i.e., no limit), or
              - a function value equals or exceeds STOPIT(3)
                (default inf, i.e., no test on function values).
       The form of the initial simplex is determined by STOPIT(4):
          STOPIT(4) = 0: regular simplex (sides of equal length, the default)
          STOPIT(4) = 1: right-angled simplex.
       Progress of the iteration is not shown if STOPIT(5) = 0 (default 1).
          STOPIT(6) indicates the direction (ie. minimization or 
                  maximization.) Default is 1, maximization.
                  set STOPIT(6)=-1 for minimization
       If a non-empty fourth parameter string SAVIT is present, then
       `SAVE SAVIT x fmax nf' is executed after each inner iteration.
       NB: x0 can be a matrix.  In the output argument, in SAVIT saves,
           and in function calls, x has the same shape as x0.
       NMSMAX(fun, x0, STOPIT, SAVIT, P1, P2,...) allows additional
       arguments to be passed to fun, via feval(fun,x,P1,P2,...).
References:
N. J. Higham, Optimization by direct search in matrix computations,
   SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993.
C. T. Kelley, Iterative Methods for Optimization, Society for Industrial
   and Applied Mathematics, Philadelphia, PA, 1999.

@end verbatim

@c ------------------------------------------------------------------

@node mdsmax
@section A multidirectional search algorithm
@mfnindex mdsmax

@subheading Helptext:

@anchor{XREFmdsmax}
@verbatim
MDSMAX  Multidirectional search method for direct search optimization.
       [x, fmax, nf] = MDSMAX(FUN, x0, STOPIT, SAVIT) attempts to
       maximize the function FUN, using the starting vector x0.
       The method of multidirectional search is used.
       Output arguments:
              x    = vector yielding largest function value found,
              fmax = function value at x,
              nf   = number of function evaluations.
       The iteration is terminated when either
              - the relative size of the simplex is <= STOPIT(1)
                (default 1e-3),
              - STOPIT(2) function evaluations have been performed
                (default inf, i.e., no limit), or
              - a function value equals or exceeds STOPIT(3)
                (default inf, i.e., no test on function values).
       The form of the initial simplex is determined by STOPIT(4):
         STOPIT(4) = 0: regular simplex (sides of equal length, the default),
         STOPIT(4) = 1: right-angled simplex.
       Progress of the iteration is not shown if STOPIT(5) = 0 (default 1).
       If a non-empty fourth parameter string SAVIT is present, then
       `SAVE SAVIT x fmax nf' is executed after each inner iteration.
       NB: x0 can be a matrix.  In the output argument, in SAVIT saves,
           and in function calls, x has the same shape as x0.
       MDSMAX(fun, x0, STOPIT, SAVIT, P1, P2,...) allows additional
       arguments to be passed to fun, via feval(fun,x,P1,P2,...).

This implementation uses 2n^2 elements of storage (two simplices), where x0
is an n-vector.  It is based on the algorithm statement in [2, sec.3],
modified so as to halve the storage (with a slight loss in readability).

References:
[1] V. J. Torczon, Multi-directional search: A direct search algorithm for
    parallel machines, Ph.D. Thesis, Rice University, Houston, Texas, 1989.
[2] V. J. Torczon, On the convergence of the multidirectional search
    algorithm, SIAM J. Optimization, 1 (1991), pp. 123-145.
[3] N. J. Higham, Optimization by direct search in matrix computations,
    SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993.
[4] N. J. Higham, Accuracy and Stability of Numerical Algorithms,
       Second edition, Society for Industrial and Applied Mathematics,
       Philadelphia, PA, 2002; sec. 20.5.

@end verbatim

@c ------------------------------------------------------------------

@node adsmax
@section An alternating directions algorithm
@mfnindex adsmax

@subheading Helptext:

@anchor{XREFadsmax}
@verbatim
ADSMAX  Alternating directions method for direct search optimization.
       [x, fmax, nf] = ADSMAX(FUN, x0, STOPIT, SAVIT, P) attempts to
       maximize the function FUN, using the starting vector x0.
       The alternating directions direct search method is used.
       Output arguments:
              x    = vector yielding largest function value found,
              fmax = function value at x,
              nf   = number of function evaluations.
       The iteration is terminated when either
              - the relative increase in function value between successive
                iterations is <= STOPIT(1) (default 1e-3),
              - STOPIT(2) function evaluations have been performed
                (default inf, i.e., no limit), or
              - a function value equals or exceeds STOPIT(3)
                (default inf, i.e., no test on function values).
       Progress of the iteration is not shown if STOPIT(5) = 0 (default 1).
       If a non-empty fourth parameter string SAVIT is present, then
       `SAVE SAVIT x fmax nf' is executed after each inner iteration.
       By default, the search directions are the co-ordinate directions.
       The columns of a fifth parameter matrix P specify alternative search
       directions (P = EYE is the default).
       NB: x0 can be a matrix.  In the output argument, in SAVIT saves,
           and in function calls, x has the same shape as x0.
       ADSMAX(fun, x0, STOPIT, SAVIT, P, P1, P2,...) allows additional
       arguments to be passed to fun, via feval(fun,x,P1,P2,...).
    Reference:
    N. J. Higham, Optimization by direct search in matrix computations,
       SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993.
    N. J. Higham, Accuracy and Stability of Numerical Algorithms,
       Second edition, Society for Industrial and Applied Mathematics,
       Philadelphia, PA, 2002; sec. 20.5.

@end verbatim

@c ------------------------------------------------------------------

@node nelder_mead_min
@section Another Nelder-Mead algorithm
@mfnindex nelder_mead_min

This function does gradient-less minimization using the Nelder-Mead
algorithm. No constraints are honoured.

@subheading Helptext:

@anchor{XREFnelder_mead_min}
@verbatim
[x0,v,nev] = nelder_mead_min (f,args,ctl) - Nelder-Mead minimization

Minimize 'f' using the Nelder-Mead algorithm. This function is inspired
from the that found in the book "Numerical Recipes".

ARGUMENTS
---------
f     : string : Name of function. Must return a real value
args  : list   : Arguments passed to f.
     or matrix : f's only argument
ctl   : vector : (Optional) Control variables, described below
     or struct

RETURNED VALUES
---------------
x0  : matrix   : Local minimum of f
v   : real     : Value of f in x0
nev : number   : Number of function evaluations

CONTROL VARIABLE : (optional) may be named arguments (i.e. "name",value
------------------ pairs), a struct, or a vector of length <= 6, where
                   NaN's are ignored. Default values are written <value>.
 OPT.   VECTOR
 NAME    POS
ftol,f  N/A    : Stopping criterion : stop search when values at simplex
                 vertices are all alike, as tested by 

                  f > (max_i (f_i) - min_i (f_i)) /max(max(|f_i|),1)

                 where f_i are the values of f at the vertices.  <10*eps>

rtol,r  N/A    : Stop search when biggest radius of simplex, using
                 infinity-norm, is small, as tested by :

             ctl(2) > Radius                                     <10*eps>

vtol,v  N/A    : Stop search when volume of simplex is small, tested by
           
             ctl(2) > Vol

crit,c ctl(1)  : Set one stopping criterion, 'ftol' (c=1), 'rtol' (c=2)
                 or 'vtol' (c=3) to the value of the 'tol' option.    <1>

tol, t ctl(2)  : Threshold in termination test chosen by 'crit'  <10*eps>

narg  ctl(3)  : Position of the minimized argument in args            <1>
maxev ctl(4)  : Maximum number of function evaluations. This number <inf>
                may be slightly exceeded.
isz   ctl(5)  : Size of initial simplex, which is :                   <1>

               { x + e_i | i in 0..N } 

               Where x == args{narg} is the initial value 
                e_0    == zeros (size (x)), 
                e_i(j) == 0 if j != i and e_i(i) == ctl(5)
                e_i    has same size as x

               Set ctl(5) to the distance you expect between the starting
               point and the minimum.

rst   ctl(6)   : When a minimum is found the algorithm restarts next to
                 it until the minimum does not improve anymore. ctl(6) is
                 the maximum number of restarts. Set ctl(6) to zero if
                 you know the function is well-behaved or if you don't
                 mind not getting a true minimum.                     <0>

verbose, v     Be more or less verbose (quiet=0)                      <0>

@end verbatim

@c ------------------------------------------------------------------

@node powell
@section Direction-set (Powell's) method
@mfnindex powell

@c powell ../inst/powell.m
@anchor{XREFpowell}
@deftypefn {Function File} {[@var{p}, @var{obj_value}, @var{convergence}, @var{iters}, @var{nevs}] =} powell (@var{f}, @var{p0}, @var{control})
Multidimensional minimization (direction-set method). Implements a direction-set (Powell's) method for multidimensional minimization of a function without calculation of the gradient [1, 2]

@subheading Arguments

@itemize @bullet
@item
@var{f}: name of function to minimize (string or handle), which should accept one input variable (see example for how to pass on additional input arguments)

@item
@var{p0}: An initial value of the function argument to minimize

@item
@var{options}: an optional structure, which can be generated by optimset, with some or all of the following fields:
@itemize @minus
@item
MaxIter: maximum iterations  (positive integer, or -1 or Inf for unlimited (default))
@item
TolFun: minimum amount by which function value must decrease in each iteration to continue (default is 1E-8)
@item
MaxFunEvals: maximum function evaluations  (positive integer, or -1 or Inf for unlimited (default))
@item
SearchDirections: an n*n matrix whose columns contain the initial set of (presumably orthogonal) directions to minimize along, where n is the number of elements in the argument to be minimized for; or an n*1 vector of magnitudes for the initial directions (defaults to the set of unit direction vectors)
@end itemize
@end itemize

@subheading Examples

@example
@group
y = @@(x, s) x(1) ^ 2 + x(2) ^ 2 + s;
o = optimset('MaxIter', 100, 'TolFun', 1E-10);
s = 1;
[x_optim, y_min, conv, iters, nevs] = powell(@@(x) y(x, s), [1 0.5], o); %pass y wrapped in an anonymous function so that all other arguments to y, which are held constant, are set
%should return something like x_optim = [4E-14 3E-14], y_min = 1, conv = 1, iters = 2, nevs = 24
@end group

@end example

@subheading Returns:

@itemize @bullet
@item
@var{p}: the minimizing value of the function argument
@item
@var{obj_value}: the value of @var{f}() at @var{p}
@item
@var{convergence}: 1 if normal convergence, 0 if not
@item
@var{iters}: number of iterations performed
@item
@var{nevs}: number of function evaluations
@end itemize

@subheading References

@enumerate
@item
Powell MJD (1964), An efficient method for finding the minimum of a function of several variables without calculating derivatives, @cite{Computer Journal}, 7 :155-162

@item
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (1992). @cite{Numerical Recipes in Fortran: The Art of Scientific Computing} (2nd Ed.). New York: Cambridge University Press (Section 10.5)
@end enumerate
@end deftypefn


@c ------------------------------------------------------------------

@node bfgsmin
@section Unconstrained BFGS algorithm
@mfnindex bfgsmin

BFGS or limited memory BFGS minimization of a function. No constraits
are honoured.

@subheading Helptext:

@anchor{XREFbfgsmin}
@verbatim
bfgsmin: bfgs or limited memory bfgs minimization of function

Usage: [x, obj_value, convergence, iters] = bfgsmin(f, args, control)

The function must be of the form
[value, return_2,..., return_m] = f(arg_1, arg_2,..., arg_n)
By default, minimization is w.r.t. arg_1, but it can be done
w.r.t. any argument that is a vector. Numeric derivatives are
used unless analytic derivatives are supplied. See bfgsmin_example.m
for methods.

Arguments:
* f: name of function to minimize (string)
* args: a cell array that holds all arguments of the function
	The argument with respect to which minimization is done
	MUST be a vector
* control: an optional cell array of 1-8 elements. If a cell
  array shorter than 8 elements is provided, the trailing elements
  are provided with default values.
	* elem 1: maximum iterations  (positive integer, or -1 or Inf for unlimited (default))
	* elem 2: verbosity
		0 = no screen output (default)
		1 = only final results
		2 = summary every iteration
		3 = detailed information
	* elem 3: convergence criterion
		1 = strict (function, gradient and param change) (default)
		0 = weak - only function convergence required
	* elem 4: arg in f_args with respect to which minimization is done (default is first)
	* elem 5: (optional) Memory limit for lbfgs. If it's a positive integer
		then lbfgs will be use. Otherwise ordinary bfgs is used
	* elem 6: function change tolerance, default 1e-12
	* elem 7: parameter change tolerance, default 1e-6
	* elem 8: gradient tolerance, default 1e-5

Returns:
* x: the minimizer
* obj_value: the value of f() at x
* convergence: 1 if normal conv, other values if not
* iters: number of iterations performed

Example: see bfgsmin_example.m

@end verbatim

@c ------------------------------------------------------------------

@node nrm
@section Newton-Raphson algorithm
@mfnindex nrm

No constraints are honoured.

@c nrm ../inst/nrm.m
@anchor{XREFnrm}
@deftypefn {Function File} {@var{xmin} =} nrm (@var{f},@var{x0})
Using @var{x0} as a starting point find a minimum of the scalar
function @var{f}.  The Newton-Raphson method is used.
@end deftypefn


@c ------------------------------------------------------------------

@node cg_min
@section A conjugate gradient method
@mfnindex cg_min

@c cg_min ../inst/cg_min.m
@anchor{XREFcg_min}
@deftypefn {Function File} {[@var{x0},@var{v},@var{nev}]} cg_min ( @var{f},@var{df},@var{args},@var{ctl} )
NonLinear Conjugate Gradient method to minimize function @var{f}.

@subheading Arguments
@itemize @bullet
@item @var{f}   : string   : Name of function. Return a real value 
@item @var{df}  : string   : Name of f's derivative. Returns a (R*C) x 1 vector 
@item @var{args}: cell     : Arguments passed to f.@*
@item @var{ctl}   : 5-vec    : (Optional) Control variables, described below
@end itemize

@subheading Returned values
@itemize @bullet
@item @var{x0}    : matrix   : Local minimum of f
@item @var{v}     : real     : Value of f in x0
@item @var{nev}   : 1 x 2    : Number of evaluations of f and of df
@end itemize

@subheading Control Variables
@itemize @bullet
@item @var{ctl}(1)       : 1 or 2 : Select stopping criterion amongst :
@item @var{ctl}(1)==0    : Default value
@item @var{ctl}(1)==1    : Stopping criterion : Stop search when value doesn't
improve, as tested by @math{ ctl(2) > Deltaf/max(|f(x)|,1) }
where Deltaf is the decrease in f observed in the last iteration
(each iteration consists R*C line searches).
@item @var{ctl}(1)==2    : Stopping criterion : Stop search when updates are small,
as tested by @math{ ctl(2) > max @{ dx(i)/max(|x(i)|,1) | i in 1..N @}}
where  dx is the change in the x that occured in the last iteration.
@item @var{ctl}(2)       : Threshold used in stopping tests.           Default=10*eps
@item @var{ctl}(2)==0    : Default value
@item @var{ctl}(3)       : Position of the minimized argument in args  Default=1
@item @var{ctl}(3)==0    : Default value
@item @var{ctl}(4)       : Maximum number of function evaluations      Default=inf
@item @var{ctl}(4)==0    : Default value
@item @var{ctl}(5)       : Type of optimization:
@item @var{ctl}(5)==1    : "Fletcher-Reves" method
@item @var{ctl}(5)==2    : "Polak-Ribiere" (Default)
@item @var{ctl}(5)==3    : "Hestenes-Stiefel" method
@end itemize

@var{ctl} may have length smaller than 4. Default values will be used if ctl is
not passed or if nan values are given.
@subheading Example:

function r=df( l )  b=[1;0;-1]; r = -( 2*l@{1@} - 2*b + rand(size(l@{1@}))); endfunction @*
function r=ff( l )  b=[1;0;-1]; r = (l@{1@}-b)' * (l@{1@}-b); endfunction @*
ll = @{ [10; 2; 3] @}; @*
ctl(5) = 3; @*
[x0,v,nev]=cg_min( "ff", "df", ll, ctl ) @*

Comment:  In general, BFGS method seems to be better performin in many cases but requires more computation per iteration
See also http://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient.
@seealso{@ref{XREFbfgsmin,,bfgsmin}}
@end deftypefn


@c ------------------------------------------------------------------

@node brent_line_min
@section Linesearch, Brent method
@mfnindex brent_line_min

@c brent_line_min ../inst/brent_line_min.m
@anchor{XREFbrent_line_min}
@deftypefn {Function File} {[@var{s},@var{v},@var{n}]} brent_line_min ( @var{f},@var{df},@var{args},@var{ctl} )
Line minimization of f along df

Finds minimum of f on line @math{ x0 + dx*w | a < w < b } by
bracketing. a and b are passed through argument ctl.

@subheading Arguments
@itemize @bullet
@item @var{f}     : string : Name of function. Must return a real value
@item @var{args}  : cell   : Arguments passed to f or RxC    : f's only argument. x0 must be at @var{args}@{ @var{ctl}(2) @}
@item @var{ctl}   : 5      : (optional) Control variables, described below.
@end itemize

@subheading Returned values
@itemize @bullet
@item @var{s}   : 1        : Minimum is at x0 + s*dx
@item @var{v}   : 1        : Value of f at x0 + s*dx
@item @var{nev} : 1        : Number of function evaluations
@end itemize

@subheading Control Variables
@itemize @bullet
@item @var{ctl}(1)       : Upper bound for error on s              Default=sqrt(eps)
@item @var{ctl}(2)       : Position of minimized argument in args  Default= 1
@item @var{ctl}(3)       : Maximum number of function evaluations  Default= inf
@item @var{ctl}(4)       : a                                       Default=-inf
@item @var{ctl}(5)       : b                                       Default= inf
@end itemize

Default values will be used if ctl is not passed or if nan values are
given.
@end deftypefn


@c ------------------------------------------------------------------

@node line_min
@section Linesearch, minimize a function along dx
@mfnindex line_min

@subheading Helptext:

@anchor{XREFline_min}
@verbatim
[a,fx,nev] = line_min (f, dx, args, narg, h, nev_max) - Minimize f() along dx

INPUT ----------
f    : string  : Name of minimized function
dx   : matrix  : Direction along which f() is minimized
args : cell    : Arguments of f
narg : integer : Position of minimized variable in args.  Default=1
h    : scalar  : Step size to use for centered finite difference
approximation of first and second derivatives. Default=1E-3.
nev_max : integer : Maximum number of function evaluations.  Default=30

OUTPUT ---------
a    : scalar  : Value for which f(x+a*dx) is a minimum (*)
fx   : scalar  : Value of f(x+a*dx) at minimum (*)
nev  : integer : Number of function evaluations

(*) The notation f(x+a*dx) assumes that args == {x}.

Reference: David G Luenberger's Linear and Nonlinear Programming

@end verbatim

@c ------------------------------------------------------------------

@node de_min
@section A differential evolution (stochastic) optimizer
@mfnindex de_min

@subheading Helptext:

@anchor{XREFde_min}
@verbatim
de_min: global optimisation using differential evolution

Usage: [x, obj_value, nfeval, convergence] = de_min(fcn, control)

minimization of a user-supplied function with respect to x(1:D),
using the differential evolution (DE) method based on an algorithm
by  Rainer Storn (http://www.icsi.berkeley.edu/~storn/code.html)
See: http://www.softcomputing.net/tevc2009_1.pdf


Arguments:  
---------------
fcn        string : Name of function. Must return a real value
control    vector : (Optional) Control variables, described below
        or struct

Returned values:
----------------
x          vector : parameter vector of best solution
obj_value  scalar : objective function value of best solution
nfeval     scalar : number of function evaluations
convergence       : 1 = best below value to reach (VTR)
                    0 = population has reached defined quality (tol)
                   -1 = some values are close to constraints/boundaries
                   -2 = max number of iterations reached (maxiter)
                   -3 = max number of functions evaluations reached (maxnfe)

Control variable:   (optional) may be named arguments (i.e. "name",value
----------------    pairs), a struct, or a vector, where
                    NaN's are ignored.

XVmin        : vector of lower bounds of initial population
               *** note: by default these are no constraints ***
XVmax        : vector of upper bounds of initial population
constr       : 1 -> enforce the bounds not just for the initial population
const        : data vector (remains fixed during the minimization)
NP           : number of population members
F            : difference factor from interval [0, 2]
CR           : crossover probability constant from interval [0, 1]
strategy     : 1 --> DE/best/1/exp           7 --> DE/best/1/bin
               2 --> DE/rand/1/exp           8 --> DE/rand/1/bin
               3 --> DE/target-to-best/1/exp 9 --> DE/target-to-best/1/bin
               4 --> DE/best/2/exp           10--> DE/best/2/bin
               5 --> DE/rand/2/exp           11--> DE/rand/2/bin
               6 --> DEGL/SAW/exp            else  DEGL/SAW/bin
refresh      : intermediate output will be produced after "refresh"
               iterations. No intermediate output will be produced
               if refresh is < 1
VTR          : Stopping criterion: "Value To Reach"
               de_min will stop when obj_value <= VTR.
               Use this if you know which value you expect.
tol          : Stopping criterion: "tolerance"
               stops if (best-worst)/max(1,worst) < tol
               This stops basically if the whole population is "good".
maxnfe       : maximum number of function evaluations
maxiter      : maximum number of iterations (generations)

      The algorithm seems to work well only if [XVmin,XVmax] covers the 
      region where the global minimum is expected.
      DE is also somewhat sensitive to the choice of the
      difference factor F. A good initial guess is to choose F from
      interval [0.5, 1], e.g. 0.8.
      CR, the crossover probability constant from interval [0, 1]
      helps to maintain the diversity of the population and is
      rather uncritical but affects strongly the convergence speed.
      If the parameters are correlated, high values of CR work better.
      The reverse is true for no correlation.
      Experiments suggest that /bin likes to have a slightly
      larger CR than /exp.
      The number of population members NP is also not very critical. A
      good initial guess is 10*D. Depending on the difficulty of the
      problem NP can be lower than 10*D or must be higher than 10*D
      to achieve convergence.

Default Values:
---------------
XVmin = [-2];
XVmax = [ 2];
constr= 0;
const = [];
NP    = 10 *D
F     = 0.8;
CR    = 0.9;
strategy = 12;
refresh  = 0;
VTR   = -Inf;
tol   = 1.e-3;
maxnfe  = 1e6;
maxiter = 1000;


Example to find the minimum of the Rosenbrock saddle:
----------------------------------------------------
Define f as:
                   function result = f(x);
                     result = 100 * (x(2) - x(1)^2)^2 + (1 - x(1))^2;
                   end
Then type:

	ctl.XVmin = [-2 -2];
	ctl.XVmax = [ 2  2];
	[x, obj_value, nfeval, convergence] = de_min (@f, ctl);

Keywords: global-optimisation optimisation minimisation

@end verbatim

@c ------------------------------------------------------------------

@node battery
@section Repeatedly call bfgsmin
@mfnindex battery

@subheading Helptext:

@anchor{XREFbattery}
@verbatim
battery.m: repeatedly call bfgs using a battery of 
start values, to attempt to find global min
of a nonconvex function

INPUTS:
func: function to mimimize
args: args of function
minarg: argument to minimize w.r.t. (usually = 1)
startvals: kxp matrix of values to try for sure (don't include all zeros, that's automatic)
max iters per start value
number of additional random start values to try

OUTPUT: theta - the best value found - NOT iterated to convergence

@end verbatim

@c ------------------------------------------------------------------

@node Residual optimization
@chapter Functions for optimization of a model function returning an array
@cindex residual optimization

Model functions whose parameters are to be optimized may return a vector
or array of values.  Either these or their differences to some constant
values (curve fitting) can be minimized in some sense, often, but not
necessarily, by minimizing the sum of their squares.  It is usually
preferable to use optimizers designed for residual optimization for this
purpose.  These can exploit information contained in the individual
elements of the returned array, which would not be possible if the user
calculated a norm (@abbr{e.g@.} sum of squares) of the elements and
performed a scalar optimization.

@menu
Optimization frontends
* nonlin_residmin::            The standard interface for non-linear
                                 residual minimization.
* nonlin_curvefit::            A convenience interface, curve fitting.

Optimization backends
* lm_svd_feasible::            L/M algorithm with SVD, constraints met
                                 throughout optimization.

Statistics frontends
* residmin_stat::              Statistics for residual minimization.
* curvefit_stat::              Statistics for curve fitting.

Statistics backends
* wls::                        Statistics for weighted least squares.

Standalone functions
* lsqlin::                     Linear least squares with linear
                                 constraints.
* leasqr::                     An older function for curve fitting.
* expfit::                     Prony's method for non-linear exponential
                                 fitting.
* polyfitinf::                 Function polyfitinf for polynomial
                                 fitting.
* wpolyfit::                   Polynomial fitting suitable for polyconf.
* polyconf::                   Confidence and prediction intervals for
                                 polynomial fitting.
* LinearRegression::           Function LinearRegression.
* wsolve::                     Another linear solver.
@end menu

@c ------------------------------------------------------------------

@node nonlin_residmin
@section Frontend nonlin_residmin for non-linear residual minimization
@mfnindex nonlin_residmin

@c include function help here
@c nonlin_residmin ../inst/nonlin_residmin.m
@anchor{XREFnonlin_residmin}
@deftypefn {Function File} {[@var{p}, @var{resid}, @var{cvg}, @var{outp}] =} nonlin_residmin (@var{f}, @var{pin})
@deftypefnx {Function File} {[@var{p}, @var{resid}, @var{cvg}, @var{outp}] =} nonlin_residmin (@var{f}, @var{pin}, @var{settings})
Frontend for nonlinear minimization of residuals returned by a model
function.

The functions supplied by the user have a minimal
interface; any additionally needed constants (e.g. observed values)
can be supplied by wrapping the user functions into anonymous
functions.

The following description applies to usage with vector-based
parameter handling. Differences in usage for structure-based
parameter handling will be explained separately.

@var{f}: function returning the array of residuals. It gets a column
vector of real parameters as argument. In gradient determination,
this function may be called with an informational second argument,
whose content depends on the function for gradient determination.

@var{pin}: real column vector of initial parameters.

@var{settings}: structure whose fields stand for optional settings
referred to below. The fields can be set by @code{optimset()}.

The returned values are the column vector of final parameters
@var{p}, the final array of residuals @var{resid}, an integer
@var{cvg} indicating if and how optimization succeeded or failed, and
a structure @var{outp} with additional information, curently with the
fields: @code{niter}, the number of iterations and
@code{user_interaction}, information on user stops (see settings).
The backend may define additional fields. If the backend supports it,
@var{outp} has a field @code{lambda} with determined Lagrange
multipliers of any constraints, seperated into subfields @code{lower}
and @code{upper} for bounds, @code{eqlin} and @code{ineqlin} for
linear equality and inequality constraints (except bounds),
respectively, and @code{eqnonlin} and @code{ineqnonlin} for general
equality and inequality constraints, respectively. @var{cvg} is
greater than zero for success and less than or equal to zero for
failure; its possible values depend on the used backend and currently
can be @code{0} (maximum number of iterations exceeded), @code{2}
(parameter change less than specified precision in two consecutive
iterations), or @code{3} (improvement in objective function -- e.g.
sum of squares -- less than specified), or @code{-1} (algorithm
aborted by a user function).

@c The following block will be cut out in the package info file.

@seealso{@ref{XREFnonlin_curvefit,,nonlin_curvefit}}
@end deftypefn


@subheading Settings

The fields of the @var{settings} structure can be set with
@ref{XREFoptimset,,optimset,octave}.

For settings common to all frontends (including these for statistics)
@mysee
@ref{Common frontend options}.

For additional settings common to all optimization frontends
@mysee
@ref{Common optimization options}.

@subsubheading Specific defaults:

@multitable {@code{Algorithm}} {"lm_svd_feasible"}
@item @code{Algorithm}:
@tab "lm_svd_feasible"
@end multitable

@subsubheading Additional settings:

@table @code
@item weights
Array of weights for the residuals. Dimensions must match.
@anchor{XREFoptiondfdp}
@item dfdp
Function computing the Jacobian of the residuals with respect to the
parameters, assuming residuals are reshaped to a column vector.
Default: real finite differences.  Will be called with the column vector
of parameters and an informational structure as arguments.  If
@code{dfdp} was specified by the user, the informational structure has
the fields @code{f}: value of residuals for current parameters, reshaped
to a column vector, @code{fixed}: logical vector indicating which
parameters are not optimized, so these partial derivatives need not be
computed and can be set to zero, @code{diffp}, @code{diff_onesided},
@code{lbound}, @code{ubound}: identical to the user settings of this
name, @code{plabels}: 1-dimensional cell-array of column-cell-arrays,
each column with labels for all parameters; the first column contains
the numerical indices of the parameters; the second and third columns,
present for structure based parameter handling,
@mysee
@ref{Parameter structures},
contain the names of the parameters and the subindices of
the parameters,
@mysee
@ref{Non-scalar parameters}, respectively.  The default
jacobian function will call the model function with the second argument
set with fields @code{f}: as the @code{f} passed to the jacobian
function, @code{plabels}: cell-array of 1x1 cell-arrays with the entries
of the column-cell-arrays of @code{plabels} as passed to the jacobian
function corresponding to current parameter, @code{side}: @code{0} for
one-sided interval, @code{1} or @code{2}, respectively, for the sides of
a two-sided interval, and @code{parallel}: logical scalar indicating
parallel computation of partial derivatives.  This information can be
useful if the model function can omit some computations depending on the
currently computed partial derivative.
@item complex_step_derivative_f
Logical scalar, default: @code{false}. Estimate Jacobian of model
function with complex step derivative approximation. Use only if you
know that your model function is suitable for this. No user function for
the Jacobian (@code{dfdp}) must be specified.
@item plot_cmd
Function enabling backend to plot results or intermediate results.  Will
be called with current computed residuals.  Default: plot nothing.  This
setting is deprecated and will disappear.  Please use
@code{user_interaction} instead (
@mysee
@ref{Common optimization options}).
@end table

@subheading Structure based parameter handling

Please
@mysee
@ref{Parameter structures}.

@subheading Backend information

Please
@mysee
@ref{Residual optimization} and choose backend from menu under
`Optimization backends'.

@c ------------------------------------------------------------------

@node nonlin_curvefit
@section Function nonlin_curvefit() for curve fitting
@mfnindex nonlin_curvefit

In curve fitting, the model function computes values from a constant set
of `independents', and the intention is to minimize the differences of
these computed values to a constant set of `observations'. This can be
done with @code{nonlin_residmin}, but it is more convenient to use
@code{nonlin_curvefit}, which cares for passing the constant
`independents' to the model function and for calculating the differences
to the constant `observations'.

However, if in some optimization problem you notice that you end up with
passing dummy-values for the `independents' and zeros for the
`observations', you can more naturally use @code{nonlin_residmin}
instead of @code{nonlin_curvefit}.

@c include function help here
@c nonlin_curvefit ../inst/nonlin_curvefit.m
@anchor{XREFnonlin_curvefit}
@deftypefn {Function File} {[@var{p}, @var{fy}, @var{cvg}, @var{outp}] =} nonlin_curvefit (@var{f}, @var{pin}, @var{x}, @var{y})
@deftypefnx {Function File} {[@var{p}, @var{fy}, @var{cvg}, @var{outp}] =} nonlin_curvefit (@var{f}, @var{pin}, @var{x}, @var{y}, @var{settings})
Frontend for nonlinear fitting of values, computed by a model
function, to observed values.

Please refer to the description of @code{nonlin_residmin}. The
differences to @code{nonlin_residmin} are the additional arguments
@var{x} (independent values, mostly, but not necessarily, an array of
the same dimensions or the same number of rows as @var{y}) and
@var{y} (array of observations), the returned value @var{fy} (final
guess for observed values) instead of @var{resid}, that the model
function has a second obligatory argument which will be set to
@var{x} and is supposed to return guesses for the observations (with
the same dimensions), and that the possibly user-supplied function
for the jacobian of the model function has also a second obligatory
argument which will be set to @var{x}.

@c The following block will be cut out in the package info file.

@seealso{@ref{XREFnonlin_residmin,,nonlin_residmin}}
@end deftypefn


@c replace the cut out text
Also, if the setting @code{user_interaction} is given, additional
information is passed to these functions,
@mysee
@ref{Common optimization options}.

@c ------------------------------------------------------------------

@node lm_svd_feasible
@section Default backend lm_svd_feasible of residual minimization
@cindex lm_svd_feasible

Levenberg/Marquardt algorithm using singular value decomposition.
Constraints must be met by the initial parameters and are attempted to
be kept met throughout the optimization.

Returned value @var{cvg} will be @code{0}, @code{1}, or @code{2}.
Returned structure @var{outp} will have the fields @code{niter} and
@code{user_interaction}.

Backend-specific defaults are: @code{MaxIter}: 20, @code{fract_prec}:
@code{zeros (size (parameters))}, @code{max_fract_change}: @code{Inf}
for all parameters. The setting @code{TolX} is not honoured.

Interpretation of @code{Display}: if set to @qcode{"iter"}, currently
some diagnostics are printed.

Specific option: @code{lm_svd_feasible_alt_s}: if falling back to nearly
gradient descent, do it more like original Levenberg/Marquardt method,
with descent in each gradient component; for testing only.


@c ------------------------------------------------------------------

@node residmin_stat
@section  Statistics for residual minimization
@mfnindex residmin_stat

@c include function help here
@c residmin_stat ../inst/residmin_stat.m
@anchor{XREFresidmin_stat}
@deftypefn {Function File} {@var{info} =} residmin_stat (@var{f}, @var{p}, @var{settings})
Frontend for computation of statistics for a residual-based
minimization.

@var{settings} is a structure whose fields can be set by
@code{optimset}. With @var{settings} the computation of certain
statistics is requested by setting the fields
@code{ret_<name_of_statistic>} to @code{true}. The respective
statistics will be returned in a structure as fields with name
@code{<name_of_statistic>}. Depending on the requested statistic and
on the additional information provided in @var{settings}, @var{f} and
@var{p} may be empty. Otherwise, @var{f} is the model function of an
optimization (the interface of @var{f} is described e.g. in
@code{nonlin_residmin}, please see there), and @var{p} is a real
column vector with parameters resulting from the same optimization.

Currently, the following statistics (or general information) can be
requested:

@code{dfdp}: Jacobian of model function with respect to parameters.

@code{covd}: Covariance matrix of data (typically guessed by applying
a factor to the covariance matrix of the residuals).

@code{covp}: Covariance matrix of final parameters.

@code{corp}: Correlation matrix of final parameters.

@c The following block will be cut out in the package info file.

@seealso{@ref{XREFcurvefit_stat,,curvefit_stat}}
@end deftypefn


@subheading Further settings

The fields of the @var{settings} structure can be set with
@ref{XREFoptimset,,optimset,octave}.

For settings common to all frontends
@mysee
@ref{Common frontend options}.

@subsubheading Additional settings:

@table @code
@item objf_type
Type of objective function of the optimization; must be specified in
many cases. This determines which backends to use. Currently, there are
only backends for the type "wls" (weighted least squares).
@item residuals
@item covd
Optional information on the result of optimization, residuals and
covariance matrix of data, respectively.
@item weights
Array of weights applied to the residuals in the previous
optimization. Dimensions must match those of the residuals.
@item dfdp
Can be set in the same way and has the same default as in
@code{nonlin_residmin} (
@mysee
@ref{nonlin_residmin}), but alternatively may
already contain the computed Jacobian of the model function at the final
parameters in matrix- or structure-form.
@item complex_step_derivative_f
Estimate Jacobian of model function with complex step derivative
approximation. Use only if you know that your model function is suitable
for this. No user function for the Jacobian (@code{dfdp}) must be
specified.
@end table

@subheading Structure based parameter handling

Please
@mysee
@ref{Parameter structures}.

@subheading Backend information

Please
@mysee
@ref{Residual optimization} and choose backend from menu under
`Statistics backends'.

@c ------------------------------------------------------------------

@node curvefit_stat
@section  Statistics for curve fitting
@mfnindex curvefit_stat

As @code{nonlin_curvefit} can be used instead of @code{nonlin_residmin}
for curve fitting (
@mysee
@ref{nonlin_curvefit},
@mysee
@ref{nonlin_residmin}),
@code{curvefit_stat} can be used instead of @code{residmin_stat}
(
@mysee
@ref{residmin_stat}) for statistics on the results of curve fitting.

@c include function help here
@c curvefit_stat ../inst/curvefit_stat.m
@anchor{XREFcurvefit_stat}
@deftypefn {Function File} {@var{info} =} curvefit_stat (@var{f}, @var{p}, @var{x}, @var{y}, @var{settings})

Frontend for computation of statistics for fitting of values,
computed by a model function, to observed values.

Please refer to the description of @code{residmin_stat}. The only
differences to @code{residmin_stat} are the additional arguments
@var{x} (independent values) and @var{y} (observations), that the
model function @var{f}, if provided, has a second obligatory argument
which will be set to @var{x} and is supposed to return guesses for
the observations (with the same dimensions), and that the possibly
user-supplied function for the jacobian of the model function has
also a second obligatory argument which will be set to @var{x}.

@seealso{@ref{XREFresidmin_stat,,residmin_stat}}
@end deftypefn


@c ------------------------------------------------------------------

@node wls
@section Statistics for weighted least squares
@cindex wls
@cindex statistics for weighted least squares

The backends for @code{objf_type == "wls"} (currently the only supported
type of objective function) compute @code{covd} (due to user request or
as a prerequisite for @code{covp} and @code{corp}) as a diagonal matrix
by assuming that the variances of data points are proportional to the
reciprocal of the squared @code{weights} and guessing the factor of
proportionality from the residuals. If @code{covp} is not defined
(e.g. because the Jacobian has no full rank), an attempt is made to
still compute its uniquely defined elements, if any. In @code{corp},
interdependent parameters can cause elements of @code{1} or @code{-1},
which in this case are not the real coefficients of correlation, but
rather indicate the direction of parameter interdependence. To be
consistent with this, an attempt is made (often not successful) to
identify parameter interdependence and mark it with elements of @code{1}
or @code{-1} in @code{corp} even if the respective elements of
@code{covp} can not be computed.

@c ------------------------------------------------------------------

@node lsqlin
@section Linear least squares with linear constraints.
@mfnindex lsqlin

(This function does not fit well into this chapter because it is
actually a special case of quadratic programming).

@c lsqlin ../inst/lsqlin.m
@anchor{XREFlsqlin}
@deftypefn {Function File} {} lsqlin (@var{C}, @var{d}, @var{A}, @var{b})
@deftypefnx {Function File} {} lsqlin (@var{C}, @var{d}, @var{A}, @var{b}, @var{Aeq}, @var{beq}, @var{lb}, @var{ub})
@deftypefnx {Function File} {} lsqlin (@var{C}, @var{d}, @var{A}, @var{b}, @var{Aeq}, @var{beq}, @var{lb}, @var{ub}, @var{x0})
@deftypefnx {Function File} {} lsqlin (@var{C}, @var{d}, @var{A}, @var{b}, @var{Aeq}, @var{beq}, @var{lb}, @var{ub}, @var{x0}, @var{options})
@deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}, @var{output}, @var{lambda}] =} lsqlin (@dots{})
Solve the linear least squares program
@example
@group
min 0.5 sumsq(C*x - d)
x
@end group
@end example
subject to
@example
@group
@var{A}*@var{x} <= @var{b},
@var{Aeq}*@var{x} = @var{beq},
@var{lb} <= @var{x} <= @var{ub}.
@end group
@end example

The initial guess @var{x0} and the constraint arguments (@var{A} and
@var{b}, @var{Aeq} and @var{beq}, @var{lb} and @var{ub}) can be set to
the empty matrix (@code{[]}) if not given. If the initial guess
@var{x0} is feasible the algorithm is faster.

@var{options} can be set with @code{optimset}, currently the only
option is @code{MaxIter}, the maximum number of iterations (default:
200).

Returned values:

@table @var
@item x
Position of minimum.

@item resnorm
Scalar value of objective as sumsq(C*x - d).

@item residual
Vector of solution residuals C*x - d.

@item exitflag
Status of solution:

@table @code
@item 0
Maximum number of iterations reached.

@item -2
The problem is infeasible.

@item 1
Global solution found.

@end table

@item output
Structure with additional information, currently the only field is
@code{iterations}, the number of used iterations.

@item lambda
Structure containing Lagrange multipliers corresponding to the
constraints.

@end table

This function calls the more general function @code{quadprog}
internally.

@seealso{@ref{XREFquadprog,,quadprog}}
@end deftypefn


@c ------------------------------------------------------------------

@node leasqr
@section An older function for curve fitting
@mfnindex leasqr

This was a popular function for curve fitting and has been enhanced to
honour constraints.  @code{nonlin_curvefit} (
@mysee
@ref{nonlin_curvefit}) does
now the same job if used with the default backend, and should be
prefered due to its more powerful interface.  The statistics returned by
@code{leasqr} can also (and partially better) be computed with
@code{curvefit_stat} (
@mysee
@ref{curvefit_stat}).  There are currently two
things which still only @code{leasqr} does:

@itemize
@item internally providing a function for plotting fits during
      optimization,
@item returning a pre-computed matrix for determining confidence
      regions.
@end itemize

@c leasqr ../inst/leasqr.m
@anchor{XREFleasqr}
@deftypefn  {Function File} {} leasqr (@var{x}, @var{y}, @var{pin}, @var{F})
@deftypefnx {Function File} {} leasqr (@var{x}, @var{y}, @var{pin}, @var{F}, @var{stol})
@deftypefnx {Function File} {} leasqr (@var{x}, @var{y}, @var{pin}, @var{F}, @var{stol}, @var{niter})
@deftypefnx {Function File} {} leasqr (@var{x}, @var{y}, @var{pin}, @var{F}, @var{stol}, @var{niter}, @var{wt})
@deftypefnx {Function File} {} leasqr (@var{x}, @var{y}, @var{pin}, @var{F}, @var{stol}, @var{niter}, @var{wt}, @var{dp})
@deftypefnx {Function File} {} leasqr (@var{x}, @var{y}, @var{pin}, @var{F}, @var{stol}, @var{niter}, @var{wt}, @var{dp}, @var{dFdp})
@deftypefnx {Function File} {} leasqr (@var{x}, @var{y}, @var{pin}, @var{F}, @var{stol}, @var{niter}, @var{wt}, @var{dp}, @var{dFdp}, @var{options})
@deftypefnx {Function File} {[@var{f}, @var{p}, @var{cvg}, @var{iter}, @var{corp}, @var{covp}, @var{covr}, @var{stdresid}, @var{Z}, @var{r2}] =} leasqr (@dots{})
Levenberg-Marquardt nonlinear regression.

Input arguments:

@table @var
@item x
Vector or matrix of independent variables.

@item y
Vector or matrix of observed values.

@item pin
Vector of initial parameters to be adjusted by leasqr.

@item F
Name of function or function handle. The function must be of the form
@code{y = f(x, p)}, with y, x, p of the form @var{y}, @var{x}, @var{pin}.

@item stol
Scalar tolerance on fractional improvement in scalar sum of squares, i.e.,
@code{sum ((@var{wt} .* (@var{y}-@var{f}))^2)}.  Set to 0.0001 if
empty or not given;

@item niter
Maximum number of iterations.  Set to 20 if empty or not given.

@item wt
Statistical weights (same dimensions as @var{y}).  These should be
set to be proportional to @code{sqrt (@var{y}) ^-1}, i.e., the
covariance matrix of the data is assumed to be proportional to
diagonal with diagonal equal to @code{(@var{wt}.^2)^-1}.  The constant of
proportionality will be estimated.  Set to @code{ones (size
(@var{y}))} if empty or not given.

@item dp
Fractional increment of @var{p} for numerical partial derivatives.  Set
to @code{0.001 * ones (size (@var{pin}))} if empty or not given.

@itemize @bullet
@item dp(j) > 0 means central differences on j-th parameter p(j).
@item dp(j) < 0 means one-sided differences on j-th parameter p(j).
@item dp(j) = 0 holds p(j) fixed, i.e., leasqr won't change initial guess: pin(j)
@end itemize

@item dFdp
Name of partial derivative function in quotes or function handle. If
not given or empty, set to @code{dfdp}, a slow but general partial
derivatives function. The function must be of the form @code{prt =
dfdp (x, f, p, dp, F [,bounds])}.  For backwards compatibility, the
function will only be called with an extra 'bounds' argument if the
'bounds' option is explicitly specified to leasqr (see dfdp.m).

@item options
Structure with multiple options. The following fields are recognized:

@table @asis
@item @qcode{fract_prec}
Column vector (same length as @var{pin})
of desired fractional precisions in parameter estimates.
Iterations are terminated if change in parameter vector (chg)
relative to current parameter estimate is less than their
corresponding elements in 'fract_prec', i.e.,
@code{all (abs (chg) < abs (options.fract_prec .* current_parm_est))} on two
consecutive iterations. Defaults to @code{zeros (size (@var{pin}))}.

@item @qcode{max_fract_change}
Column vector (same length as @var{pin}) of maximum fractional step
changes in parameter vector.
Fractional change in elements of parameter vector is constrained to
be at most 'max_fract_change' between sucessive iterations, i.e.,
@code{abs (chg(i)) = abs (min([chg(i), options.max_fract_change(i) * current param estimate]))}.
Defaults to @code{Inf * ones (size (@var{pin}))}.

@item @qcode{inequc}
Cell-array containing up to four entries,
two entries for linear inequality constraints and/or one or two
entries for general inequality constraints.  Initial parameters
must satisfy these constraints.  Either linear or general
constraints may be the first entries, but the two entries for
linear constraints must be adjacent and, if two entries are given
for general constraints, they also must be adjacent.  The two
entries for linear constraints are a matrix (say m) and a vector
(say v), specifying linear inequality constraints of the form
`m.' * parameters + v >= 0'. If the constraints are just bounds,
it is suggested to specify them in 'options.bounds' instead,
since then some sanity tests are performed, and since the
function 'dfdp.m' is guarantied not to violate constraints during
determination of the numeric gradient only for those constraints
specified as 'bounds' (possibly with violations due to a certain
inaccuracy, however, except if no constraints except bounds are
specified). The first entry for general constraints must be a
differentiable vector valued function (say h), specifying general
inequality constraints of the form `h (p[, idx]) >= 0'; p is the
column vector of optimized paraters and the optional argument idx
is a logical index. h has to return the values of all constraints
if idx is not given, and has to return only the indexed
constraints if idx is given (so computation of the other
constraints can be spared). If a second entry for general
constraints is given, it must be a function (say dh) which
returnes a matrix whos rows contain the gradients of the
constraint function h with respect to the optimized parameters.
It has the form jac_h = dh (vh, p, dp, h, idx[, bounds]); p is
the column vector of optimized parameters, and idx is a logical
index --- only the rows indexed by idx must be returned (so
computation of the others can be spared). The other arguments of
dh are for the case that dh computes numerical gradients: vh is
the column vector of the current values of the constraint
function h, with idx already applied. h is a function h (p) to
compute the values of the constraints for parameters p, it will
return only the values indexed by idx. dp is a suggestion for
relative step width, having the same value as the argument 'dp'
of leasqr above. If bounds were specified to leasqr, they are
provided in the argument bounds of dh, to enable their
consideration in determination of numerical gradients. If dh is
not specified to leasqr, numerical gradients are computed in the
same way as with 'dfdp.m' (see above). If some constraints are
linear, they should be specified as linear constraints (or
bounds, if applicable) for reasons of performance, even if
general constraints are also specified.

@item @qcode{bounds}
Two-column-matrix, one row for each
parameter in @var{pin}. Each row contains a minimal and maximal value
for each parameter. Default: [-Inf, Inf] in each row. If this
field is used with an existing user-side function for 'dFdp'
(see above) the functions interface might have to be changed.

@item @qcode{equc}
Equality constraints, specified the same
way as inequality constraints (see field 'options.inequc').
Initial parameters must satisfy these constraints.
Note that there is possibly a certain inaccuracy in honoring
constraints, except if only bounds are specified.
@emph{Warning}: If constraints (or bounds) are set, returned guesses
of @var{corp}, @var{covp}, and @var{Z} are generally invalid, even if
no constraints
are active for the final parameters. If equality constraints are
specified, @var{corp}, @var{covp}, and @var{Z} are not guessed at all.

@item @qcode{cpiv}
Function for complementary pivot algorithm
for inequality constraints. Defaults to cpiv_bard.  No different
function is supplied.

@end table

For backwards compatibility, @var{options} can also be a matrix whose
first and second column contains the values of @qcode{fract_prec} and
@qcode{max_fract_change}, respectively.

@end table

Output:

@table @var
@item f
Column vector of values computed: f = F(x,p).

@item p
Column vector trial or final parameters, i.e, the solution.

@item cvg
Scalar: = 1 if convergence, = 0 otherwise.

@item iter
Scalar number of iterations used.

@item corp
Correlation matrix for parameters.

@item covp
Covariance matrix of the parameters.

@item covr
Diag(covariance matrix of the residuals).

@item stdresid
Standardized residuals.

@item Z
Matrix that defines confidence region (see comments in the source).

@item r2
Coefficient of multiple determination, intercept form.

@end table

Not suitable for non-real residuals.

References:
Bard, Nonlinear Parameter Estimation, Academic Press, 1974.
Draper and Smith, Applied Regression Analysis, John Wiley and Sons, 1981.

@end deftypefn


@c ------------------------------------------------------------------

@node expfit
@section Prony's method for non-linear exponential fitting
@mfnindex expfit

@subheading Helptext:

@anchor{XREFexpfit}
@verbatim
USAGE  [alpha,c,rms] = expfit( deg, x1, h, y )

Prony's method for non-linear exponential fitting

Fit function:   \sum_1^{deg} c(i)*exp(alpha(i)*x)

Elements of data vector y must correspond to
equidistant x-values starting at x1 with stepsize h

The method is fully compatible with complex linear
coefficients c, complex nonlinear coefficients alpha
and complex input arguments y, x1, non-zero h .
Fit-order deg  must be a real positive integer.

Returns linear coefficients c, nonlinear coefficients
alpha and root mean square error rms. This method is
known to be more stable than 'brute-force' non-linear
least squares fitting.

Example
   x0 = 0; step = 0.05; xend = 5; x = x0:step:xend;
   y = 2*exp(1.3*x)-0.5*exp(2*x);
   error = (rand(1,length(y))-0.5)*1e-4;
   [alpha,c,rms] = expfit(2,x0,step,y+error)

 alpha =
   2.0000
   1.3000
 c =
   -0.50000
    2.00000
 rms = 0.00028461

The fit is very sensitive to the number of data points.
It doesn't perform very well for small data sets.
Theoretically, you need at least 2*deg data points, but
if there are errors on the data, you certainly need more.

Be aware that this is a very (very,very) ill-posed problem.
By the way, this algorithm relies heavily on computing the
roots of a polynomial. I used 'roots.m', if there is
something better please use that code.

Demo for a complex fit-function:
deg= 2; N= 20; x1= -(1+i), x= linspace(x1,1+i/2,N).';
h = x(2) - x(1)
y= (2+i)*exp( (-1-2i)*x ) + (-1+3i)*exp( (2+3i)*x );
A= 5e-2; y+= A*(randn(N,1)+randn(N,1)*i); % add complex noise
[alpha,c,rms]= expfit( deg, x1, h, y )

@end verbatim

@c ------------------------------------------------------------------

@node polyfitinf
@section Function polyfitinf for polynomial fitting.
@mfnindex polyfitinf

@subheading Helptext:

@anchor{XREFpolyfitinf}
@verbatim
function [A,REF,HMAX,H,R,EQUAL] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT,REF0)

  Best polynomial approximation in discrete uniform norm

  INPUT VARIABLES:

  M       : degree of the fitting polynomial
  N       : number of data points
  X(N)    : x-coordinates of data points
  Y(N)    : y-coordinates of data points
  K       : character of the polynomial:
                  K = 0 : mixed parity polynomial
                  K = 1 : odd polynomial  ( X(1) must be >  0 )
                  K = 2 : even polynomial ( X(1) must be >= 0 )
  EPSH    : tolerance for leveling. A useful value for 24-bit
            mantissa is EPSH = 2.0E-7
  MAXIT   : upper limit for number of exchange steps
  REF0(M2): initial alternating set ( N-vector ). This is an
            OPTIONAL argument. The length M2 is given by:
                  M2 = M + 2                      , if K = 0
                  M2 = integer part of (M+3)/2    , if K = 1
                  M2 = 2 + M/2 (M must be even)   , if K = 2

  OUTPUT VARIABLES:

  A       : polynomial coefficients of the best approximation
            in order of increasing powers:
                  p*(x) = A(1) + A(2)*x + A(3)*x^2 + ...
  REF     : selected alternating set of points
  HMAX    : maximum deviation ( uniform norm of p* - f )
  H       : pointwise approximation errors
	R		: total number of iterations
  EQUAL   : success of failure of algorithm
                  EQUAL=1 :  succesful
                  EQUAL=0 :  convergence not acheived
                  EQUAL=-1:  input error
                  EQUAL=-2:  algorithm failure

  Relies on function EXCH, provided below.

  Example: 
  M = 5; N = 10000; K = 0; EPSH = 10^-12; MAXIT = 10;
  X = linspace(-1,1,N);   % uniformly spaced nodes on [-1,1]
  k=1; Y = abs(X).^k;     % the function Y to approximate
  [A,REF,HMAX,H,R,EQUAL] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT);
  p = polyval(A,X); plot(X,Y,X,p) % p is the best approximation

  Note: using an even value of M, e.g., M=2, in the example above, makes
  the algorithm to fail with EQUAL=-2, because of collocation, which
  appears because both the appriximating function and the polynomial are
  even functions. The way aroung it is to approximate only the right half
  of the function, setting K = 2 : even polynomial. For example: 

N = 10000; K = 2; EPSH = 10^-12; MAXIT = 10;  X = linspace(0,1,N);
for i = 1:2
    k = 2*i-1; Y = abs(X).^k;
    for j = 1:4
        M = 2^j;
        [~,~,HMAX] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT);
        approxerror(i,j) = HMAX;
    end
end
disp('Table 3.1 from Approximation theory and methods, M.J.D.POWELL, p. 27');
disp(' ');
disp('            n          K=1          K=3'); 
disp(' '); format short g;
disp([(2.^(1:4))' approxerror']);

  ALGORITHM:

  Computation of the polynomial that best approximates the data (X,Y)
  in the discrete uniform norm, i.e. the polynomial with the  minimum
  value of max{ | p(x_i) - y_i | , x_i in X } . That polynomial, also
  known as minimax polynomial, is obtained by the exchange algorithm,
  a finite iterative process requiring, at most,
     n
   (   ) iterations ( usually p = M + 2. See also function EXCH ).
     p
  since this number can be very large , the routine  may not converge
  within MAXIT iterations . The  other possibility of  failure occurs
  when there is insufficient floating point precision  for  the input
  data chosen.

  CREDITS: This routine was developed and modified as 
  computer assignments in Approximation Theory courses by 
  Prof. Andrew Knyazev, University of Colorado Denver, USA.

  Team Fall 98 (Revision 1.0):
          Chanchai Aniwathananon
          Crhistopher Mehl
          David A. Duran
          Saulo P. Oliveira

  Team Spring 11 (Revision 1.1): Manuchehr Aminian

  The algorithm and the comments are based on a FORTRAN code written
  by Joseph C. Simpson. The code is available on Netlib repository:
  http://www.netlib.org/toms/501
  See also: Communications of the ACM, V14, pp.355-356(1971)

  NOTES:

  1) A may contain the collocation polynomial
  2) If MAXIT is exceeded, REF contains a new reference set
  3) M, EPSH and REF can be altered during the execution
  4) To keep consistency to the original code , EPSH can be
  negative. However, the use of REF0 is *NOT* determined by
  EPSH< 0, but only by its inclusion as an input parameter.

  Some parts of the code can still take advantage of vectorization.  

  Revision 1.0 from 1998 is a direct human translation of 
  the FORTRAN code http://www.netlib.org/toms/501
  Revision 1.1 is a clean-up and technical update.  
  Tested on MATLAB Version 7.11.0.584 (R2010b) and 
  GNU Octave Version 3.2.4

@end verbatim

@c ------------------------------------------------------------------

@node wpolyfit
@section Polynomial fitting suitable for polyconf
@mfnindex wpolyfit

@c wpolyfit ../inst/wpolyfit.m
@anchor{XREFwpolyfit}
@deftypefn {Function File} {[@var{p}, @var{s}] =} wpolyfit (@var{x}, @var{y}, @var{dy}, @var{n})
Return the coefficients of a polynomial @var{p}(@var{x}) of degree
@var{n} that minimizes
@c ########################################################
@c These lines must be written without space at start to work around
@c a bug in html generation.
@iftex
@tex
$$
\sum_{i=1}^N (p(x_i) - y_i)^2
$$
@end tex
@end iftex
@ifnottex
@code{sumsq (p(x(i)) - y(i))},
@end ifnottex
@c ########################################################
to best fit the data in the least squares sense.  The standard error
on the observations @var{y} if present are given in @var{dy}.

The returned value @var{p} contains the polynomial coefficients 
suitable for use in the function polyval.  The structure @var{s} returns
information necessary to compute uncertainty in the model.

To compute the predicted values of y with uncertainty use
@example
[y,dy] = polyconf(p,x,s,'ci');
@end example
You can see the effects of different confidence intervals and
prediction intervals by calling the wpolyfit internal plot
function with your fit:
@example
feval('wpolyfit:plt',x,y,dy,p,s,0.05,'pi')
@end example
Use @var{dy}=[] if uncertainty is unknown.

You can use a chi^2 test to reject the polynomial fit:
@example
p = 1-chi2cdf(s.normr^2,s.df);
@end example
p is the probability of seeing a chi^2 value higher than that which 
was observed assuming the data are normally distributed around the fit.
If p < 0.01, you can reject the fit at the 1% level.

You can use an F test to determine if a higher order polynomial 
improves the fit:
@example
[poly1,S1] = wpolyfit(x,y,dy,n);
[poly2,S2] = wpolyfit(x,y,dy,n+1);
F = (S1.normr^2 - S2.normr^2)/(S1.df-S2.df)/(S2.normr^2/S2.df);
p = 1-f_cdf(F,S1.df-S2.df,S2.df);
@end example
p is the probability of observing the improvement in chi^2 obtained
by adding the extra parameter to the fit.  If p < 0.01, you can reject 
the lower order polynomial at the 1% level.

You can estimate the uncertainty in the polynomial coefficients 
themselves using
@example
dp = sqrt(sumsq(inv(s.R'))'/s.df)*s.normr;
@end example
but the high degree of covariance amongst them makes this a questionable
operation.
@end deftypefn

@deftypefn {Function File} {[@var{p}, @var{s}, @var{mu}] =} wpolyfit (...)

If an additional output @code{mu = [mean(x),std(x)]} is requested then 
the @var{x} values are centered and normalized prior to computing the fit.
This will give more stable numerical results.  To compute a predicted 
@var{y} from the returned model use
@code{y = polyval(p, (x-mu(1))/mu(2)}
@end deftypefn

@deftypefn {Function File} {} wpolyfit (...)

If no output arguments are requested, then wpolyfit plots the data,
the fitted line and polynomials defining the standard error range.

Example
@example
x = linspace(0,4,20);
dy = (1+rand(size(x)))/2;
y = polyval([2,3,1],x) + dy.*randn(size(x));
wpolyfit(x,y,dy,2);
@end example
@end deftypefn

@deftypefn {Function File} {} wpolyfit (..., 'origin')

If 'origin' is specified, then the fitted polynomial will go through
the origin.  This is generally ill-advised.  Use with caution.

Hocking, RR (2003). Methods and Applications of Linear Models.
New Jersey: John Wiley and Sons, Inc.

@c Will be cut out in optims info file and replaced with the same
@c refernces explicitely there, since references to core Octave
@c functions are not automatically transformed from here to there.
@seealso{@ref{XREFpolyconf,,polyconf}}
@end deftypefn


See also @ref{XREFpolyfit,,polyfit,octave}.

@c ------------------------------------------------------------------

@node polyconf
@section Confidence and prediction intervals for polynomial fitting
@mfnindex polyconf

@subheading Helptext:

@anchor{XREFpolyconf}
@verbatim
[y,dy] = polyconf(p,x,s)

  Produce prediction intervals for the fitted y. The vector p 
  and structure s are returned from polyfit or wpolyfit. The 
  x values are where you want to compute the prediction interval.

polyconf(...,['ci'|'pi'])

  Produce a confidence interval (range of likely values for the
  mean at x) or a prediction interval (range of likely values 
  seen when measuring at x).  The prediction interval tells
  you the width of the distribution at x.  This should be the same
  regardless of the number of measurements you have for the value
  at x.  The confidence interval tells you how well you know the
  mean at x.  It should get smaller as you increase the number of
  measurements.  Error bars in the physical sciences usually show 
  a 1-alpha confidence value of erfc(1/sqrt(2)), representing
  one standandard deviation of uncertainty in the mean.

polyconf(...,1-alpha)

  Control the width of the interval. If asking for the prediction
  interval 'pi', the default is .05 for the 95% prediction interval.
  If asking for the confidence interval 'ci', the default is
  erfc(1/sqrt(2)) for a one standard deviation confidence interval.

Example:
 [p,s] = polyfit(x,y,1);
 xf = linspace(x(1),x(end),150);
 [yf,dyf] = polyconf(p,xf,s,'ci');
 plot(xf,yf,'g-;fit;',xf,yf+dyf,'g.;;',xf,yf-dyf,'g.;;',x,y,'xr;data;');
 plot(x,y-polyval(p,x),';residuals;',xf,dyf,'g-;;',xf,-dyf,'g-;;');

@end verbatim

@c ------------------------------------------------------------------

@node LinearRegression
@section Function LinearRegression
@mfnindex LinearRegression

@c LinearRegression ../inst/LinearRegression.m
@anchor{XREFLinearRegression}
@deftypefn {Function File} {} LinearRegression (@var{F}, @var{y})
@deftypefnx {Function File} {} LinearRegression (@var{F}, @var{y}, @var{w})
@deftypefnx {Function File} {[@var{p}, @var{e_var}, @var{r}, @var{p_var}, @var{fit_var}] =} LinearRegression (@dots{})


general linear regression

determine the parameters p_j  (j=1,2,...,m) such that the function
f(x) = sum_(j=1,...,m) p_j*f_j(x) is the best fit to the given values
y_i by f(x_i) for i=1,...,n, i.e. minimize
sum_(i=1,...,n)(y_i-sum_(j=1,...,m) p_j*f_j(x_i))^2 with respect to p_j

parameters:  
@itemize
@item @var{F} is an n*m matrix with the values of the basis functions at
the support points. In column j give the values of f_j at the points
x_i  (i=1,2,...,n)
@item @var{y} is a column vector of length n with the given values
@item @var{w} is a column vector of length n with the weights of
the data points. 1/w_i is expected to be proportional to the
estimated uncertainty in the y values. Then the weighted expression
sum_(i=1,...,n)(w_i^2*(y_i-f(x_i))^2) is minimized.
@end itemize

return values:
@itemize
@item @var{p} is the vector of length m with the estimated values
of the parameters
@item @var{e_var} is the vector of estimated variances of the
provided y values. If weights are provided, then the product
e_var_i * w^2_i is assumed to be constant.
@item @var{r} is the weighted norm of the residual
@item @var{p_var} is the vector of estimated variances of the parameters p_j
@item @var{fit_var} is the vector of the estimated variances of the
fitted function values f(x_i)
@end itemize

To estimate the variance of the difference between future y values
and fitted y values use the sum of @var{e_var} and @var{fit_var}

Caution:  
do NOT request @var{fit_var} for large data sets, as a n by n matrix is
generated

@c Will be cut out in optims info file and replaced with the same
@c refernces explicitely there, since references to core Octave
@c functions are not automatically transformed from here to there.
@end deftypefn


See also @ref{XREFols,,ols,octave}, @ref{XREFgls,,gls,octave},
@ref{XREFregress,,regress,octave}, @ref{XREFleasqr,,leasqr},
@ref{XREFnonlin_curvefit,,nonlin_curvefit},
@ref{XREFpolyfit,,polyfit,octave}, @ref{XREFwpolyfit,,wpolyfit},
@ref{XREFexpfit,,expfit}.

@c ------------------------------------------------------------------

@node wsolve
@section Function wsolve, another linear solver
@mfnindex wsolve

@subheading Helptext:

@anchor{XREFwsolve}
@verbatim
[x,s] = wsolve(A,y,dy)

Solve a potentially over-determined system with uncertainty in
the values. 

    A x = y +/- dy

Use QR decomposition for increased accuracy.  Estimate the 
uncertainty for the solution from the scatter in the data.

The returned structure s contains

   normr = sqrt( A x - y ), weighted by dy
   R such that R'R = A'A
   df = n-p, n = rows of A, p = columns of A

See polyconf for details on how to use s to compute dy.
The covariance matrix is inv(R'*R).  If you know that the
parameters are independent, then uncertainty is given by
the diagonal of the covariance matrix, or 

   dx = sqrt(N*sumsq(inv(s.R'))')

where N = normr^2/df, or N = 1 if df = 0.

Example 1: weighted system

   A=[1,2,3;2,1,3;1,1,1]; xin=[1;2;3]; 
   dy=[0.2;0.01;0.1]; y=A*xin+randn(size(dy)).*dy;
   [x,s] = wsolve(A,y,dy);
   dx = sqrt(sumsq(inv(s.R'))');
   res = [xin, x, dx]

Example 2: weighted overdetermined system  y = x1 + 2*x2 + 3*x3 + e

   A = fullfact([3,3,3]); xin=[1;2;3];
   y = A*xin; dy = rand(size(y))/50; y+=dy.*randn(size(y));
   [x,s] = wsolve(A,y,dy);
   dx = s.normr*sqrt(sumsq(inv(s.R'))'/s.df);
   res = [xin, x, dx]

Note there is a counter-intuitive result that scaling the
uncertainty in the data does not affect the uncertainty in
the fit.  Indeed, if you perform a monte carlo simulation
with x,y datasets selected from a normal distribution centered
on y with width 10*dy instead of dy you will see that the
variance in the parameters indeed increases by a factor of 100.
However, if the error bars really do increase by a factor of 10
you should expect a corresponding increase in the scatter of 
the data, which will increase the variance computed by the fit.

@end verbatim

@c ------------------------------------------------------------------

@node Zero finders
@chapter Functions for finding the zero of a nonlinear user function
@cindex zero finders

There is only one dedicated zero finder in the optim package, which is
just a vectorized version of Octaves fzero
(
@mysee
@ref{XREFfzero,,fzero,octave}).

@menu
* vfzero::                A vectorized version of fzero.
@end menu

@c ------------------------------------------------------------------

@node vfzero
@section A vectorized version of fzero
@mfnindex vfzero

@c vfzero ../inst/vfzero.m
@anchor{XREFvfzero}
@deftypefn  {Function File} {} vfzero (@var{fun}, @var{x0})
@deftypefnx {Function File} {} vfzero (@var{fun}, @var{x0}, @var{options})
@deftypefnx {Function File} {[@var{x}, @var{fval}, @var{info}, @var{output}] =} vfzero (@dots{})
A variant of @code{fzero}. Finds a zero of a vector-valued
multivariate function where each output element only depends on the
input element with the same index (so the Jacobian is diagonal).

@var{fun} should be a handle or name of a function returning a column
vector.  @var{x0} should be a two-column matrix, each row specifying
two points which bracket a zero of the respective output element of
@var{fun}.

If @var{x0} is a single-column matrix 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{vfzero}
recognizes these options: @code{"FunValCheck"}, @code{"OutputFcn"},
@code{"TolX"}, @code{"MaxIter"}, @code{"MaxFunEvals"}. For a
description of these options, see optimset.

On exit, the function returns @var{x}, the approximate zero and
@var{fval}, the function value thereof. @var{info} is a column vector
of exit flags  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 nfev Number of function evaluations.

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

@item brackety A two-column matrix with the final bracketing of the
zero along the y-axis.
@end itemize
@end deftypefn


@c ------------------------------------------------------------------

@node Gradient functions
@chapter Functions for numerical approximation of gradients and Hessians
@cindex gradient functions

You should not usually need to use these functions directly or pass them
as arguments. They should be chosen and used by optimizer functions,
possibly subject to their configuration options.

@menu
* dfpdp::                 Direct user interface to default numerical
                            gradient method of new frontends.
* deriv::                 Higher order numerical derivatives.
* numgradient::           Another numerical gradient function.
* numhessian::            Numerical Hessian function.
* cdiff::                 A string, yielding the numerical gradient if
                            evaluated.
* jacobs::                Complex step derivatives.
@end menu

@c ------------------------------------------------------------------

@node dfpdp
@section Direct user interface to default numerical gradient method of new frontends
@mfnindex dfpdp

@subheading Helptext:

@anchor{XREFdfpdp}
@verbatim
function jac = dfpdp (p, func[, hook])

Returns Jacobian of func (p) with respect to p with finite
differencing. The optional argument hook is a structure which can
contain the following fields at the moment:

hook.f: value of func(p) for p as given in the arguments

hook.diffp: positive vector of fractional steps from given p in
finite differencing (actual steps may be smaller if bounds are
given). The default is .001 * ones (size (p)).

hook.diff_onesided: logical vector, indexing elements of p for
which only one-sided differences should be computed (faster); even
if not one-sided, differences might not be exactly central if
bounds are given. The default is false (size (p)).

hook.fixed: logical vector, indexing elements of p for which zero
should be returned instead of the guessed partial derivatives
(useful in optimization if some parameters are not optimized, but
are 'fixed').

hook.lbound, hook.ubound: vectors of lower and upper parameter
bounds (or -Inf or +Inf, respectively) to be respected in finite
differencing. The consistency of bounds is not checked.

@end verbatim

@c ------------------------------------------------------------------

@node deriv
@section Higher order numerical derivatives
@mfnindex deriv

@c deriv ../inst/deriv.m
@anchor{XREFderiv}
@deftypefn {Function File} {@var{dx} =} deriv (@var{f}, @var{x0})
@deftypefnx {Function File} {@var{dx} =} deriv (@var{f}, @var{x0}, @var{h})
@deftypefnx {Function File} {@var{dx} =} deriv (@var{f}, @var{x0}, @var{h}, @var{O})
@deftypefnx {Function File} {@var{dx} =} deriv (@var{f}, @var{x0}, @var{h}, @var{O}, @var{N})
Calculate derivate of function @var{f}.

@var{f} must be a function handle or the name of a function that takes @var{x0}
and returns a variable of equal length and orientation. @var{x0} must be a
numeric vector or scalar.

@var{h} defines the step taken for the derivative calculation. Defaults to 1e-7.

@var{O} defines the order of the calculation. Supported values are 2 (h^2 order)
or 4 (h^4 order). Defaults to 2.

@var{N} defines the derivative order. Defaults to the 1st derivative of the
function. Can be up to the 4th derivative.

Reference: Numerical Methods for Mathematics, Science, and Engineering by
John H. Mathews.
@end deftypefn


@c ------------------------------------------------------------------

@node numgradient
@section Another numerical gradient function
@mfnindex numgradient

@subheading Helptext:

@anchor{XREFnumgradient}
@verbatim
numgradient(f, {args}, minarg)

Numeric central difference gradient of f with respect
to argument "minarg".
* first argument: function name (string)
* second argument: all arguments of the function (cell array)
* third argument: (optional) the argument to differentiate w.r.t.
        (scalar, default=1)

"f" may be vector-valued. If "f" returns
an n-vector, and the argument is a k-vector, the gradient
will be an nxk matrix

Example:
function a = f(x);
        a = [x'*x; 2*x];
endfunction
numgradient("f", {ones(2,1)})
ans =

  2.00000  2.00000
  2.00000  0.00000
  0.00000  2.00000



@end verbatim

@c ------------------------------------------------------------------

@node numhessian
@section Numerical Hessian function
@mfnindex numhessian

@subheading Helptext:

@anchor{XREFnumhessian}
@verbatim
numhessian(f, {args}, minarg)

Numeric second derivative of f with respect
to argument "minarg".
* first argument: function name (string)
* second argument: all arguments of the function (cell array)
* third argument: (optional) the argument to differentiate w.r.t.
        (scalar, default=1)

If the argument
is a k-vector, the Hessian will be a kxk matrix

function a = f(x, y)
        a = x'*x + log(y);
endfunction

numhessian("f", {ones(2,1), 1})
ans =

    2.0000e+00   -7.4507e-09
   -7.4507e-09    2.0000e+00

Now, w.r.t. second argument:
numhessian("f", {ones(2,1), 1}, 2)
ans = -1.0000



@end verbatim

@c ------------------------------------------------------------------

@node cdiff
@section A string, yielding the numerical gradient if evaluated
@mfnindex cdiff

@subheading Helptext:

@anchor{XREFcdiff}
@verbatim
c = cdiff (func,wrt,N,dfunc,stack,dx) - Code for num. differentiation
  = "function df = dfunc (var1,..,dvar,..,varN) .. endfunction

Returns a string of octave code that defines a function 'dfunc' that
returns the derivative of 'func' with respect to it's 'wrt'th
argument.

The derivatives are obtained by symmetric finite difference.

dfunc()'s return value is in the same format as that of  ndiff()

func  : string : name of the function to differentiate

wrt   : int    : position, in argument list, of the differentiation
                 variable.                                Default:1

N     : int    : total number of arguments taken by 'func'. 
                 If N=inf, dfunc will take variable argument list.
                                                        Default:wrt

dfunc : string : Name of the octave function that returns the
                  derivatives.                   Default:['d',func]

stack : string : Indicates whether 'func' accepts vertically
                 (stack="rstack") or horizontally (stack="cstack")
                 arguments. Any other string indicates that 'func'
                 does not allow stacking.                Default:''

dx    : real   : Step used in the symmetric difference scheme.
                                                 Default:10*sqrt(eps)

See also : ndiff, eval, todisk

@end verbatim

@c ------------------------------------------------------------------

@node jacobs
@section Complex step derivatives
@mfnindex jacobs

@c jacobs ../inst/jacobs.m
@anchor{XREFjacobs}
@deftypefn {Function File} {Df =} jacobs (@var{x}, @var{f})
@deftypefnx {Function File} {Df =} jacobs (@var{x}, @var{f}, @var{hook})
Calculate the jacobian of a function using the complex step method.

Let @var{f} be a user-supplied function. Given a point @var{x} at
which we seek for the Jacobian, the function @command{jacobs} returns
the Jacobian matrix @code{d(f(1), @dots{}, df(end))/d(x(1), @dots{},
x(n))}. The function uses the complex step method and thus can be
applied to real analytic functions.

The optional argument @var{hook} is a structure with additional options. @var{hook}
can have the following fields:
@itemize @bullet
@item
@code{h} - can be used to define the magnitude of the complex step and defaults
to 1e-20; steps larger than 1e-3 are not allowed.
@item
@code{fixed} - is a logical vector internally usable by some optimization
functions; it indicates for which elements of @var{x} no gradient should be
computed, but zero should be returned.
@end itemize

For example:

@example
@group
f = @@(x) [x(1)^2 + x(2); x(2)*exp(x(1))];
Df = jacobs ([1, 2], f)
@end group
@end example
@end deftypefn


@c ------------------------------------------------------------------

@node Helper functions
@chapter Functions for algebraic tasks common to optimization problems
@cindex helper functions

@menu
* cpiv_bard::             A complementary pivoting algorithm.
* gjp::                   Gauss-Jordan pivoting.
@end menu

@c ------------------------------------------------------------------

@node cpiv_bard
@section A complementary pivoting algorithm
@mfnindex cpiv_bard

@subheading Helptext:

@anchor{XREFcpiv_bard}
@verbatim
[lb, idx, ridx, mv] = cpiv_bard (v, m[, incl])

v: column vector; m: matrix; incl (optional): index. length (v)
must equal rows (m). Finds column vectors w and l with w == v + m *
l, w >= 0, l >= 0, l.' * w == 0. Chooses idx, w, and l so that
l(~idx) == 0, l(idx) == -inv (m(idx, idx)) * v(idx), w(idx) roughly
== 0, and w(~idx) == v(~idx) + m(idx, ~idx).' * l(idx). idx indexes
at least everything indexed by incl, but l(incl) may be < 0. lb:
l(idx) (column vector); idx: logical index, defined above; ridx:
~idx & w roughly == 0; mv: [m, v] after performing a Gauss-Jordan
'sweep' (with gjp.m) on each diagonal element indexed by idx.
Except the handling of incl (which enables handling of equality
constraints in the calling code), this is called solving the
'complementary pivot problem' (Cottle, R. W. and Dantzig, G. B.,
'Complementary pivot theory of mathematical programming', Linear
Algebra and Appl. 1, 102--125. References for the current
algorithm: Bard, Y.: Nonlinear Parameter Estimation, p. 147--149,
Academic Press, New York and London 1974; Bard, Y., 'An eclectic
approach to nonlinear programming', Proc. ANU Sem. Optimization,
Canberra, Austral. Nat. Univ.).

@end verbatim

@c ------------------------------------------------------------------

@node gjp
@section Gauss-Jordan pivoting
@mfnindex gjp

@subheading Helptext:

@anchor{XREFgjp}
@verbatim
m = gjp (m, k[, l])

m: matrix; k, l: row- and column-index of pivot, l defaults to k.

Gauss-Jordon pivot as defined in Bard, Y.: Nonlinear Parameter
Estimation, p. 296, Academic Press, New York and London 1974. In
the pivot column, this seems not quite the same as the usual
Gauss-Jordan(-Clasen) pivot. Bard gives Beaton, A. E., 'The use of
special matrix operators in statistical calculus' Research Bulletin
RB-64-51 (1964), Educational Testing Service, Princeton, New Jersey
as a reference, but this article is not easily accessible. Another
reference, whose definition of gjp differs from Bards by some
signs, is Clarke, R. B., 'Algorithm AS 178: The Gauss-Jordan sweep
operator with detection of collinearity', Journal of the Royal
Statistical Society, Series C (Applied Statistics) (1982), 31(2),
166--168.

@end verbatim

@c ------------------------------------------------------------------

@node Documentation
@chapter Function optim_doc to view documentation
@mfnindex optim_doc

@c optim_doc ../inst/optim_doc.m
@anchor{XREFoptim_doc}
@deftypefn {Function File} {} optim_doc ()
@deftypefnx {Function File} {} optim_doc (@var{keyword})
Show optim package documentation.

Runs the info viewer Octave is configured with on the documentation
in info format of the installed optim package. Without argument, the
top node of the documentation is displayed. With an argument, the
respective index entry is searched for and its node displayed.

@end deftypefn


@c ------------------------------------------------------------------

@node Compatibility functions
@chapter Traditional functions
@cindex compatibility functions

@menu
* linprog::                    Linear programming.
* quadprog::                   Quadratic programming.
* lsqnonlin::                  Non-linear residual minimization.
* lsqcurvefit::                Curve fitting.
* nlinfit::                    Non-linear regression.
* fmincon::                    Non-linear scalar optimization.
@end menu

@c ------------------------------------------------------------------

@node linprog
@section Linear programming
@mfnindex linprog

This function works by calling @code{glpk} of core Octave.

@c linprog ../inst/linprog.m
@anchor{XREFlinprog}
@deftypefn{Function File} {@var{x} =} linprog (@var{f}, @var{A}, @var{b})
@deftypefnx{Function File} {@var{x} =} linprog (@var{f}, @var{A}, @var{b}, @var{Aeq}, @var{beq})
@deftypefnx{Function File} {@var{x} =} linprog (@var{f}, @var{A}, @var{b}, @var{Aeq}, @var{beq}, @var{lb}, @var{ub})
@deftypefnx{Function File} {[@var{x}, @var{fval}] =} linprog (@dots{})
Solve a linear problem.

Finds

@example
min (f' * x)
@end example

(both f and x are column vectors) subject to

@example
@group
A   * x <= b
Aeq * x  = beq
lb <= x <= ub
@end group
@end example

If not specified, @var{Aeq} and @var{beq} default to empty matrices.

If not specified, the lower bound @var{lb} defaults to minus infinite
and the upper bound @var{ub} defaults to infinite.

@c Will be cut out in optims info file and replaced with the same
@c refernces explicitely there, since references to core Octave
@c functions are not automatically transformed from here to there.
@end deftypefn


See also @ref{XREFglpk,,glpk,octave}.

@c ------------------------------------------------------------------

@node quadprog
@section Quadratic programming
@mfnindex quadprog

This function is similar to @code{qp} of core Octave.

@c quadprog ../inst/quadprog.m
@anchor{XREFquadprog}
@deftypefn {Function File} {} quadprog (@var{H}, @var{f})
@deftypefnx {Function File} {} quadprog (@var{H}, @var{f}, @var{A}, @var{b})
@deftypefnx {Function File} {} quadprog (@var{H}, @var{f}, @var{A}, @var{b}, @var{Aeq}, @var{beq})
@deftypefnx {Function File} {} quadprog (@var{H}, @var{f}, @var{A}, @var{b}, @var{Aeq}, @var{beq}, @var{lb}, @var{ub})
@deftypefnx {Function File} {} quadprog (@var{H}, @var{f}, @var{A}, @var{b}, @var{Aeq}, @var{beq}, @var{lb}, @var{ub}, @var{x0})
@deftypefnx {Function File} {} quadprog (@var{H}, @var{f}, @var{A}, @var{b}, @var{Aeq}, @var{beq}, @var{lb}, @var{ub}, @var{x0}, @var{options})
@deftypefnx {Function File} {[@var{x}, @var{fval}, @var{exitflag}, @var{output}, @var{lambda}] =} quadprog (@dots{})
Solve the quadratic program
@example
@group
min 0.5 x'*H*x + x'*f
 x
@end group
@end example
subject to
@example
@group
@var{A}*@var{x} <= @var{b},
@var{Aeq}*@var{x} = @var{beq},
@var{lb} <= @var{x} <= @var{ub}.
@end group
@end example

The initial guess @var{x0} and the constraint arguments (@var{A} and
@var{b}, @var{Aeq} and @var{beq}, @var{lb} and @var{ub}) can be set to
the empty matrix (@code{[]}) if not given.  If the initial guess
@var{x0} is feasible the algorithm is faster.

@var{options} can be set with @code{optimset}, currently the only
option is @code{MaxIter}, the maximum number of iterations (default:
200).

Returned values:

@table @var
@item x
Position of minimum.

@item fval
Value at the minimum.

@item exitflag
Status of solution:

@table @code
@item 0
Maximum number of iterations reached.

@item -2
The problem is infeasible.

@item -3
The problem is not convex and unbounded

@item 1
Global solution found.

@item 4
Local solution found.
@end table

@item output
Structure with additional information, currently the only field is
@code{iterations}, the number of used iterations.

@item lambda
Structure containing Lagrange multipliers corresponding to the
constraints. For equality constraints, the sign of the multipliers
is chosen to satisfy the equation
@example
0.5 H * x + f + A' * lambda_inequ + Aeq' * lambda_equ = 0 .
@end example
If lower and upper bounds are equal, or so close to each other that
they are considered equal by the algorithm, only one of these
bounds is considered active when computing the solution, and a
positive lambda will be placed only at this bound.

@end table

This function calls Octave's @code{__qp__} back-end algorithm internally.
@end deftypefn


@c ------------------------------------------------------------------

@node lsqnonlin
@section Non-linear residual minimization
@mfnindex lsqnonlin

This function is for Matlab compatibility. It attempts to work like
@code{lsqnonlin} by calling @ref{nonlin_residmin}.

@c lsqnonlin ../inst/lsqnonlin.m
@anchor{XREFlsqnonlin}
@deftypefn {Function File} {} lsqnonlin (@var{fun}, @var{x0})
@deftypefnx {Function File} {} lsqnonlin (@var{fun}, @var{x0}, @var{lb}, @var{ub})
@deftypefnx {Function File} {} lsqnonlin (@var{fun}, @var{x0}, @var{lb}, @var{ub}, @var{options})
@deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}, @var{output}, @var{lambda}, @var{jacobian}] =} lsqnonlin (@dots{})
Solve nonlinear least-squares (nonlinear data-fitting) problems
@example
@group
min [EuclidianNorm(f(x))] .^ 2
 x   
@end group
@end example

@var{fun} computes residuals from given parameters. The initial
guess of the parameters @var{x0} must be provided while the bounds
@var{lb} and @var{ub}) can be set to the empty matrix (@code{[]})
if not given.

@code{lsqnonlin} may also be called with a single structure
argument with the fields @code{fun}, @code{x0}, @code{lb},
@code{ub}, and @code{options}, resembling the separate input
arguments above. For compatibility reasons, field @code{fun} may
also be called @code{objective}. Additionally, the structure must
have the field @code{solver}, set to @qcode{"lsqnonlin"}.

@var{options} can be set with @code{optimset}. Follwing Matlab compatible options
are recognized:

@code{Algorithm}
   String specifying backend algorithm. Currently available "lm_svd_feasible"
   only. 

@code{TolFun}
   Minimum fractional improvement in objective function in an iteration 
   (termination criterium). Default: 1e-6. 

@code{TypicalX}
   Typical values of x. Default: 1.

@code{MaxIter}
   Maximum number of iterations allowed. Default: 400.

@code{Jacobian}
   If set to "on", the function @var{fun} must return a second output
   containing a user-specified Jacobian. The Jacobian is computed using
   finite differences otherwise. Default: "off"

@code{FinDiffType}
   "centered" or "forward" (Default) type finite differences estimation.

@code{FinDiffRelStep}
   Step size factor. The default is sqrt(eps) for forward finite differences,
   and eps^(1/3) for central finite differences

@code{OutputFcn}
   One or more user-defined functions, either as a function handle or as a
   cell array of function handles that an optimization function calls at each
   iteration. The function definition has the following form:

@code{stop = outfun(x, optimValues, state)}
   
@code{x} is the point computed at the current iteration.
@code{optimValues} is a structure containing data from the current
   iteration in the following fields:
   "iteration"- number of current iteration.
   "residual"- residuals.
@code{state} is the state of the algorithm: "init" at start,
   "iter" after each iteration and "done" at the end.

@code{Display}
   String indicating the degree of verbosity. Default: "off". 
   Currently only supported values are "off" (no messages) and "iter" 
   (some messages after each iteration).    

Returned values:

@table @var
@item x
Position of minimum.

@item resnorm
Scalar value of objective as squared EuclidianNorm(f(x)).

@item residual
Value of solution residuals f(x).

@item exitflag
Status of solution:

@table @code
@item 0
Maximum number of iterations reached.

@item 2
Change in x was less than the specified tolerance.

@item 3
Change in the residual was less than the specified tolerance.

@item -1
Output function terminated the algorithm.
@end table

@item output
Structure with additional information, currently the only field is
@code{iterations}, the number of used iterations.

@item lambda
Structure containing Lagrange multipliers at the solution @var{x} sepatared by constraint type (@var{lb} and @var{ub}).

@item jacobian
m-by-n matrix, where @var{jacobian(i,j)} is the partial derivative of @var{fun(i)} with respect to @var{x(j)}
Default: lsqnonlin approximates the Jacobian using finite differences. If @code{Jacobian} is set to "on" in 
@var{options} then @var{fun} must return a second argument providing a user-sepcified Jacobian .
@end table

This function is a compatibility wrapper. It calls the more general @code{nonlin_residmin} function internally.

@seealso{@ref{XREFlsqcurvefit,,lsqcurvefit}, @ref{XREFnonlin_residmin,,nonlin_residmin}, @ref{XREFnonlin_curvefit,,nonlin_curvefit}}
@end deftypefn


@c ------------------------------------------------------------------

@node lsqcurvefit
@section Curve fitting
@mfnindex lsqcurvefit

This function is for Matlab compatibility. It attempts to work like
@code{lsqcurvefit} by calling @ref{nonlin_curvefit}.

@c lsqcurvefit ../inst/lsqcurvefit.m
@anchor{XREFlsqcurvefit}
@deftypefn {Function File} {} lsqcurvefit (@var{fun}, @var{x0}, @var{xdata}, @var{ydata})
@deftypefnx {Function File} {} lsqcurvefit (@var{fun}, @var{x0}, @var{xdata}, @var{ydata}, @var{lb}, @var{ub})
@deftypefnx {Function File} {} lsqcurvefit (@var{fun}, @var{x0}, @var{xdata}, @var{ydata}, @var{lb}, @var{ub}, @var{options})
@deftypefnx {Function File} {} lsqcurvefit (@var{problem})
@deftypefnx {Function File} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}, @var{output}, @var{lambda}, @var{jacobian}] =} lsqcurvefit (@dots{})
Solve nonlinear least-squares (nonlinear data-fitting) problems
@example
@group
min [EuclidianNorm (f(x, xdata) - ydata)] .^ 2
 x
@end group
@end example

The first four input arguments must be provided with non-empty
initial guess @var{x0}. For a given input @var{xdata}, @var{ydata}
is the observed output. @var{ydata} must be the same size as the
vector (or matrix) returned by @var{fun}. The optional bounds
@var{lb} and @var{ub} should be the same size as @var{x0}.

@code{lsqcurvefit} may also be called with a single structure
argument with the fields @code{fun}, @code{x0}, @code{xdata},
@code{ydata}, @code{lb}, @code{ub}, and @code{options}, resembling
the separate input arguments above. For compatibility reasons,
field @code{fun} may also be called @code{objective}. Additionally,
the structure must have the field @code{solver}, set to
@qcode{"lsqcurvefit"}.

@var{options} can be set with @code{optimset}.
Follwing Matlab compatible options
are recognized:

@code{Algorithm}
   String specifying backend algorithm. Currently available "lm_svd_feasible"
   only.

@code{TolFun}
   Minimum fractional improvement in objective function in an iteration
   (termination criterium). Default: 1e-6.

@code{TypicalX}
   Typical values of x. Default: 1.

@code{MaxIter}
   Maximum number of iterations allowed. Default: 400.

@code{Jacobian}
   If set to "on", the function @var{fun} must return a second output
   containing a user-specified Jacobian. The Jacobian is computed using
   finite differences otherwise. Default: "off"

@code{FinDiffType}
   "centered" or "forward" (Default) type finite differences estimation.

@code{FinDiffRelStep}
   Step size factor. The default is sqrt(eps) for forward finite differences,
   and eps^(1/3) for central finite differences

@code{OutputFcn}
   One or more user-defined functions, either as a function handle or as a
   cell array of function handles that an optimization function calls at each
   iteration. The function definition has the following form:

@code{stop = outfun(x, optimValues, state)}

@code{x} is the point computed at the current iteration.
@code{optimValues} is a structure containing data from the current
   iteration in the following fields:
   "iteration"- number of current iteration.
   "residual"- residuals.
@code{state} is the state of the algorithm: "init" at start,
   "iter" after each iteration and "done" at the end.

@code{Display}
   String indicating the degree of verbosity. Default: "off".
   Currently only supported values are "off" (no messages) and "iter"
   (some messages after each iteration).

Returned values:

@table @var
@item x
Coefficients to best fit the nonlinear function fun(x,xdata) to the observed values ydata.

@item resnorm
Scalar value of objective as squared EuclidianNorm(f(x)).

@item residual
Value of solution residuals f(x).

@item exitflag
Status of solution:

@table @code
@item 0
Maximum number of iterations reached.

@item 2
Change in x was less than the specified tolerance.

@item 3
Change in the residual was less than the specified tolerance.

@item -1
Output function terminated the algorithm.
@end table

@item output
Structure with additional information, currently the only field is
@code{iterations}, the number of used iterations.

@item lambda
Structure containing Lagrange multipliers at the solution @var{x} sepatared by constraint type (@var{lb} and @var{ub}).

@item jacobian
m-by-n matrix, where @var{jacobian}(i,j) is the partial derivative of @var{fun(i)} with respect to @var{x(j)}
If @code{Jacobian} is set to "on" in @var{options} then @var{fun} must return a second argument providing a user-sepcified Jacobian. Otherwise, lsqnonlin approximates the Jacobian using finite differences.
@end table

This function is a compatibility wrapper. It calls the more general @code{nonlin_curvefit} function internally.

@seealso{@ref{XREFlsqnonlin,,lsqnonlin}, @ref{XREFnonlin_residmin,,nonlin_residmin}, @ref{XREFnonlin_curvefit,,nonlin_curvefit}}
@end deftypefn


@c ------------------------------------------------------------------

@node nlinfit
@section Non-linear regression
@mfnindex nlinfit

This function is for Matlab compatibility. It attempts to work like
@code{nlinfit} by calling @ref{nonlin_curvefit} and @ref{curvefit_stat}.

@c nlinfit ../inst/nlinfit.m
@anchor{XREFnlinfit}
@deftypefn {Function File} {} nlinfit (@var{X}, @var{Y}, @var{modelfun}, @var{beta0})
@deftypefnx {Function File} {} nlinfit (@var{X}, @var{Y}, @var{modelfun}, @var{beta0}, @var{options})
@deftypefnx {Function File} {} nlinfit (@dots{}, @var{Name}, @var{Value})
@deftypefnx {Function File} {[@var{beta}, @var{R}, @var{J}, @var{CovB}, @var{MSE}] =} nlinfit (@dots{})
Nonlinear Regression.

@example
@group
min [EuclidianNorm (Y - modelfun (beta, X))] ^ 2
beta
@end group
@end example

@var{X} is a matrix of independents, @var{Y} is the observed output and @var{modelfun} is the nonlinear regression model function.
@var{modelfun} should be specified as a function handle, which
accepts two inputs: an array of coefficients and an array of
independents -- in that order.
The first four input arguments must be provided with non-empty initial guess of the coefficients @var{beta0}.
@var{Y} and @var{X} must be the same size as the vector (or matrix) returned by @var{fun}.
@var{options} is a structure containing estimation algorithm options. It can be set using @code{statset}.
Follwing Matlab compatible options are recognized:

@code{TolFun}
   Minimum fractional improvement in objective function in an iteration
   (termination criterium). Default: 1e-6.

@code{MaxIter}
   Maximum number of iterations allowed. Default: 400.

@code{DerivStep}
   Step size factor. The default is eps^(1/3) for finite differences gradient
   calculation.

@code{Display}
   String indicating the degree of verbosity. Default: "off".
   Currently only supported values are "off" (no messages) and "iter"
   (some messages after each iteration).

Optional @var{Name}, @var{Value} pairs can be provided to set additional options.
Currently the only applicable name-value pair is 'Weights', w,
where w is the array of real positive weight factors for the
squared residuals.

Returned values:

@table @var
@item beta
Coefficients to best fit the nonlinear function modelfun (beta, X) to the observed values Y.

@item R
Value of solution residuals: @code{modelfun (beta, X) - Y}.
If observation weights are specified then @var{R} is the array of
weighted residuals: @code{sqrt (weights) .* modelfun (beta, X) - Y}.

@item J
A matrix where @code{J(i,j)} is the partial derivative of @code{modelfun(i)} with respect to @code{beta(j)}.
If observation weights are specified, then @var{J} is the weighted
model function Jacobian: @code{diag (sqrt (weights)) * J}.

@item CovB

Estimated covariance matrix of the fitted coefficients.

@item MSE
Scalar valued estimate of the variance of error term. If the model Jacobian is full rank, then MSE = (R' * R)/(N-p),
where N is the number of observations and p is the number of estimated coefficients.
@end table

This function is a compatibility wrapper. It calls the more general @code{nonlin_curvefit}
and @code{curvefit_stat} functions internally.

@seealso{@ref{XREFnonlin_residmin,,nonlin_residmin}, @ref{XREFnonlin_curvefit,,nonlin_curvefit}, @ref{XREFresidmin_stat,,residmin_stat}, @ref{XREFcurvefit_stat,,curvefit_stat}}
@end deftypefn


@c ------------------------------------------------------------------

@node fmincon
@section Non-linear scalar optimization
@mfnindex fmincon

@c fmincon ../inst/fmincon.m
@anchor{XREFfmincon}
@deftypefn {Function File} {} fmincon (@var{objf}, @var{x0})
@deftypefnx {Function File} {} fmincon (@var{objf}, @var{x0}, @var{A}, @var{b})
@deftypefnx {Function File} {} fmincon (@var{objf}, @var{x0}, @var{A}, @var{b}, @var{Aeq}, @var{beq})
@deftypefnx {Function File} {} fmincon (@var{objf}, @var{x0}, @var{A}, @var{b}, @var{Aeq}, @var{beq}, @var{lb}, @var{ub})
@deftypefnx {Function File} {} fmincon (@var{objf}, @var{x0},  @var{A}, @var{b}, @var{Aeq}, @var{beq}, @var{lb}, @var{ub}, @var{nonlcon})
@deftypefnx {Function File} {} fmincon (@var{objf}, @var{x0}, @var{A}, @var{b}, @var{Aeq}, @var{beq}, @var{lb}, @var{ub}, @var{nonlcon}, @var{options})
@deftypefnx {Function File} {} fmincon (@var{problem})
@deftypefnx {Function File} {[@var{x}, @var{fval}, @var{cvg}, @var{outp}] =} fmincon (@dots{})
Compatibility frontend for nonlinear minimization of a scalar
objective function.

This function is for Matlab compatibility and provides a subset of
the functionality of @code{nonlin_min}.

@var{objf}: objective function. It gets a column vector of real
parameters as argument.

@var{x0}: real column vector of initial parameters.

@var{A}, @var{b}: Inequality constraints of the parameters @code{p}
with @code{A * p - b <= 0}.

@var{Aeq}, @var{beq}: Equality constraints of the parameters @code{p}
with @code{A * p - b = 0}.

@var{lb}, @var{ub}: Bounds of the parameters @code{p} with @code{lb <= p <= ub}.

@var{nonlcon}: Nonlinear constraints. Function returning the
current values of nonlinear inequality constraints (constrained to
@code{<= 0}) in the first output and the current values of nonlinear
equality constraints in the second output.

@var{options}: structure whose fields stand for optional settings
referred to below. The fields can be set by @code{optimset()}.

An argument can be set to @code{[]} to indicate that its value is
not set.

@code{fmincon} may also be called with a single structure
argument with the fields @code{objective}, @code{x0}, @code{Aineq},
@code{bineq}, @code{Aeq}, @code{beq}, @code{lb}, @code{ub},
@code{nonlcon} and @code{options}, resembling
the separate input arguments above. Additionally,
the structure must have the field @code{solver}, set to
@qcode{"fmincon"}.

The returned values are the column vector of final parameters
@var{x}, the final value of the objective function @var{fval}, an
integer @var{cvg} indicating if and how optimization succeeded or
failed, and a structure @var{outp} with additional information,
curently with possible fields: @code{iterations}, the number of
iterations, @code{funcCount}, the number of objective function
calls (indirect calls by gradient function not counted),
@code{constrviolation}, the maximum of the constraint
violations. The backend may define additional fields. @var{cvg} is
greater than zero for success and less than or equal to zero for
failure; its possible values depend on the used backend and
currently can be @code{0} (maximum number of iterations exceeded),
@code{1} (success without further specification of criteria),
@code{2} (parameter change less than specified precision in two
consecutive iterations), @code{3} (improvement in objective
function less than specified), @code{-1} (algorithm aborted by a
user function), or @code{-4} (algorithm got stuck).

@subsubheading Options:

@table @code

@item Algorithm
@code{interior-point}, @code{sqp}, and @code{sqp-legacy} are
mapped to optims @code{lm_feasible} algorithm (the default) to
satisfy constraints throughout the optimization. @code{active-set}
is mapped to @code{octave_sqp}, which may perform better if
constraints only need to be satisfied for the result. Other
algorithms are available with @code{nonlin_min}.

@item OutputFcn
Similar to the setting @code{user_interaction} --- see
@code{optim_doc()}. Differently, @code{OutputFcn} returns only one
output argument, the  @var{stop} flag.

@item GradObj
If set to @code{"on"}, @var{objf} must return the gradient of the
objective function as a second output. The default is @code{"off"}.

@item GradConstr
If set to @code{"on"}, @var{nonlcon} must return the Jacobians of
the inequality- and equality-constraints as third and fourth
output, respectively.

@item HessianFcn
If set to @code{"objective"}, @var{objf} must not only return the
gradient as the second, but also the Hessian as the third output.

@item Display, FinDiffRelStep, FinDiffType, TypicalX, MaxIter, TolFun, TolX,
See documentation of these options in @code{optim_doc()}.

@end table

For description of individual backends, type
@code{optim_doc ("scalar optimization")} and choose the backend in
the menu.

@end deftypefn


@c ------------------------------------------------------------------

@node Common frontend options
@chapter Options common to all frontends
@cindex common options

All frontends for optimization and for result statistics
(@ref{nonlin_min}, @ref{nonlin_residmin}, @ref{nonlin_curvefit},
@ref{residmin_stat}, @ref{curvefit_stat})accept the following options,
settable with @ref{XREFoptimset,,optimset,octave}.

These options are handled within the frontend.

@table @code
@item FinDiffRelStep
Column vector (or scalar, for all parameters) of fractional intervals
supposed to be used by gradient or Jacobian functions performing finite
differencing. Default: @code{.002 * ones (size (parameters))} for
central intervals and @code{.001 * ones (size (parameters))} for
one-sided intervals. The default function for finite differencing won't
let the absolute interval width get smaller than
@code{abs (FinDiffRelStep .* TypicalX} (see below).
@item diffp
Can be used alternatively to @code{FinDiffRelStep}, but for central
intervals twice the specified value will be used for backwards compatibility.
@item diff_onesided
Logical column vector (or scalar, for all parameters) indicating the
parameters for which one-sided intervals (instead of central intervals)
should be used by gradient or Jacobian functions performing finite
differencing. Default: @code{false (size (parameters))}.
@item FinDiffType
Can be used alternatively to @code{diff_onesided}, but always applies to
all parameters at once. Possible values: @code{"central"} (central
intervals) or @code{"forward"} (one-sided intervals).
@item TypicalX
Column vector (or scalar, for all parameters) whose absolute value
specifies minimal absolute parameter values for computation of intervals
in finite differencing by gradient or Jacobian functions (see
@code{FinDiffRelStep}). Default: 0.0001. Must not be zero.
@item cstep
Scalar step size for complex step derivative approximation of gradients
or Jacobians. Default: 1e-20.
@item parallel_local
Logical or numeric scalar, default: @code{false}. If the @code{parallel}
package, @code{version >= 2.0.5}, is loaded, estimate gradients of
objective function and Jacobians of model function and of constraints in
parallel processes. If @code{parallel_local} is set to an integer
@code{> 1}, this is number of parallel processes; if it is @code{<= 1},
the number of processes will be the number of available processor cores.
Works for default (real) finite differences and for complex step
derivatives. Due to overhead, a speed advantage can only be expected if
objective function, model function or constraint functions are time
consuming enough. Additionally, this setting is also passed to the
individual optimization backends, which may also consider this option
(see documentation of backends). If this option is equivalent to
@code{true}, a warning (ID: @code{optim:parallel_local}) will be issued
if no @code{parallel} package of a correct version is loaded.
@item parallel_net
Empty (default) or a parallel connections object, see function
@code{pconnect} of the @code{parallel} package. If not empty, estimate
gradients of objective function and Jacobians of model function and of
constraints using parallel processing in a network of machines. The
considerations regarding a speed advantage are similar to those for
option @code{parallel_local}.
@item fixed
Logical column vector indicating which parameters are not optimized, but
kept to their inital value.
@end table

@c ------------------------------------------------------------------

@node Common optimization options
@chapter Options common to all optimization frontends
@cindex common optimization options

All frontends for optimization (@ref{nonlin_min}, @ref{nonlin_residmin},
@ref{nonlin_curvefit}) accept the following options, settable with
@ref{XREFoptimset,,optimset,octave}.

@subheading Settings handled within the frontend

@table @code
@item Algorithm
String specifying the backend.
@item complex_step_derivative_inequc,
@item complex_step_derivative_equc
Logical scalars, default: @code{false}. Estimate Jacobian of general
inequality constraints and equality constraints, respectively, with
complex step derivative approximation. Use only if you know that your
function of general inequality constraints or function of general
equality constraints, respectively, is suitable for this. No user
function for the respective Jacobian must be specified.
@end table

@subheading Settings passed to the backend

Which of these options are actually honored is noted in the descriptions
of the individual backends.

@table @code
@item lbound,
@item ubound
Column vectors of lower and upper bounds for parameters.  Default:
@code{-Inf} and @code{+Inf}, respectively.  The bounds are non-strict,
i.e. parameters are allowed to be exactly equal to a bound.  The default
function for gradients or Jacobians will respect bounds (but no further
inequality constraints) in finite differencing if the backend respects
bounds even during the course of optimization.
@item inequc
Further inequality constraints.  Cell-array containing up to four
entries, two entries for linear inequality constraints and/or one or two
entries for general inequality constraints.  Either linear or general
constraints may be the first entries, but the two entries for linear
constraints must be adjacent and, if two entries are given for general
constraints, they also must be adjacent.  The two entries for linear
constraints are a matrix (say @code{m}) and a vector (say @code{v}),
specifying linear inequality constraints of the form @code{m.' *
parameters + v >= 0}.  The first entry for general constraints must be a
differentiable column-vector valued function (say @code{h}), specifying
general inequality constraints of the form @code{h (p[, idx]) >= 0};
@code{p} is the column vector of optimized paraters and the optional
argument @code{idx} is a logical index.  @code{h} has to return the
values of all constraints if @code{idx} is not given.  It may choose to
return only the indexed constraints if @code{idx} is given (so
computation of the other constraints can be spared); in this case, the
additional setting @code{f_inequc_idx} has to be set to @code{true}.  In
gradient determination, this function may be called with an
informational third argument, whose content depends on the function for
gradient determination.  If a second entry for general inequality
constraints is given, it must be a function computing the jacobian of
the constraints with respect to the parameters.  For this function, the
description of the setting @code{dfdp},
@mysee
@ref{XREFoptiondfdp,,dfdp},
applies, with 2 exceptions: 1) it is called with 3 arguments since it
has an additional argument @code{idx}, a logical index, at second
position, indicating which rows of the jacobian must be returned (if the
function chooses to return only indexed rows, the additional setting
@code{df_inequc_idx} has to be set to @code{true}).  2) the default
jacobian function calls @code{h} with 3 arguments, since the argument
@code{idx} is also supplied.  Note that specifying linear constraints as
general constraints will generally waste performance, even if further,
non-linear, general constraints are also specified.
@item f_inequc_idx,
@item df_inequc_idx
Indicate that functions for general inequality constraints or their
jacobian, respectively, return only the values or derivatives for the
indexed parameters.  See description of setting @code{inequc} above.
@item equc
Equality constraints.  Specified the same way as inequality constraints
(see @code{inequc} above).
@item f_equc_idx,
@item df_equc_idx
As @code{f_inequc_idx} and @code{df_inequc_idx} above, but for equality
constraints.
@item cpiv
Function for complementary pivoting, usable in algorithms for
constraints.  Default: @code{cpiv_bard}.  Only the default function is
supplied with the package.
@item TolFun
Minimum fractional improvement in objective function (e.g. sum of
squares) in an iteration (termination criterium).  Default: .0001.
@item TolX
Minimum fractional change in a norm of the parameters in an iteration
(termination criterium).  Default: backend specific.
@item MaxIter
Maximum number of iterations (termination criterium).  Default:
backend-specific.
@item fract_prec
Column Vector, minimum fractional changes of corresponding parameters in
an iteration (termination criterium if violated in two consecutive
iterations).  Default: backend-specific.
@item max_fract_change
Column Vector, enforced maximum fractional changes in corresponding
parameters in an iteration. Default: backend-specific.
@item Display
String indicating the degree of verbosity. Default:
@qcode{"off"}. Possible values are currently @qcode{"off"} (no messages)
and @qcode{"iter"} (some messages after each iteration).  Support of
this setting and its exact interpretation are backend-specific.
@item debug
Logical scalar, default: @code{false}. Will be passed to the backend,
which might print debugging information if @code{true}.
@item FunValCheck
If @qcode{"on"}, the output of user functions will be sanity-checked.
Default: @qcode{"off"}.
@item user_interaction
@c This setting has deliberately not been named as its Matlab equivalent
@c `OutputFcn' since it differs from the latter by requiring the
@c functions to return _two_ outputs. The rationale for the difference
@c is that information about the reason for a user-stop should be
@c possible to pass in the output. The second output can't be made
@c optional without possibly altering the warning state under which the
@c user-function runs.
Handle to a user function or cell-array with a number of these.
Functions must have this interface:
@example
[@var{stop}, @var{info}] = some_user_function (@var{p}, @var{vals},
                                               @var{state});
@end example
If @var{stop} is @code{true}, the algorithm stops.  In @var{info}
information about the reason for stopping can be returned in a free
format.  @var{info} can be set to be empty, but it must be set.  Note
that this is different from the otherwise similar Matlab setting
@code{OutputFcn}.  The functions will be called by the algorithms at the
start with @var{state} set to @qcode{init}, after each iteration with
@var{state} set to @qcode{iter}, and at the end with @var{state} set to
@qcode{done}.  @var{p} contains the current parameters, and @var{vals}
is a structure with other current values, the possible fields are
currently:
@table @code
@item iteration
number of the current iteration,
@item fval
value of objective function (for scalar optimization),
@item residual
residuals (for residual-based optimization),
@item model_y
in @code{nonlin_curvefit}, the output of the model function,
@item observations
in @code{nonlin_curvefit}, the constant observations,
@item model_x
in @code{nonlin_curvefit}, the constant argument @var{x}.
@end table
Information about the output of these functions when they were called
the last time (possibly causing a stop) will be contained in the output
@var{outp} of the frontend in field @code{user_interaction}.  Subfield
@code{stop} is a vector containing the @var{stop} outputs of each
function, subfield @code{info} is a cell-array containing the output
@var{info} of each function.  In the case of a stop, the output
@var{cvg} of the frontent will be @code{-1}.
@end table

@c ------------------------------------------------------------------

@node Parameter structures
@chapter Handling of structures of optimized parameters
@cindex parameter structures

It can be convenient not to handle the optimized parameters as elements
of a vector, but as named fields of a structure.  The frontends
@code{nonlin_residmin}, @code{nonlin_curvefit}, @code{residmin_stat},
@code{curvefit_stat}, and @code{nonlin_min} can accept parameter
information in structure form, and can pass the parameters as a
structure to user functions, although the backends still handle the
parameters as vectors.

To use this feature, the initial parameters must be given in structure
form, or the setting @code{param_order} must be given, a cell-array with
names of the parameters.  If both is done, only the parameters in
structure fields named in @code{param_order} will be optimized.  If
there are still some non-structure-based configuration settings or user
functions, specifying @code{param_order} is mandatory even if the
initial parameters are given in structure form.

If the initial parameters are a structure, the parameters being the
optimization result will also be returned as a structure.

@menu
* Structure-based user functions:: Specify which user functions accept
                             parameter structures.
* Structure-based gradients and Hessians::    Format of returned values of
                             structure-based gradient and Hessian functions.
* Structure-based linear constraints:: Specify structure-based linear
                             constraints.
* Structure-based configuration settings:: Parameter-related
                             configuration settings in structure form.
* Non-scalar parameters::  Handling named parameter arrays.
@end menu

@c ------------------------------------------------------------------

@node Structure-based user functions
@section Specify which user functions accept parameter structures
@cindex structure-based user functions

The frontend must be told which user functions accept parameter
structures by setting corresponding settings to @code{true}. The names
of these settings depend on which user functions are applicable to the
respective frontend and are shown in the following table.

@multitable {inequality constraints} {@code{hessian_objf_pstruct}} {@code{nonlin_residmin},}
@headitem User function @tab Setting @tab Frontends
@item Objective function
@tab @code{objf_pstruct}
@tab @code{nonlin_min}
@item Gradient
@tab @code{grad_objf_pstruct}
@tab @code{nonlin_min}
@item Hessian
@tab @code{hessian_objf_pstruct}
@tab @code{nonlin_min}
@item Model function
@tab @code{f_pstruct}
@tab @code{nonlin_residmin}, @code{nonlin_curvefit}, @code{residmin_stat}, @code{curvefit_stat}
@item Jacobian
@tab @code{df_pstruct}
@tab @code{nonlin_residmin}, @code{nonlin_curvefit}, @code{residmin_stat}, @code{curvefit_stat}
@item General inequality constraints
@tab @code{f_inequc_pstruct}
@tab @code{nonlin_min}, @code{nonlin_residmin}, @code{nonlin_curvefit}
@item Jacobian of general inequality constraints
@tab @code{df_inequc_pstruct}
@tab @code{nonlin_min}, @code{nonlin_residmin}, @code{nonlin_curvefit}
@item General equality constraints
@tab @code{f_equc_pstruct}
@tab @code{nonlin_min}, @code{nonlin_residmin}, @code{nonlin_curvefit}
@item Jacobian of general equality constraints
@tab @code{df_equc_pstruct}
@tab @code{nonlin_min}, @code{nonlin_residmin}, @code{nonlin_curvefit}
@end multitable

@c ------------------------------------------------------------------

@node Structure-based gradients and Hessians
@section Format of returned values of structure-based gradient and Hessian functions
@cindex structure-based gradients
@cindex structure-based Hessians

Structure-based gradient or Jacobian functions, including Jacobians of
general constraints, must return the partial derivatives as fields of a
structure under the respective parameter names.  For gradients, the
partial derivatives are scalar.  For Jacobians, the partial derivatives
must be column vectors.

Structure-based Hessian functions must return the 2nd derivatives as
subfields in a two-level structure of parameter names.  For example, if
the parameter names are @code{a} and @code{b}, the 2nd derivative with
respect to @code{a} and @code{b} must be in the field
@code{returned_structure.a.b} or @code{returned_structure.b.a} (there is
no need to specify both).

@c ------------------------------------------------------------------

@node Structure-based linear constraints
@section Specify structure-based linear constraints
@cindex structure-based linear constraints

Linear constraints, otherwise specified with a matrix and a vector@c
@c (@xref{})
, can be adapted to structure-based parameter handling by specifying,
instead of a matrix, a structure containing the rows of the matrix in
fields under the respective parameter names.  In this case, rows
containing only zeros need not be given.

@c ------------------------------------------------------------------

@node Structure-based configuration settings
@section Parameter-related configuration settings in structure form
@cindex structure-based configuration settings

The vector-based settings @code{lbound}, @code{ubound}, @code{fixed},
@code{diffp}, @code{diff_onesided}, @code{fract_prec}, and
@code{max_fract_change} can be replaced by the setting
@code{param_config}.  It is a structure that can contain fields
corresponding to parameter names.  For each such field, there may be
subfields with the same names as the above vector-based settings, but
containing a scalar value for the respective parameter.

For example, if the parameters are named @code{a} and @code{b}, instead
of specifying

@example
settings = optimset ("lbound", [-Inf; 0],
                     "diff_onesided", [true; true]);
@end example

one can specify

@example
pconf.b.lbound = 0;
pconf.a.diff_onesided = true;
pconf.b.diff_onesided = true;
settings = optimset ("param_config", pconf);
@end example

If @code{param_config} is specified, none of the above vector-based
settings may be used.

@c ------------------------------------------------------------------

@node Non-scalar parameters
@section Handling named parameter arrays
@cindex non-scalar parameters
@cindex named parameter arrays

Parameters in named structure fields are allowed to be non-scalar real
arrays.  In this case, their dimensions must be given by the setting
@code{param_dims}, a cell-array of dimension vectors, each containing at
least two dimensions; if not given, dimensions are taken from the
initial parameters, if these are given in a structure.

If there are any vector-based settings or not structure-based linear
constraints, they must correspond to an order of parameters defined as
follows:

All named parameter arrays are reshaped to vectors. Then, all
parameters, scalars and vectors, are concatenated in the order of
parameter names, given by the user.

Structure-based settings or structure-based initial parameters must
contain arrays with dimensions reshapable to those of the respective
parameters.

@c ------------------------------------------------------------------

@node Additional parameters
@chapter Passing additional parameters to user functions
@cindex additional parameters

Optimizers often require the user to supply functions (@abbr{e.g@.}
objective functions, model functions, constraint functions).  The
interface of these functions --- arguments and returned values --- is
defined by the optimizer.  Often, a user function needs additional
arguments, not covered by the defined interface, which are constant
throughout the optimization.  These can be supplied by wrapping the user
function into an anonymous function.  @xref{Anonymous
Functions,,,octave}, for further explanation and examples.

There are also some older optimizers in the optim package, written when
anonymous functions were not available in Octave.  Some of these offer
an interface to user functions which is itself able to pass additional
constant variables in arbitrary argument positions.  Newer optimizers
should not be written this way, since this is an unnecessary
complication.

Though it is possible to use global variables to pass additional data to
user functions, this is not recommended since it introduces the
possibility of name conflicts within the pool of global variables.


@c ------------------------------------------------------------------

@node Function index
@unnumbered Index of functions in optim

@printindex mfn

@c ------------------------------------------------------------------

@node Concept index
@unnumbered Concept index

@printindex cp

@bye
