lmfnlsq.m

计算复变方程的根

function [xf, SS, cnt, res, XY] = LMFnlsq(varargin)
% LMFNLSQ   Solve a set of (over)determined nonlinear equations
% in least squares sense
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% A solution is obtained by a Fletcher's version of the Levenberg-Maquardt
% algoritm for minimization of a sum of squares of equation residuals.
% The main domain of LMFnlsq applications is in curve fitting during
% processing of experimental data.
%
% [Xf, Ssq, CNT, Res, XY] = LMFnlsq(FUN,Xo,Options)
% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
% FUN     is a function handle or a function M-file name (string) that
%         evaluates m-vector of equation residuals. Residuals are defined
%         as   res = FUN(x) - y,  where y is m-vector of given values.
% Xo      is n-vector of initial guesses of solution, unknown parameters,
% Options is an optional set of Name/Value pairs of control parameters
%         of the algorithm. It may also be preset by calling:
%         Options = LMFnlsq; %   for setting default values for Options,
%         Options = LMFnlsq('default');  or by a set of Name/Value pairs:
%         Options = LMFnlsq('Name',Value, ... ), or updating the Options
%                   set by calling
%         Options = LMFnlsq(Options,'Name',Value, ...).
%
%    Name   Values {default}         Description
% 'Display'      {0}        Display control
%                             0   no display
%                             k   display initial and each k-th iteration;
% 'Printf'    name|handle   Function for displaying of results; {@printit}
% 'Jacobian'  name|handle   Jacobian matrix function; {@finjac}
% 'FunTol'      {1e-7}      norm(FUN(x),1) stopping tolerance;
% 'XTol'        {1e-4}      norm(x-xold,1) stopping tolerance;
% 'MaxIter'     {100}       Maximum number of iterations;
% 'ScaleD'                  Scale control:
%               value         D = eye(m)*value;
%               vector        D = diag(vector);
%                {[]}         D(k,k) = JJ(k,k) for JJ(k,k)>0, or
%                                    = 1 otherwise,
%                                      where JJ = J.'*J
% 'Trace'        {0}        Tracing control:
%                             0 = don't save iteration results x^(k)
%                             nonzero = save iteration results x^(k)
% 'Lambda'       {0}        Initial value of parameter lambda
%
% A user may supply his own functions for building Jacobian matrix and for
% displaying intermediate results (corresponding to 'Jacobian' and
% 'Printf' options respectively).
% Not defined fields of the Options structure are filled by default values.
%
% Output Arguments:
%   Xf    final solution approximation
%   Ssq   sum of squares of residuals
%   Cnt     >0          count of iterations
%           -MaxIter,   no converge in MaxIter iterations
%   Res   number of calls of FUN and Jacobian matrix
%   XY    points of intermediate results in iterations
%
% Forms of function callings:
%   LMFnlsq                             %   Display help to LMFnlsq
%   Options = LMFnlsq;                  %   Settings of Options
%   Options = LMFnlsq('default');       %   The same as  Options = LMFnlsq;
%   Options = LMFnlsq(Name1,Value1,Name2,Value2,