matlab中用fsolve解非线性方程组用的是什么原理？

Question

解非线性方程组的方法有很多，比如直接降维、搜索（用最小二乘、牛顿迭代及最优化法）、连续法等等
直接降维操作较难，求解时间长；牛顿迭代有局部收敛性；最优化必须给出真实解的初始值；连续发需要构造同伦方程。

请教：matlab中fsolve语句用的是什么方法呢？（个人认为是最优化法，但对它们的了解都不是很清楚），是否能详细介绍一下最优化法和连续法的原理呢？？？

这个问题以前提问过可是没有满意答案，请高手不吝赐教！谢谢！！！！

Answer 1

看看fsolve的源代码：

>> type fsolve

function [x,FVAL,EXITFLAG,OUTPUT,JACOB] = fsolve(FUN,x,options,varargin)
%FSOLVE solves systems of nonlinear equations of several variables.
%
% FSOLVE attempts to solve equations of the form:
%
% F(X)=0 where F and X may be vectors or matrices.
%
% X=FSOLVE(FUN,X0) starts at the matrix X0 and tries to solve the
% equations in FUN. FUN accepts input X and returns a vector (matrix) of
% equation values F evaluated at X.
%
% X=FSOLVE(FUN,X0,OPTIONS) solves the equations with the default optimization
% parameters replaced by values in the structure OPTIONS, an argument
% created with the OPTIMSET function. See OPTIMSET for details. Used
% options are Display, TolX, TolFun, DerivativeCheck, Diagnostics,
% FunValCheck, Jacobian, JacobMult, JacobPattern, LineSearchType,
% NonlEqnAlgorithm, MaxFunEvals, MaxIter, PlotFcns, OutputFcn,
% DiffMinChange and DiffMaxChange, LargeScale, MaxPCGIter,
% PrecondBandWidth, TolPCG, and TypicalX. Use the Jacobian option to
% specify that FUN also returns a second output argument J that is the
% Jacobian matrix at the point X. If FUN returns a vector F of m
% components when X has length n, then J is an m-by-n matrix where J(i,j)
% is the partial derivative of F(i) with respect to x(j). (Note that the
% Jacobian J is the transpose of the gradient of F.)
%
% X = FSOLVE(PROBLEM) solves system defined in PROBLEM. PROBLEM is a
% structure with the function FUN in PROBLEM.objective, the start point
% in PROBLEM.x0, the options structure in PROBLEM.options, and solver
% name 'fsolve' in PROBLEM.solver. Use this syntax to solve at the
% command line a problem exported from OPTIMTOOL. The structure PROBLEM
% must have all the fields.
%
% [X,FVAL]=FSOLVE(FUN,X0,...) returns the value of the equations FUN at X.
%
% [X,FVAL,EXITFLAG]=FSOLVE(FUN,X0,...) returns an EXITFLAG that describes the
% exit condition of FSOLVE. Possible values of EXITFLAG and the corresponding
% exit conditions are
%
% 1 FSOLVE converged to a solution X.
% 2 Change in X smaller than the specified tolerance.
% 3 Change in the residual smaller than the specified tolerance.
% 4 Magnitude of search direction smaller than the specified tolerance.
% 0 Maximum number of function evaluations or iterations reached.
% -1 Algorithm terminated by the output function.
% -2 Algorithm seems to be converging to a point that is not a root.
% -3 Trust region radius became too small.
% -4 Line search cannot sufficiently decrease the residual along the current
% search direction.
%
% [X,FVAL,EXITFLAG,OUTPUT]=FSOLVE(FUN,X0,...) returns a structure OUTPUT
% with the number of iterations taken in OUTPUT.iterations, the number of
% function evaluations in OUTPUT.funcCount, the algorithm used in OUTPUT.algorithm,
% the number of CG iterations (if used) in OUTPUT.cgiterations, the first-order
% optimality (if used) in OUTPUT.firstorderopt, and the exit message in
% OUTPUT.message.
%
% [X,FVAL,EXITFLAG,OUTPUT,JACOB]=FSOLVE(FUN,X0,...) returns the
% Jacobian of FUN at X.
%
% Examples
% FUN can be specified using @:
% x = fsolve(@myfun,[2 3 4],optimset('Display','iter'))
%
% where myfun is a MATLAB function such as:
%
% function F = myfun(x)
% F = sin(x);
%
% FUN can also be an anonymous function:
%
% x = fsolve(@(x) sin(3*x),[1 4],optimset('Display','off'))
%
% If FUN is parameterized, you can use anonymous functions to capture the
% problem-dependent parameters. Suppose you want to solve the system of
% nonlinear equations given in the function myfun, which is parameterized
% by its second argument c. Here myfun is an M-file function such as
%
% function F = myfun(x,c)
% F = [ 2*x(1) - x(2) - exp(c*x(1))
% -x(1) + 2*x(2) - exp(c*x(2))];
%
% To solve the system of equations for a specific value of c, first assign the
% value to c. Then create a one-argument anonymous function that captures
% that value of c and calls myfun with two arguments. Finally, pass this anonymous
% function to FSOLVE:
%
% c = -1; % define parameter first
% x = fsolve(@(x) myfun(x,c),[-5;-5])
%
% See also OPTIMSET, LSQNONLIN, @, INLINE.

% Copyright 1990-2006 The MathWorks, Inc.
% $Revision: 1.41.4.12 $ $Date: 2006/05/19 20:18:49 $

% ------------Initialization----------------

defaultopt = struct('Display','final','LargeScale','off',...
'NonlEqnAlgorithm','dogleg',...
'TolX',1e-6,'TolFun',1e-6,'DerivativeCheck','off',...
'Diagnostics','off','FunValCheck','off',...
'Jacobian','off','JacobMult',[],...% JacobMult set to [] by default
'JacobPattern','sparse(ones(Jrows,Jcols))',...
'MaxFunEvals','100*numberOfVariables',...
'DiffMaxChange',1e-1,'DiffMinChange',1e-8,...
'PrecondBandWidth',0,'TypicalX','ones(numberOfVariables,1)',...
'MaxPCGIter','max(1,floor(numberOfVariables/2))', ...
'TolPCG',0.1,'MaxIter',400,...
'LineSearchType','quadcubic','OutputFcn',[],'PlotFcns',[]);

% If just 'defaults' passed in, return the default options in X
if nargin==1 && nargout <= 1 && isequal(FUN,'defaults')
x = defaultopt;
return
end

if nargin < 3, options=[]; end

% Detect problem structure input
if nargin == 1
if isa(FUN,'struct')
[FUN,x,options] = separateOptimStruct(FUN);
else % Single input and non-structure.
error('optim:fsolve:InputArg','The input to FSOLVE should be either a structure with valid fields or consist of at least two arguments.');
end
end

if nargin == 0
error('optim:fsolve:NotEnoughInputs','FSOLVE requires at least two input arguments.')
end

% Check for non-double inputs
if ~isa(x,'double')
error('optim:fsolve:NonDoubleInput', ...
'FSOLVE only accepts inputs of data type double.')
end

LB = []; UB = [];
xstart=x(:);
numberOfVariables=length(xstart);

large = 'large-scale';
medium = 'medium-scale: line search';
dogleg = 'trust-region dogleg';

switch optimget(options,'Display',defaultopt,'fast')
case {'off','none'}
verbosity = 0;
case 'iter'
verbosity = 2;
case 'final'
verbosity = 1;
case 'testing'
verbosity = Inf;
otherwise
verbosity = 1;
end
diagnostics = isequal(optimget(options,'Diagnostics',defaultopt,'fast'),'on');
gradflag = strcmp(optimget(options,'Jacobian',defaultopt,'fast'),'on');
% 0 means large-scale trust-region, 1 means medium-scale algorithm
mediumflag = strcmp(optimget(options,'LargeScale',defaultopt,'fast'),'off');
funValCheck = strcmp(optimget(options,'FunValCheck',defaultopt,'fast'),'on');
switch optimget(options,'NonlEqnAlgorithm',defaultopt,'fast')
case 'dogleg'
algorithmflag = 1;
case 'lm'
algorithmflag = 2;
case 'gn'
algorithmflag = 3;
otherwise
algorithmflag = 1;
end
mtxmpy = optimget(options,'JacobMult',defaultopt,'fast');
if isequal(mtxmpy,'atamult')
warning('optim:fsolve:NameClash', ...
['Potential function name clash with a Toolbox helper function:\n' ...
'Use a name besides ''atamult'' for your JacobMult function to\n' ...
'avoid errors or unexpected results.'])
end

% Convert to inline function as needed
if ~isempty(FUN) % will detect empty string, empty matrix, empty cell array
funfcn = lsqfcnchk(FUN,'fsolve',length(varargin),funValCheck,gradflag);
else
error('optim:fsolve:InvalidFUN', ...
['FUN must be a function name, valid string expression, or inline object;\n' ...
' or, FUN may be a cell array that contains these type of objects.'])
end

JAC = [];
x(:) = xstart;
switch funfcn{1}
case 'fun'
fuser = feval(funfcn{3},x,varargin{:});
f = fuser(:);
nfun=length(f);
case 'fungrad'
[fuser,JAC] = feval(funfcn{3},x,varargin{:});
f = fuser(:);
nfun=length(f);
case 'fun_then_grad'
fuser = feval(funfcn{3},x,varargin{:});
f = fuser(:);
JAC = feval(funfcn{4},x,varargin{:});
nfun=length(f);
otherwise
error('optim:fsolve:UndefinedCalltype','Undefined calltype in FSOLVE.')
end

if gradflag
% check size of JAC
[Jrows, Jcols]=size(JAC);
if isempty(mtxmpy)
% Not using 'JacobMult' so Jacobian must be correct size
if Jrows~=nfun || Jcols~=numberOfVariables
error('optim:fsolve:InvalidJacobian', ...
['User-defined Jacobian is not the correct size:\n' ...
' the Jacobian matrix should be %d-by-%d.'],nfun,numberOfVariables)
end
end
else
Jrows = nfun;
Jcols = numberOfVariables;
end

XDATA = []; YDATA = []; caller = 'fsolve';

% Choose what algorithm to run: determine (i) OUTPUT.algorithm and
% (ii) if and only if OUTPUT.algorithm = medium, also option.LevenbergMarquardt.
% Option LevenbergMarquardt is used internally; it's not user settable. For
% this reason we change this option directly, for speed; users should use
% optimset.
if ~mediumflag
if nfun >= numberOfVariables
% large-scale method and enough equations (as many as variables)
OUTPUT.algorithm = large;
else
% large-scale method and not enough equations - switch to medium-scale algorithm
warning('optim:fsolve:FewerFunsThanVars', ...
['Large-scale method requires at least as many equations as variables;\n' ...
' using line-search method instead.'])
OUTPUT.algorithm = medium;
options.LevenbergMarquardt = 'off';
end
else
if algorithmflag == 1 && nfun == numberOfVariables
OUTPUT.algorithm = dogleg;
elseif algorithmflag == 1 && nfun ~= numberOfVariables
warning('optim:fsolve:NonSquareSystem', ...
['Default trust-region dogleg method of FSOLVE cannot\n handle non-square systems; ', ...
'using Gauss-Newton method instead.']);
OUTPUT.algorithm = medium;
options.LevenbergMarquardt = 'off';
elseif algorithmflag == 2
OUTPUT.algorithm = medium;
options.LevenbergMarquardt = 'on';
else % algorithmflag == 3
OUTPUT.algorithm = medium;
options.LevenbergMarquardt = 'off';
end
end

if diagnostics > 0
% Do diagnostics on information so far
constflag = 0; gradconstflag = 0; non_eq=0;non_ineq=0;lin_eq=0;lin_ineq=0;
confcn{1}=[];c=[];ceq=[];cGRAD=[];ceqGRAD=[];
hessflag = 0; HESS=[];
diagnose('fsolve',OUTPUT,gradflag,hessflag,constflag,gradconstflag,...
mediumflag,options,defaultopt,xstart,non_eq,...
non_ineq,lin_eq,lin_ineq,LB,UB,funfcn,confcn,f,JAC,HESS,c,ceq,cGRAD,ceqGRAD);

end

% Execute algorithm
if isequal(OUTPUT.algorithm, large)
if ~gradflag
Jstr = optimget(options,'JacobPattern',defaultopt,'fast');
if ischar(Jstr)
if isequal(lower(Jstr),'sparse(ones(jrows,jcols))')
Jstr = sparse(ones(Jrows,Jcols));
else
error('optim:fsolve:InvalidJacobPattern', ...
'Option ''JacobPattern'' must be a matrix if not the default.')
end
end
else
Jstr = [];
end
computeLambda = 0;
[x,FVAL,LAMBDA,JACOB,EXITFLAG,OUTPUT,msg]=...
snls(funfcn,x,LB,UB,verbosity,options,defaultopt,f,JAC,XDATA,YDATA,caller,...
Jstr,computeLambda,varargin{:});
elseif isequal(OUTPUT.algorithm, dogleg)
% trust region dogleg method
Jstr = [];
[x,FVAL,JACOB,EXITFLAG,OUTPUT,msg]=...
trustnleqn(funfcn,x,verbosity,gradflag,options,defaultopt,f,JAC,...
Jstr,varargin{:});
else
% line search (Gauss-Newton or Levenberg-Marquardt)
[x,FVAL,JACOB,EXITFLAG,OUTPUT,msg] = ...
nlsq(funfcn,x,verbosity,options,defaultopt,f,JAC,XDATA,YDATA,caller,varargin{:});
end

Resnorm = FVAL'*FVAL; % assumes FVAL still a vector
if EXITFLAG > 0 % if we think we converged:
if Resnorm > sqrt(optimget(options,'TolFun',defaultopt,'fast'))
OUTPUT.message = ...
sprintf(['Optimizer appears to be converging to a minimum that is not a root:\n' ...
'Sum of squares of the function values is > sqrt(options.TolFun).\n' ...
'Try again with a new starting point.']);
if verbosity > 0
disp(OUTPUT.message)
end
EXITFLAG = -2;
else
OUTPUT.message = msg;
if verbosity > 0
disp(OUTPUT.message);
end
end
else
OUTPUT.message = msg;
if verbosity > 0
disp(OUTPUT.message);
end
end

% Reset FVAL to shape of the user-function output, fuser
FVAL = reshape(FVAL,size(fuser));

Answer 2

简单地说，matlab中fsolve语句数值效果较好，采用的解法是将方程组转化为最小二乘问题，调用指令lsqnonlin求解，所以，它参数的选取和优化指令的用法是一致的。
最优化，原理上说到底都是要从一个初值开始，选择搜索的方向与步长。参数的不同选取，使得算法出现不同。例如Levenberg-Marquardt如果选择'on',搜索方向就是用Levenberg-Marquardt法，如果选择'off',搜索方向就是用Gauss-Newton法.