MATLAB实现方程求根【数学建模、科学计算算法】.zip

function [x,FVAL,EXITFLAG,OUTPUT,JACOB] = fsolve(FUN,x,options,varargin) %优化工具箱函数,可求多元非线性方程组的实根. 用法与fzero类似。 %例 先写一个M函数rooteg4fun.m % function y=rooteg4fun(x) % y(1)=4*x(1)-x(2)+exp(x(1))/10-1; % y(2)=-x(1)+4*x(2)+x(1).^2/8; % 使用 % [x,f,h]=fsolve('rooteg4fun',[0,0]) %初值x(1)=0,x(2)=0 % x返回解向量，f返回误差向量，h>0表明算法收敛 % 注意：方程变量必须拼成一个向量变量，即用x(1),x(2),... % %FSOLVE Solves nonlinear equations by a least squares method. % % FSOLVE solves 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 described in FUN. FUN is usually an M-file which returns % an evaluation of the equations for a particular value of X: F=FUN(X). % % X=FSOLVE(FUN,X0,OPTIONS) minimizes 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, Jacobian, % JacobPattern, LineSearchType, LevenbergMarquardt, MaxFunEvals, MaxIter, % DiffMinChange and DiffMaxChange, LargeScale, MaxPCGIter, PrecondBandWidth, % TolPCG, TypicalX. Use the Jacobian option to specify that FUN may be called % with two output arguments where the second, J, is the Jacobian matrix: % [F,J] = feval(FUN,X). If FUN returns a vector (matrix) 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(FUN,X0,OPTIONS,P1,P2,...) passes the problem-dependent % parameters P1,P2,... directly to the function FUN: FUN(X,P1,P2,...). % Pass an empty matrix for OPTIONS to use the default values. % % [X,FVAL]=FSOLVE(FUN,X0,...) returns the value of the objective function % at X. % % [X,FVAL,EXITFLAG]=FSOLVE(FUN,X0,...) returns a string EXITFLAG that % describes the exit condition of FSOLVE. % If EXITFLAG is: % > 0 then FSOLVE converged to a solution X. % 0 then the maximum number of function evaluations was reached. % < 0 then FSOLVE did not converge to a solution. % % [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, and the first-order % optimality (if used) in OUTPUT.firstorderopt. % % [X,FVAL,EXITFLAG,OUTPUT,JACOB]=FSOLVE(FUN,X0,...) returns the % Jacobian of FUN at X. % Copyright (c) 1990-98 by The MathWorks, Inc. % $Revision: 1.26 $ $Date: 1998/10/22 19:28:31 $ % Andy Grace 7-9-90. % Grandfathered FSOLVE call for Optimization Toolbox versions prior to 2.0: % [X,OPTIONS]=FSOLVE(FUN,X0,OPTIONS,GRADFUN,P1,P2,...) % % ------------Initialization---------------- defaultopt = optimset('display','final','LargeScale','on', ... 'TolX',1e-6,'TolFun',1e-6,'DerivativeCheck','off',... 'Jacobian','off','MaxFunEvals','100*numberOfVariables',... 'Diagnostics','off',... 'DiffMaxChange',1e-1,'DiffMinChange',1e-8,... 'PrecondBandWidth',0,'TypicalX','ones(numberOfVariables,1)','MaxPCGIter','max(1,floor(numberOfVariables/2))', ... 'TolPCG',0.1,'MaxIter',400,'JacobPattern',[], ... 'LineSearchType','quadcubic','LevenbergMarq','off'); % 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 < 2, error('FSOLVE requires two input arguments');end if nargin < 3, options=[]; end % These are added so that we can have the same code as in lsqnonlin which % actually has upper and lower bounds. LB = []; UB = []; %[x,FVAL,EXITFLAG,OUTPUT,JACOB] = fsolve(FUNin,x,options,varargin) % Note: don't send varargin in as a comma separated list!! numargin = nargin; numargout = nargout; [calltype, GRADFUN, varargin] = parse_call(FUN,options,numargin,numargout,varargin); if isequal(calltype,'new') % fsolve version 2.* xstart=x(:); numberOfVariables=length(xstart); large = 'large-scale'; medium = 'medium-scale'; l = []; u = []; options = optimset(defaultopt,options); switch optimget(options,'display') 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','off'),'on'); gradflag = strcmp(optimget(options,'Jacobian'),'on'); line_search = strcmp(optimget(options,'largescale','off'),'off'); % 0 means trust-region, 1 means line-search % Convert to inline function as needed if ~isempty(FUN) % will detect empty string, empty matrix, empty cell array [funfcn, msg] = fprefcnchk(FUN,'fsolve',length(varargin),gradflag); else errmsg = sprintf('%s\n%s', ... 'FUN must be a function name, valid string expression, or inline object;', ... ' or, FUN may be a cell array that contains these type of objects.'); error(errmsg) end x(:) = xstart; switch funfcn{1} case 'fun' fuser = feval(funfcn{3},x,varargin{:}); f = fuser(:); nfun=length(f); JAC = zeros(nfun,numberOfVariables); 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('Undefined calltype in FSOLVE'); end % check size of JAC [Jrows, Jcols]=size(JAC); if Jrows~=nfun | Jcols ~=numberOfVariables errstr = sprintf('%s\n%s%d%s%d\n',... 'User-defined Jacobian is not the correct size:',... ' the Jacobian matrix should be ',nfun,'-by-',numberOfVariables); error(errstr); end YDATA = []; caller = 'fsolve'; % trustregion and enough equations (as many as variables) if ~line_search & nfun >= numberOfVariables OUTPUT.algorithm = large; % trust region and not enough equations -- switch to line_search elseif ~line_search & nfun < numberOfVariables warnstr = sprintf('%s\n%s\n', ... 'Large-scale method requires at least as many equations as variables; ',... ' switching to line-search method instead.'); warning(warnstr); OUTPUT.algorithm = medium; % line search and no bounds elseif line_search & isempty(l) & isempty(u) OUTPUT.algorithm = medium; % line search and bounds and enough equations, switch to trust region elseif line_search & (~isempty(LB) | ~isempty(UB)) & nfun >= numberOfVariables warnstr = sprintf('%s\n%s\n', ... 'Line-search method does not handle bound constraints; ',... ' switching to trust-region method instead.'); warning(warnstr); OUTPUT.algorithm = large; % can't handle this one: elseif line_search & (~isempty(LB) | ~isempty(UB)) & nfun < numberOfVariables errstr = sprintf('%s\n%s\n%s\n', ... 'Line-search method does not handle bound constraints ',... ' and trust-region method requires at least as many equations as variables; ',... ' aborting.'); error(errstr); 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=[]; msg = diagnose('fsolv