Broyden方法求解非线性方程组的Matlab实现
注:matlab代码来自网络,仅供学习参考。
1. 把以下代码复制在一个.m文件上
function [sol, it_hist, ierr] = brsola(x,f,tol, parms) % Broyden's Method solver, globally convergent
% solver for f(x) = 0, Armijo rule, one vector storage %
% This code es with no guarantee or warranty of any kind. %
% function [sol, it_hist, ierr] = brsola(x,f,tol,parms) %
% inputs:
% initial iterate = x % function = f
% tol = [atol, rtol] relative/absolute
% error tolerances for the nonlinear iteration % parms = [maxit, maxdim]
% maxit = maxmium number of nonlinear iterations % default = 40
% maxdim = maximum number of Broyden iterations % before restart, so maxdim-1 vectors are % stored
% default = 40 %
% output:
% sol = solution
% it_hist(maxit,3) = scaled l2 norms of nonlinear residuals % for the iteration, number function evaluations, % and number of steplength reductions % ierr = 0 upon successful termination % ierr = 1 if after maxit iterations
% the termination criterion is not satsified. % ierr = 2 failure in the line search. The iteration % is terminated if too many steplength reductions % are taken. % %
% internal parameter:
% debug = turns on/off iteration statistics display as % the iteration progresses
%
% alpha = 1.d-4, parameter to measure sufficient decrease %
% maxarm = 10, maximum number of steplength reductions before % failure is reported %
% set the debug parameter, 1 turns display on, otherwise off %
debug=1; %
% initialize it_hist, ierr, and set the iteration parameters %
ierr = 0; maxit=40; maxdim=39; it_histx=zeros(maxit,3); maxarm=10; %
if nargin == 4
maxit=parms(1); maxdim=parms(2)-1; end
rtol=tol(2); atol=tol(1); n = length(x); fnrm=1; itc=0; nbroy=0; %
% evaluate f at the initial iterate % pute the stop tolerance %
f0=feval(f,x); fc=f0;
fnrm=norm(f0)/sqrt(n); it_hist(itc+1)=fnrm;
it_histx(itc+1,1)=fnrm; it_histx(itc+1,2)=0; it_histx(itc+1,3)=0; fnrmo=1;
stop_tol=atol + rtol*fnrm;
outstat(itc+1, :) = [itc fnrm 0 0]; %
% terminate on entry? %
if fnrm < stop_tol sol=x; return end %
% initialize the iteration history storage matrices %
stp=zeros(n,maxdim);
stp_nrm=zeros(maxdim,1); lam_rec=ones(maxdim,1); %
% Set the initial step to -F, pute the step norm %
lambda=1;
stp(:,1) = -fc;
stp_nrm(1)=stp(:,1)'*stp(:,1); %
% main iteration loop %
while(itc < maxit) %
nbroy=nbroy+1; %
% keep track of successive residual norms and % the iteration counter (itc) %
fnrmo=fnrm; itc=itc+1; %
% pute the new point, test for termination before % adding to iteration history %
xold=x; lambda=1; iarm=0; lrat=.5; alpha=1.d-4; x = x + stp(:,nbroy); fc=feval(f,x);
fnrm=norm(fc)/sqrt(n);
ff0=fnrmo*fnrmo; ffc=fnrm*fnrm; lamc=lambda; % %
% Line search, we assume that the Broyden direction is an % ineact Newton direction. If the line search fails to
% find sufficient decrease after maxarm steplength reductions % brsola returns with failure. %
% Three-point parabolic line search %
while fnrm >= (1 - lambda*alpha)*fnrmo && iarm < maxarm % lambda=lambda*lrat; if iarm==0
lambda=lambda*lrat; else
lambda=parab3p(lamc, lamm, ff0, ffc, ffm);
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