mmfsolve.m

一种数值方法解非线性方程组,可用于工程计算

function [xf,fval,exitflag,iters]=mmfsolve(varargin)
%MMFSOLVE Solve a Set of Nonlinear Equations.
% [Xf,Fval,ExitFlag,Iters]=MMFSOLVE(FUN,Xo,Options) finds a zero of the
% vector function FUN(X) starting from the initial guess Xo. FUN is a
% function handle or a function M-file name that evaluates FUN(X). Use
% anonymous functions or nested functions to pass additional parameters to
% FUN. Options is an optional structure that defines algorithm behavior. If
% Options are not given default options are used.
%
% Options=MMFSOLVE('Name',Value,...) sets values in Options structure
% based on the Name/Value pairs:
%
% Name        Values {default}   Description
% 'Display'   ['on' {'off'}]     Display iteration information
% 'Jacobian'  {'broyden'}        Broyden's Jacobian approximation
% 'Jacobian'   'finite'          Finite Difference Jacobian
% 'Jacobian'   @JNAME            Analytic Jacobian function handle
% 'FunTol'     {1e-7}            NORM(FUN(X),1) stopping tolerance
% 'MaxIter'    {100}             Maximum number of iterations
% 'MaxStep'    value             Maximum step size in X allowed
%                                if 0, algorithm chooses MaxStep
% 'Scale'     ['on' {'off'}]     Scale algorithm using Xo
%
% Options=MMFSOLVE('default') returns the default options structure.
%
% Options=MMFSOLVE(Options,'Name',Value,...) updates the Options structure
% with new parameter values.
%
% Output Arguments:
% Xf = final solution approximation
% Fval = FUN(Xf)
% ExitFlag = termination code:
%            1 = normal return, 2 = two steps too small
%            3 = line search failure, 4 = too many iterations
%            5 = five steps too big, 6 = stuck at minimizer
% Iters = iteration count

% Reference: Algorithm D6.1.3 in "Numerical Methods for Unconstrained
% Optimization and Nonlinear Equations," by Dennis and Schnabel,
% Siam's Classics in Applied Mathematics, no. 16, (2)nd printing 1996.

% D.C. Hanselman, University of Maine, Orono, ME 04469
% [e-mail]
% Mastering MATLAB 7
% 2005-10-10
% 2005-12-21 fixed bugs noted by Rajeev Kumar
% 2006-05-03 fixed bug noted by Jason Antenucci

if nargin==1 && strcmpi('default',varargin(1))  % set default parameters
   xf.Display='off';
   xf.Jacobian='broyden';
   xf.MaxIter=100;
   xf.MaxStep=0; % to be determined by the algorithm
   xf.Scale='off';
   xf.FunTol=1e-7;
   return
end
if isstruct(varargin{1}) % check if Options=MMFSOLVE(Options,'Name','Value',...)
   if ~isfield(varargin{1},'Jacobian')
      error('Options Structure not Correct for MMFSOLVE.')
   end
   xf=varargin{1};
   for i=2:2:nargin-1
      name=varargin{i};
      if ~ischar(name)
         error('Parameter Names Must be Strings.')
      end
      name=lower(name(isletter(name)));
      value=varargin{i+1};
      if     strncmp(name,'d',1),    xf.Display=value;
      elseif strncmp(name,'f',1),    xf.FunTol=value(1);
      elseif strncmp(name,'j',1),    xf.Jacobian=value;
      elseif strncmp(name,'maxi',4), xf.MaxIter=value(1);
      elseif strncmp(name,'maxs',4), xf.MaxStep=value(1);
      elseif strncmp(name,'s',1),    xf.Scale=value;
      else   disp(['Unknown Parameter Name --> ' name])
      end
   end
   return
end
if ischar(varargin{1}) % check for Options=MMFSOLVE('Name','Value',...)
   Pnames=char('display','funtol','jacobian','maxiter','maxstep','scale');
   if strncmpi(varargin{1},Pnames,length(varargin{1}))
      xf=mmfsolve('default');  % get default values
      xf=mmfsolve(xf,varargin{:});
      return
   end
end
% MMFSOLVE(FUN,Xo,Options,P1,P2,...)

FUN=varargin{1};
if ~(isvarname(FUN) || isa(FUN,'function_handle'))
   error('FUN Must be a Function Handle or M-file Name.')
end
xc=varargin{2};
if nargin>2 && isstruct(varargin{3})
   options=varargin{3};
else
   options=mmfsolve('default');
end

if ~isstruct(options) || ~isfield(options,'Jacobian')
   error('Options Structure not Correct for MMFSOLVE.')
end

% Steps 2 and 3
[options,errmsg]=local_neinck(FUN,xc,options);
xf=xc;
error(errmsg)

% Step 4
iters=0;

% Step 5
[fc,FVc]=local_nefn(FUN,xc,options);

% Steps 6 and 7
consecmax=0;
if max(options.Sf.*abs(FVc))<=options.FunTol/100
   xf=xc;
   fval=FVc;
   exitflag=0;
   return
end
if options.analjac  % analytic Jacobian
   Jc=feval(options.Jacobian,xc);
else
   Jc=local_fdjac(FUN,xc,FVc,options);
end
gc=Jc'*(FVc.*(options.Sf.^2));

% Step 8
% been there, done that

% Step 9
restart=1;
exitflag=0;
fval=FVc;

% Step 10
while exitflag==0
   iters=iters+1;
   
   Sn=local_nemodel(FVc,Jc,gc,options);
   
   [retcode,xplus,fplus,FVplus,maxtaken]=...
      local_linesearch(FUN,xc,fc,gc,Sn,options);
   
   if retcode~=1 || restart
      if options.analjac
         Jc=feval(options.Jacobian,xc);
      elseif strcmp(options.Jacobian,'finite')
         Jc=local_fdjac(FUN,xc,FVplus,options);
      else
         Jc=local_broyunfac(Jc,xc,xplus,FVc,FVplus,options);
      end
      gc=Jc'*(FVplus.*(options.Sf.^2));
      
      [consecmax,exitflag]=local_nestop(xc,xplus,FVplus,fplus,gc,...
         retcode,iters,maxtaken,consecmax,options);
   end
   if retcode==1 ||...
         (exitflag==2 && ~restart && strcmpi(options.Jacobian,'broyden'))
      Jc=local_fdjac(FUN,xc,FVc,options);
      gc=Jc'*(FVc.*(options.Sf.^2));
      restart=1;
   else
      if exitflag>0
         xf=xplus;
         fval=FVc;
      else
         restart=0;
      end
      xc=xplus;
      fc=fplus;
      FVc=FVplus;
      if strcmp(options.Display,'on')
         fprintf(' Iteration = %.0f,  Current X:\n',iters)
         disp(xc')
      end
   end
end
if strcmp(options.Display,'on')
   switch exitflag
   case 1, disp('Successful Solution.')
   case 2, disp('2 Steps Too Small.')
   case 3, disp('Line Search Failure.')
   case 4, disp('Too Many Iterations.')
   case 5, disp('5 Steps Too Big.')
   case 6, disp('Stuck at Minimizer.')
   end
end
%--------------------------------------------------------------------------
%---------------------------------------------------------------------------
function s=local_sign(x)
s=sign(x)+(x==0);
%---------------------------------------------------------------------------	
function Sn=local_nemodel(FVc,Jc,gc,options)
n=length(FVc);
M=diag(options.Sf)*Jc;
[M,M1,M2,sing]=local_qrdecomp(M);

if sing==0
   for j=2:n
      M(1:j-1,j)=M(1:j-1,j)/options.Sx(j);
   end
   M2=M2.*options.iSx;
   est=local_condest(M,M2);
else
   est=0;
end

if sing==1 || est>1./sqrt(eps)
   H=Jc'*diag(options.Sf);
   H=H*H';
   
   Hnorm=options.iSx(1)*abs(H(1,:))*options.iSx;
   for i=2:n
      Hper=abs(H(:,i))'*options.iSx + abs(H(i,:))*options.iSx;
      Hper=options.iSx(i)*Hper;
      Hnorm=max(Hnorm,Hper);
   end
   H=H+sqrt(n*eps)*Hnorm*(diag(options.Sx)^2);
   
   M=local_choldecomp(H);
   Sn=-M'\(M\gc);
else
   for j=2:n
      M(1:j-1,j)=M(1:j-1,j)*options.Sx(j);
   end
   M2=M2.*options.Sx;
   Sn=-options.Sf.*FVc;
   Sn=local_qrsolve(M,M1,M2,Sn);
end
%---------------------------------------------------------------------------	
function [L,maxadd]=local_choldecomp(H)
minl=0;
maxoffl=sqrt(max(diag(H)));
minl2=sqrt(eps)*maxoffl;

maxadd=0;
n=size(H,1);
L=zeros(n);
for j=1:n
   if j==1
      L(j,j)=H(j,j);
   else
      L(j,j)=H(j,j)-L(j,1:j-1)*L(j,1:j-1)';
   end
   minljj=0;
   for i=j+1:n
      if (j==1)
         L(i,j)=H(j,i);
      else
         L(i,j)=H(j,i)-L(i,1:j-1)*L(j,1:j-1)';
      end
      minljj=max(abs(L(i,j)),minljj);
   end
   minljj=max(minljj/maxoffl,minl);
   if (L(j,j)>minljj^2)   % Normal Cholesky
      L(j,j)=sqrt(L(j,j));
   else                   % Augment H(j,j)
      minljj=max(minl2,minljj);
      maxadd=max(maxadd,(minljj^2-L(j,j)));
      L(j,j)=minljj;
   end
   for i=j+1:n
      L(i,j)=L(i,j)/L(j,j);
   end
end
%---------------------------------------------------------------------------	
function est=local_condest(M,M2)
n=length(M2);
p=zeros(n,1);
pm=p;
x=p;

est=norm(triu(M)-diag(diag(M))+diag(M2),1);
x(1)=1/M2(1);
p(2:n)=M(1,2:n)*x(1);

for j=2:n
   xp=(+1-p(j))/M2(j);
   xm=(-1-p(j))/M2(j);
   temp=abs(xp);
   tempm=abs(xm);
   for i=j+1:n
      pm(i)=p(i)+M(j,i)*xm;
      tempm=tempm+abs(pm(i))/abs(M2(i));
      p(i)=p(i)+M(j,i)*xp;
      temp=temp+abs(p(i))/abs(M2(i));
   end
   if temp>=tempm
      x(j)=xp;
   else
      x(j)=xm; p(j+1:n)=pm(j+1:n);
   end
end

est=est/norm(x,1);
x=local_rsolve(M,M2,x);
est=est*norm(x,1);
%---------------------------------------------------------------------------	
function [M,M1,M2,sing]=local_qrdecomp(M)
n=size(M,1);
M1=zeros(n,1);
M2=M1;
sing=0;

for k=1:n-1
   eta=max(abs(M(k:n,k)));
   if eta==0
      M1(k)=0;
      M2(k)=0;
      sing=1;
   else
      M(k:n,k)=M(k:n,k)/eta;
      sigma=local_sign(M(k,k))*norm(M(k:n,k));
      M(k,k)=M(k,k)+sigma;
      M1(k)=sigma*M(k,k);
      M2(k)=-eta*sigma;
      tau=(M(k:n,k)'*M(k:n,(k+1):n))/M1(k);
      M(k:n,(k+1):n)=M(k:n,(k+1):n)-M(k:n,k)*tau;
   end
end
M2(n)=M(n,n);
if M(n,n)==0
   sing=1;
end
%---------------------------------------------------------------------------	
function b=local_qrsolve(M,M1,M2,b)
n=length(M2);
for j=1:n-1
   tau=(M(j:n,j)'*b(j:n))/M1(j);
   b(j:n)=b(j:n)-tau*M(j:n,j);
end
b=local_rsolve(M,M2,b);
%---------------------------------------------------------------------------	
function b=local_rsolve(M,M2,b)
n=length(M2);
b(n)=b(n)/M2(n);  % rsolve
for i=n-1:-1:1
   b(i)=(b(i)-M(i,i+1:n)*b((i+1):n))/M2(i);
end
%---------------------------------------------------------------------------	
function Jc=local_broyunfac(Jc,xc,xplus,FVc,FVplus,options)
s=xplus-xc;
Sx=options.Sx;
denom=norm(Sx.*s)^2;
tempi=FVplus-FVc-Jc*s;
i=find(abs(tempi)<options.eta*(abs(FVplus)+abs(FVc)));
tempi(i)=zeros(length(i),1);
Jc=Jc+tempi*(s.*(Sx.*Sx))'/denom;
%---------------------------------------------------------------------------	
function [consecmax,exitflag]=local_nestop(xc,xplus,FVplus,fplus,gc,...
   retcode,iters,maxtaken,consecmax,options)
exitflag=0;
if retcode==1
   exitflag=3;
elseif max(options.Sf.*abs(FVplus))<=options.FunTol
   exitflag=1;
elseif max(abs(xplus-xc)./max(abs(xplus),options.iSx)) <= options.steptol
   exitflag=2;
elseif iters>=options.MaxIter
   exitflag=4;
elseif maxtaken
   consecmax=consecmax+1;
   if consecmax==5
      exitflag=5;
   end
else
   consecmax=0;
   if ~strcmp(options.Jacobian,'broyden')
      if max(abs(gc).*max(abs(xplus),options.iSx)/max(fplus,length(xc)/2)) ...
            <= options.mintol
         exitflag=6;
      end
   end
end
%---------------------------------------------------------------------------	
function [retcode,xplus,fplus,FVplus,maxtaken]=...
   local_linesearch(FUN,xc,fc,gc,p,options,varargin)
n=length(xc);
xplus=zeros(n,1);
fplus=0;
maxtaken=0;
retcode=2;
alpha=1e-4;

newtlen=norm(options.Sx .* p);
if newtlen>options.MaxStep
   p=p*options.MaxStep/newtlen;
   newtlen=options.MaxStep;
end
initslope=gc'*p;
rellength=max(abs(p)./max(abs(xc),options.iSx));
minlambda=options.steptol/rellength;
lambda=1;
while retcode>1
   
   xplus=xc+lambda*p;
   [fplus,FVplus]=local_nefn(FUN,xplus,options,varargin{:});
   if fplus<=fc+alpha*lambda*initslope
      retcode=0;
      maxtaken=(lambda==1)&(newtlen>0.99*options.MaxStep);
   elseif lambda<minlambda
      retcode=1;
      xplus=xc;
   else
      if lambda==1
         if strcmp(options.Display,'on')
            disp('Quadratic Line Search')
         end
         lambdatemp=-initslope/(2*(fplus-fc-initslope));
      else
         if strcmp(options.Display,'on')
            disp('Cubic Line Search')
         end
         ilp2=lambdaprev^(-2);
         il2=lambda^(-2);
         ab=1/(lambda-lambdaprev) * ...
            [il2 -ilp2;-lambdaprev*il2 lambda*ilp2]*...
            [fplus-fc-lambda*initslope;fplusprev-fc-lambdaprev*initslope];
         disc=ab(2)^2-3*ab(1)*initslope;
         if abs(ab(1))<=eps
            lambdatemp=-initslope/(2*ab(2));
         else
            lambdatemp=(-ab(2)+sqrt(disc))/(3*ab(1));
         end
      end
      lambdatemp=min(lambdatemp,lambda/2);
      lambdaprev=lambda;
      fplusprev=fplus;
      lambda=max(lambda/10,lambdatemp);
   end
end
%---------------------------------------------------------------------------	
function Jc=local_fdjac(FUN,xc,Fc,options)
n=length(Fc);
sqrteta=sqrt(options.eta);
Jc=zeros(n);

for i=1:n
   Xc=xc;
   stepsize=sqrteta*max(abs(xc(i)),options.iSx(i))*local_sign(xc(i));
   Xc(i)=Xc(i)+stepsize;
   Fi=feval(FUN,Xc);
   Jc(:,i)=(Fi-Fc)/stepsize;
end
%---------------------------------------------------------------------------	
function [fplus,FVplus]=local_nefn(FUN,xc,options)
FVplus=feval(FUN,xc);
fplus=sum((options.Sf.*FVplus).^2)/2;
%---------------------------------------------------------------------------
function [options,errmsg]=local_neinck(FUN,xc,options)
n=size(xc,1);
f=feval(FUN,xc);
if strcmp(options.Scale,'off')
   options.Sx=ones(n,1);
   options.iSx=ones(n,1);
   options.Sf=ones(n,1);
else
   options.iSx=local_sign(xc);
   options.Sx=1./options.iSx;
   options.Sf=1./local_sign(abs(f));
end
if options.MaxStep==0 % find default maximum step
   options.MaxStep=1e3*max(norm(options.Sx.*xc),norm(diag(options.Sx)));
end
if ischar(options.Jacobian)
   options.analjac=~(strcmpi(options.Jacobian,'broyden') ||...
      strcmpi(options.Jacobian,'finite'));
else
   options.analjac=true;
   if ~isa(options.Jacobian,'function_handle')
      error('Jacobian Must be Function Handle.')
   end
end
options.eta=eps;
options.steptol=eps^(2/3);
options.mintol=eps^(2/3);
errmsg='';
if size(xc,2)>1
   errmsg='Xo Must be a Column Vector.';
end
if n<2
   errmsg='Two or More Variables in X Required.';
end
if size(f,1)~=n
   errmsg='Xo and FUN(Xo) Must Return Equal Length Column Vectors.';
end
%---------------------------------------------------------------------------