levmar.m

matlab语言实现非线性最小二乘法的曲线拟合LM算法

function [x,histout,costdata] = levmar(x0,f,tol,maxit)
%
% C. T. Kelley, Dec 14, 1997
%
% This code comes with no guarantee or warranty of any kind.
%
% function [x,histout,costdata] = levmar(x0,f,tol,maxit)
%
% Levenberg-Marquardt code, trust region control of LM parameter
%
%
% Input: x0 = initial iterate
%        f = r^T r/2 = objective function,
%            the calling sequence for f should be
%            [fout,gout,jac]=f(x) where fout=f(x) is a scalar
%              gout = jac^T r = grad f(x) is a COLUMN vector
%              and jac = r' = Jacobian of r is an M x N matrix
%        tol = termination criterion norm(grad) < tol
%        maxit = maximum iterations (optional) default = 100
%
% Output: x = solution
%         histout = iteration history   
%             Each row of histout is      
%       [norm(grad), f, number of stepsize cuts, iteration count] 
%       costdata = [num f, num grad, num hess] (for levmar, num hess=0)
%
% At this stage all iteration parameters are hardwired in the code.
%
%
maxarm=10;
if nargin < 4
maxit=100; 
end
itc=1; xc=x0; 
[fc,gc,jac]=feval(f,xc);
nu0=.001d0;
mvar=length(gc); nvar=length(xc);
nfun=1; ngrad=1; numh=0;
numf=1; numg=1; numh=0;
ithist(1,1)=norm(gc); ithist(1,2) = fc; ithist(1,4)=itc-1; ithist(1,3)=0; 
ithist(itc,:)
nu=norm(gc);
while(norm(gc) > tol & itc <= maxit)
        itc=itc+1;
        hc=(jac'*jac)+ nu*eye(nvar);
        dc=hc\gc;
        xt=xc-dc; 
        [xp,nup,idid]=trtestlm(f,xc,xt,fc,jac,gc,nu,nu0);
        if idid > 20
           error('too many iterations in TR solve')
        end
        xc=xp; nu=nup; numf=numf+idid;
        if idid > 1
	[fc,gc,jac]=feval(f,xc); numf = numf+1; numg=numg+1;
        end
	ithist(itc,1)=norm(gc); ithist(itc,2) = fc; 
	ithist(itc,4)=itc-1; ithist(itc,3)=idid;
        ithist(itc,:)
end
x=xc; histout=ithist(1:itc,:);
costdata=[numf, numg, numh];
%
%
%
function [xp,nup,idid]=trtestlm(f,xc,xt,fc,jac,gc,nu,nu0);
mu0=.1; mulow=.25; muhigh=.75; mup=2.d0; mdown=.5d0;
z=xc; nvar=length(z); numft=0; numgt=0; itr=0;
while z==xc & itr <= 20
	ft=feval(f,xt); 
        itr=itr+1;
	s=xt-xc; ared=fc-ft; pred=-gc'*s;
        rat = ared/pred;
	if rat < mu0
                nu=max(nu*mup,nu0);
		hc=(jac'*jac)+ nu*eye(nvar);
                dc=hc\gc;
                xt=xc-dc;
	elseif rat < mulow
		z=xt; nu=max(nu*mup,nu0);
	else
		z=xt;
                if rat > muhigh
		nu=mdown*nu;
                if nu < nu0 nu=0.d0; end
                end
	end
end
nup=nu; xp=z; idid=itr;