function [sol, it_hist, ierr] = nsola(x,f,tol, parms)
% Newton-Krylov solver, globally convergent
% solver for f(x) = 0
%
% Inexact-Newton-Armijo iteration
%
% Eisenstat-Walker forcing term
%
% Parabolic line search via three point interpolation.
%
% C. T. Kelley, June 29, 1994
% parms made truly optional, Feb 12, 1996
%
% This code comes with no guarantee or warranty of any kind.
%
% function [sol, it_hist, ierr] = nsola(x,f,tol,parms)
%
% inputs:
% initial iterate = x
% function = f
% tol = [atol, rtol] relative/absolute
% error tolerances for the nonlinear iteration
% parms = [maxit, maxitl, etamax, lmeth, restart_limit]
% maxit = maxmium number of nonlinear iterations
% default = 40
% maxitl = maximum number of inner iterations before restart
% in GMRES(m), m=maxitl
% default = 40
%
% For iterative methods other than GMRES(m) maxitl
% is the upper bound on linear iterations.
%
% |etamax| = Maximum error tolerance for residual in inner
% iteration. The inner iteration terminates
% when the relative linear residual is
% smaller than eta*| F(x_c) |. eta is determined
% by the modified Eisenstat-Walker formula if etamax > 0.
% If etamax < 0, then eta = |etamax| for the entire
% iteration.
% default: etamax=.9
%
% lmeth = choice of linear iterative method
% 1 (GMRES), 2 GMRES(m),
% 3 (BICGSTAB), 4 (TFQMR)
% default = 1 (GMRES, no restarts)
%
% restart_limit = max number of restarts for GMRES if
% lmeth = 2
% default=20
%
% output:
% sol = solution
% it_hist(maxit,3) = scaled l2 norms of nonlinear residuals
% for the iteration, of number function evaluations,
% and number of steplength reductions
% ierr = 0 upon successful termination
% ierr = 1 if either after maxit iterations
% the termination criterion is not satsified
% or the ratio of successive nonlinear residuals
% exceeds 1. In this latter case, the iteration
% is terminted.
% ierr = 2 failure in the line search. The iteration
% is terminated if too many steplength reductions
% are taken.
%
%
% internal parameters:
% debug = turns on/off iteration statistics display as
% the iteration progresses
%
% alpha = 1.d-4, parameter to measure sufficient decrease
%
% sigma0 = .1, sigma1=.5, safeguarding bounds for the linesearch
%
% maxarm = 50, maximum number of steplength reductions before
% failure is reported
%
%
% Requires dirder.m, fdkrylov.m, parab3p.m, fdgmres.m,
% fdcgstab.m, fdtfqmr.m, givapp.m
%
%
%
%
% set the debug parameter, 1 turns display on, otherwise off
%
debug=1;
%
% set alpha, sigma0, sigma1, maxarm, and restart_limit
%
alpha = 1.d-4; sigma0=.1; sigma1=.5; maxarm = 50;
%
% initialize it_hist, ierr, and set the iteration parameters
%
gamma=.9;
ierr = 0; maxit=40; lmaxit=40; etamax=.9; it_histx=zeros(maxit,3);
lmeth=1; restart_limit=20;
%
% initialize parameters for the iterative methods
%
gmparms=[abs(etamax), lmaxit];
if nargin == 4
maxit=parms(1); lmaxit=parms(2); etamax=parms(3);
gmparms=[abs(etamax), lmaxit];
if length(parms)>=4
lmeth=parms(4);
end
if length(parms)==5
gmparms=[abs(etamax), lmaxit, parms(5), 1];
end
end
%
rtol=tol(2); atol=tol(1); n = length(x); fnrm=1; itc=0;
%
% evaluate f at the initial iterate
% compute the stop tolerance
%
f0=feval(f,x);
fnrm=norm(f0)/sqrt(n);
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 0];
%
% main iteration loop
%
while(fnrm > stop_tol & itc < maxit)
%
% keep track of the ratio (rat = fnrm/frnmo)
% of successive residual norms and
% the iteration counter (itc)
%
rat=fnrm/fnrmo;
fnrmo=fnrm;
itc=itc+1;
[step, errstep, inner_it_count,inner_f_evals]=...
fdkrylov(f0, f, x, gmparms, lmeth);
%
% The line search starts here.
%
xold=x;
lambda=1; lamm=1; lamc=lambda; iarm=0;
xt = x + lambda*step;
ft=feval(f,xt);
nft=norm(ft); nf0=norm(f0); ff0=nf0*nf0; ffc=nft*nft; ffm=nft*nft;
while nft >= (1 - alpha*lambda) * nf0;
%
% apply the three point parabolic model
%
if iarm == 0
lambda=sigma1*lambda;
else
lambda=parab3p(lamc, lamm, ff0, ffc, ffm);
end
%
% update x; keep the books on lambda
%
xt=x+lambda*step;
lamm=lamc;
lamc=lambda;
%
% keep the books on the function norms
%
ft=feval(f,xt);
nft=norm(ft);
ffm=ffc;
ffc=nft*nft;
iarm=iarm+1;
if iarm > maxarm
disp(' Armijo failure, too many reductions ');
ierr=2;
disp(outstat)
it_hist=it_histx(1:itc+1,:);
sol=xold;
return;
end
end
x=xt;
f0=ft;
%
% end of line search
%
fnrm=norm(f0)/sqrt(n);
it_histx(itc+1,1)=fnrm;
%
% How many function evaluations did this iteration require?
%
it_histx(itc+1,2)=it_histx(itc,2)+inner_f_evals+iarm+1;
if(itc == 1) it_histx(itc+1,2) = it_histx(itc+1,2)+1; end;
it_histx(itc+1,3)=iarm;
%
rat=fnrm/fnrmo;
%
% adjust eta as per Eisenstat-Walker
%
if etamax > 0
etaold=gmparms(1);
etanew=gamma*rat*rat;
if gamma*etaold*etaold > .1
etanew=max(etanew,gamma*etaold*etaold);
end
gmparms(1)=min([etanew,etamax]);
gmparms(1)=max(gmparms(1),.5*stop_tol/fnrm);
end
%
outstat(itc+1, :) = [itc fnrm inner_it_count rat iarm];
%
if debug==1
disp(outstat(itc+1,:))
end
% end while
end
sol=x;
it_hist=it_histx(1:itc+1,:);
if debug==1
disp(outstat)
it_hist=it_histx(1:itc+1,:);
end
%
% on failure, set the error flag
%
if fnrm > stop_tol
ierr = 1;
end
