solve_chol

 (Modified) Newton method via Cholesky factorization (with inertia control)
 Symmetric approximate minimum degree permutation is possible to employ
 Solves the system "matrix*dir=rhs"
 OUTPUT
   nFlag ... 0..OK, 1..Cholesky/inertia failed
 'obj' is used only for option settings (and to store last perturbation)

0001 function [dir,nFlag]=solve_chol(obj,matrix,rhs)
0002 % (Modified) Newton method via Cholesky factorization (with inertia control)
0003 % Symmetric approximate minimum degree permutation is possible to employ
0004 % Solves the system "matrix*dir=rhs"
0005 % OUTPUT
0006 %   nFlag ... 0..OK, 1..Cholesky/inertia failed
0007 % 'obj' is used only for option settings (and to store last perturbation)
0008 
0009 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0010 
0011 
0012   start_time=cputime;
0013   [n m] = size(matrix);
0014   mat_nnz=nnz(matrix);
0015 
0016   LMUPDATE = obj.allopts.pert_update;
0017   LMLOW = obj.allopts.pert_min;
0018   PERT_TRY_MAX = obj.allopts.pert_try_max;
0019   ORDERING = obj.allopts.chol_ordering;
0020   %LMFASTER = obj.allopts.pert_faster; % && issparse(matrix); ??
0021   LMFASTER = 0; % && issparse(matrix); ??
0022 
0023   if (ORDERING && issparse(matrix))
0024     if (~isempty(obj.chol_factperm))
0025       obj.print(4,Inf,'Reusing ordering');
0026       p=obj.chol_factperm;
0027     else
0028       obj.print(4,Inf,'Computing ordering');
0029       p=symamd(matrix);
0030       obj.chol_factperm=p;
0031     end
0032   else
0033     p=1:n;
0034   end
0035   rev(p)=1:n;  % inverse permutation
0036 
0037   matrix=matrix(p,p);
0038   [factor, status] = chol(matrix);
0039   chol_try=0;
0040   lambda=max(obj.chol_lmlast/LMUPDATE, LMLOW);  % lambda to try if it fails now
0041   lambda_last=0.;
0042   while (status~=0 && chol_try<PERT_TRY_MAX)
0043     obj.print(4,Inf,'Chol fact failed (%d), new pert: %e',status,lambda);
0044     if (LMFASTER)
0045       % find negative curvature vector from the last factorization
0046       %k = status-1;
0047       %R11 = factor(1:status-1,1:status-1);
0048       %R12 = factor(1:status-1,status:n);
0049       %v_nc = [R11 \ R12(:,1); -1; zeros(n-k-1,1)];
0050       R12e1 = factor(1:status-1,status);
0051       v_nc = [factor(1:status-1,1:status-1) \ R12e1; -1; zeros(n-status,1)];
0052       v_nc = v_nc/norm(v_nc);      % negative curvature vector
0053       % perturbated matrix must be positive definite, e.i., v_nc'*(matrix+lambda*I)*v_nc>0
0054       % it gives: lambda > -v_nc'*matrix*v_nc since v_nc is normed
0055       negcrv = -v_nc'*matrix*v_nc;
0056       cnt=0;
0057       while (lambda-lambda_last <= negcrv)
0058         lambda=lambda*LMUPDATE;
0059         cnt=cnt+1;
0060       end
0061       obj.print(5,Inf,'Chol: pert_faster: skipped %d steps: pert=%e -> %e',cnt,lambda_last,lambda);
0062     end
0063     matrix=matrix + (lambda-lambda_last)*speye(n,n);
0064     [factor, status] = chol(matrix);
0065     chol_try = chol_try+1;
0066     lambda_last=lambda;
0067     lambda=lambda*LMUPDATE;
0068   end
0069 
0070   if (chol_try>0)
0071     obj.chol_lmlast=lambda_last;
0072   end
0073 
0074   % update statistics
0075   time_fact = cputime - start_time;
0076   obj.stats_time_fact_last=obj.stats_time_fact_last+time_fact;
0077   obj.initer_last = obj.initer_last+chol_try+1;
0078 
0079   if (status~=0)
0080     obj.print(3,Inf,'Chol fact failure (%d), last pert: %e, giving up',status,lambda_last);
0081     nFlag = 1;
0082     dir=zeros(n,1);
0083     return;
0084   end
0085 
0086   dir=factor \ (factor' \ rhs(p));
0087   dir=dir(rev);
0088   time_total=cputime - start_time;
0089   nFlag=0;
0090 
0091   obj.print(4,Inf,'Chol fact OK in %fs, total %fs, no pert=%i, pert=%.4e, nnz=%d (dim %dx%d)',time_fact, time_total, chol_try, lambda_last, nnz(factor),n,n);
0092   if (lambda_last>0)
0093     obj.print(3,4,'Chol fact: OK, pert=%.4e',lambda_last);
0094   end
0095 
0096