111 lines
3.4 KiB
Mathematica
111 lines
3.4 KiB
Mathematica
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%%*********************************************************************
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%% schursysolve: solve
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%% [H U][x1] = [b1]
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%% [V' -E][x2] [b2]
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%% via the Schur complement method as follows:
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%% SMWmat*x2 = V'*inv(H)*b1 - b2, where SMWmat = V'*inv(H)*U + E,
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%% x1 = inv(H)*(b1 - U*x2).
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%%
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%% Here H is symmetric positive definite
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%% and U, V are low-rank, possibly [].
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%%
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%% [x,solve_ok,SMWmat] = schursysolve(H,L,U,V,E,b,matfct_options,SMWmat)
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%%
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%% if matfct_options == 'chol'
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%% L is the Cholesky factor of H computed by Matlab's rountine
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%% Note H = L'*L
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%% On entry, H has already been overwritten by L.
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%% elseif matfct_options == 'spchol'
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%% L contains the sparse LDL' factorization of H computed
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%% via the sparse Cholesky routines in Spchol
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%% Note H = L*D*L'
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%% end
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%%
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%% SDPT3: version 3.0
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%% Copyright (c) 1997 by
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%% K.C. Toh, M.J. Todd, R.H. Tutuncu
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%% Last modified: 2 Feb 01
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%%*********************************************************************
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function [x,solve_ok,SMWmat] = schursysolve(H,L,U,V,E,b,matfct_options,SMWmat)
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if (nargin <= 7); SMWmat = []; end;
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ismtU = isempty(U);
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m = length(H); n = size(U,2);
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if (length(b) < m+n)
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b = [b; zeros(m+n-length(b),1)];
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end
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%%
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if ismtU
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solve_ok = 1;
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x = linslv(L,b,matfct_options);
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else
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solve_ok = -1;
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if isempty(SMWmat)
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n = size(U,2);
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HinvU = zeros(size(U));
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for i = 1:n
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HinvU(:,i) = linslv(L,U(:,i),matfct_options);
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end
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[SMWmat.l,SMWmat.u,SMWmat.p] = lu(E+V'*HinvU);
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end
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b1 = b(1:m); b2 = b(m+[1:n]);
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btmp = V'*linslv(L,b1,matfct_options) - b2;
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x2 = SMWmat.u \ (SMWmat.l \ (SMWmat.p*btmp));
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x1 = linslv(L,b1-U*x2,matfct_options);
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x = [x1; x2];
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%%
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%% Iterative refinement. This step is crucial to ensure
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%% that computed solution is sufficiently accuraete.
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%%
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rtol = 1e-4;
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if length(b) > 5000;
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maxit = 50;
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else
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maxit = 25;
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end
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bnorm = max(1,norm(b));
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for k = 1:maxit
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x0 = x;
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if strcmp(matfct_options,'chol');
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tmp = L*x1; Hx1 = (tmp'*L)';
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elseif strcmp(matfct_options,'spchol');
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Hx1 = H*x1;
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end
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r1 = b1 - (Hx1 + U*x2);
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r2 = b2 - (V'*x1 - E*x2);
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%%
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rrnrm(k) = sqrt(norm(r1)^2 + norm(r2)^2)/bnorm;
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if (rrnrm(k) < rtol); solve_ok = 1; break; end;
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if (k > 1) & (rrnrm(k)/rrnrm(k-1) > 0.8);
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x = x0; break;
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end
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rtmp = V'*linslv(L,r1,matfct_options) -r2;
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d2 = SMWmat.u \ (SMWmat.l \ (SMWmat.p*rtmp));
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d1 = linslv(L,r1-U*d2,matfct_options);
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x1 = x1 + d1;
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x2 = x2 + d2;
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x = [x1; x2];
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end
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end
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%%
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%%*************************************************************************
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%% linslv: Solve a linear system H*x = b
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%% where H is symmetric positive definite.
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%%
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%% L is the Cholesky factor of H.
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%%*************************************************************************
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function x = linslv(L,b,matfct_options)
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if strcmp(matfct_options,'chol')
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if issparse(b); b = full(b); end;
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tmp = mexbsolve(L,b,2);
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x = mexbsolve(L,tmp,1);
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elseif strcmp(matfct_options,'spchol')
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x = bwblkslvfun(L, fwblkslvfun(L,b) ./ L.d);
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end
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%%
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%%*************************************************************************
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