220 lines
7.6 KiB
Mathematica
220 lines
7.6 KiB
Mathematica
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%%*****************************************************************************
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%% checkdepconst: compute AAt to determine if the
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%% constraint matrices Ak are linearly independent.
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%%
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%% [At,b,y,idxB,neardepconstr,feasible,AAt] = checkdepconstr(blk,At,b,y,rmdepconstr);
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%%
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%% rmdepconstr = 1, if want to remove dependent columns in At
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%% = 0, otherwise.
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%%
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%% idxB = indices of linearly independent columns of original At.
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%% neardepconstr = 1 if there is nearly dependent columns in At
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%% = 0, otherwise.
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%% Note: the definition of "nearly dependent" is dependent on the
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%% threshold used to determine the small diagonal elements in
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%% the LDLt factorization of A*At.
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%%
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%% SDPT3: version 3.1
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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: 16 Sep 2004
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%%*****************************************************************************
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function [At,b,y,idxB,neardepconstr,feasible,AAt] = ...
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checkdepconstr(blk,At,b,y,rmdepconstr);
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global existlowrank printlevel
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global nnzmatold
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%%
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%% compute AAt
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%%
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m = length(b);
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AAt = sparse(m,m); numdencol = 0; UU = [];
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for p = 1:size(blk,1)
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pblk = blk(p,:);
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if strcmp(pblk{1},'s');
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m1 = size(At{p,1},2); m2 = m - m1;
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if (m2 > 0)
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if (m2 > 0.3*m); AAt = full(AAt); end
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dd = At{p,3};
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lenn = size(dd,1);
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idxD = [0; find(diff(dd(:,1))); lenn];
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ss = [0, cumsum(pblk{3})];
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if (existlowrank)
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AAt(1:m1,1:m1) = AAt(1:m1,1:m1) + At{p,1}'*At{p,1};
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for k = 1:m2
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idx = [ss(k)+1 : ss(k+1)];
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idx2 = [idxD(k)+1: idxD(k+1)];
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ii = dd(idx2,2)-ss(k); %% undo cumulative indexing
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jj = dd(idx2,3)-ss(k);
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len = pblk{3}(k);
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Dk = spconvert([ii,jj,dd(idx2,4); len,len,0]);
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tmp = svec(pblk,At{p,2}(:,idx)*Dk*At{p,2}(:,idx)');
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tmp2 = AAt(1:m1,m1+k) + (tmp'*At{p,1})';
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AAt(1:m1,m1+k) = tmp2;
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AAt(m1+k,1:m1) = tmp2';
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end
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end
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DD = spconvert([dd(:,2:4); sum(pblk{3}),sum(pblk{3}),0]);
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VtVD = (At{p,2}'*At{p,2})*DD;
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VtVD2 = VtVD'.*VtVD;
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for k = 1:m2
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idx0 = [ss(k)+1 : ss(k+1)];
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%%tmp = VtVD(idx0,:)' .* VtVD(:,idx0);
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tmp = VtVD2(:,idx0);
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tmp = tmp*ones(length(idx0),1);
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tmp3 = AAt(m1+[1:m2],m1+k) + mexqops(pblk{3},tmp,ones(length(tmp),1),1);
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AAt(m1+[1:m2],m1+k) = tmp3;
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end
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else
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AAt = AAt + abs(At{p,1})'*abs(At{p,1});
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end
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else
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decolidx = checkdense(At{p,1}');
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%% checkdense punts if the matrix has too many dense columns
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%% in this case we make the whole matrix dense
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if ~isempty(decolidx);
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n2 = size(At{p,1},1);
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dd = ones(n2,1);
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len= length(decolidx);
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if (len < 0.5*sum(pblk{2})) || (sum(pblk{2}) >= 20)
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dd(decolidx) = zeros(len,1);
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AAt = AAt + (abs(At{p,1})' *spdiags(dd,0,n2,n2)) *abs(At{p,1});
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tmp = At{p,1}(decolidx,:)';
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UU = [UU, tmp];
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numdencol = numdencol + len;
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end
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else
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if (nnz(At{p,1})>0.2*numel(At{p,1}))
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Atp=full(abs(At{p,1}));
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else
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Atp=abs(At{p,1});
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end
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AAt = AAt + Atp'*Atp;
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end
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end
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end
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if (numdencol > 0) & (printlevel)
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fprintf('\n number of dense column in A = %d',numdencol);
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end
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numdencol = size(UU,2);
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%%
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%%
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%%
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feasible = 1; neardepconstr = 0;
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if ~issparse(AAt); AAt = sparse(AAt); end
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nnzmatold = mexnnz(AAt);
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rho = 1e-15;
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diagAAt = diag(AAt);
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%%mexschurfun(AAt,rho*max(diagAAt,1)); %% does not work for 2015b
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AAt = mexschurfun(AAt,rho*max(diagAAt,1));
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[L.R,indef,L.perm] = chol(AAt,'vector');
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L.d = full(diag(L.R)).^2;
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if (indef)
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msg = 'AAt is not pos. def.';
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idxB = [1:m];
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neardepconstr = 1;
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if (printlevel); fprintf('\n checkdepconstr: %s',msg); end
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return;
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end
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%%
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%% find independent rows of A
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%%
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dd = zeros(m,1);
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idxB = [1:m]';
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dd(L.perm) = abs(L.d);
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idxN = find(dd < 1e-13*mean(L.d));
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ddB = dd(setdiff([1:m],idxN));
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ddN = dd(idxN);
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if ~isempty(ddN) & ~isempty(ddB) & (min(ddB)/max(ddN) < 10)
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%% no clear separation of elements in dd
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%% do not label constraints as dependent
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idxN = [];
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end
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if ~isempty(idxN)
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neardepconstr = 1;
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if (printlevel)
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fprintf('\n number of nearly dependent constraints = %1.0d',length(idxN));
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end
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if (numdencol==0)
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if (rmdepconstr)
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idxB = setdiff([1:m]',idxN);
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if (printlevel)
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fprintf('\n checkdepconstr: removing dependent constraints...');
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end
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[W,resnorm] = findcoeffsub(blk,At,idxB,idxN);
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tol = 1e-8;
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if (resnorm > sqrt(tol))
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idxB = [1:m]';
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neardepconstr = 0;
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if (printlevel)
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fprintf('\n checkdepconstr: basis rows cannot be reliably identified,');
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fprintf(' abort removing nearly dependent constraints');
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end
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return;
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end
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tmp = W'*b(idxB) - b(idxN);
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nnorm = norm(tmp)/max(1,norm(b));
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if (nnorm > tol)
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feasible = 0;
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if (printlevel)
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fprintf('\n checkdepconstr: inconsistent constraints exist,');
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fprintf(' problem is infeasible.');
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end
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else
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feasible = 1;
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for p = 1:size(blk,1)
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At{p,1} = At{p,1}(:,idxB);
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end
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b = b(idxB);
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y = y(idxB);
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AAt = AAt(idxB,idxB);
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end
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else
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if (printlevel)
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fprintf('\n To remove these constraints,');
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fprintf(' re-run sqlp.m with OPTIONS.rmdepconstr = 1.');
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end
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end
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else
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if (printlevel)
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fprintf('\n warning: the sparse part of AAt may be nearly singular.');
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end
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end
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end
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%%*****************************************************************************
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%% findcoeffsub:
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%%
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%% [W,resnorm] = findcoeffsub(blk,At,idXB,idXN);
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%%
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%% idXB = indices of independent columns of At.
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%% idxN = indices of dependent columns of At.
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%%
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%% AB = At(:,idxB); AN = At(:,idxN) = AB*W
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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 [W,resnorm] = findcoeffsub(blk,At,idxB,idxN);
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AB = []; AN = [];
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for p = 1:size(blk,1)
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AB = [AB; At{p,1}(:,idxB)];
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AN = [AN; At{p,1}(:,idxN)];
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end
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[m,n] = size(AB);
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%%
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%%-----------------------------------------
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%% find W so that AN = AB*W
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%%-----------------------------------------
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%%
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[L,U,P,Q] = lu(sparse(AB));
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rhs = P*AN;
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Lhat = L(1:n,:);
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W = Q*( U \ (Lhat \ rhs(1:n,:)));
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resnorm = norm(AN-AB*W,'fro')/max(1,norm(AN,'fro'));
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%%*****************************************************************************
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