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137 lines
4.7 KiB
137 lines
4.7 KiB
function [L,U,Q] = luq(A,do_pivot,tol)
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% PURPOSE: calculates the following decomposition
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%
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% A = L |Ubar 0 | Q
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% |0 0 |
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%
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% where Ubar is a square invertible matrix
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% and matrices L, Q are invertible.
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%
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% ---------------------------------------------------
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% USAGE: [L,U,Q] = luq(A,do_pivot,tol)
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% INPUT:
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% A a sparse matrix
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% do_pivot = 1 with column pivoting
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% = 0 without column pivoting
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% tol uses the tolerance tol in separating zero and
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% nonzero values
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%
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% OUTPUT:
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% L,U,Q matrices
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%
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% COMMENTS:
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% based on lu decomposition
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%
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% Copyright (c) Pawel Kowal (2006)
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% All rights reserved
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% LREM_SOLVE toolbox is available free for noncommercial academic use only.
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% pkowal3@sgh.waw.pl
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[n,m] = size(A);
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if ~issparse(A)
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A = sparse(A);
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end
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%--------------------------------------------------------------------------
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% SPECIAL CASES
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%--------------------------------------------------------------------------
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if size(A,1)==0
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L = speye(n);
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U = A;
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Q = speye(m);
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return;
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end
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if size(A,2)==0
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L = speye(n);
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U = A;
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Q = speye(m);
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return;
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end
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%--------------------------------------------------------------------------
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% LU DECOMPOSITION
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%--------------------------------------------------------------------------
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if do_pivot
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[L,U,P,Q] = lu(A);
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Q = Q';
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else
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[L,U,P] = lu(A);
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Q = speye(m);
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end
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p = size(A,1)-size(L,2);
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LL = [sparse(n-p,p);speye(p)];
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L = [P'*L P(n-p+1:n,:)'];
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U = [U;sparse(p,m)];
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%--------------------------------------------------------------------------
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% FINDS ROWS WITH ZERO AND NONZERO ELEMENTS ON THE DIAGONAL
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%--------------------------------------------------------------------------
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if size(U,1)==1 || size(U,2)==1
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S = U(1,1);
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else
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S = diag(U);
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end
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I = find(abs(S)>tol);
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Jl = (1:n)';
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Jl(I) = [];
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Jq = (1:m)';
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Jq(I) = [];
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Ubar1 = U(I,I);
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Ubar2 = U(Jl,Jq);
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Qbar1 = Q(I,:);
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Lbar1 = L(:,I);
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%--------------------------------------------------------------------------
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% ELININATES NONZEZO ELEMENTS BELOW AND ON THE RIGHT OF THE
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% INVERTIBLE BLOCK OF THE MATRIX U
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%
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% UPDATES MATRICES L, Q
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%--------------------------------------------------------------------------
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if ~isempty(I)
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Utmp = U(I,Jq);
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X = Ubar1'\U(Jl,I)';
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Ubar2 = Ubar2-X'*Utmp;
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Lbar1 = Lbar1+L(:,Jl)*X';
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X = Ubar1\Utmp;
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Qbar1 = Qbar1+X*Q(Jq,:);
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Utmp = [];
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X = [];
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end
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%--------------------------------------------------------------------------
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% FINDS ROWS AND COLUMNS WITH ONLY ZERO ELEMENTS
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%--------------------------------------------------------------------------
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I2 = find(max(abs(Ubar2),[],2)>tol);
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I5 = find(max(abs(Ubar2),[],1)>tol);
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I3 = Jl(I2);
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I4 = Jq(I5);
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Jq(I5) = [];
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Jl(I2) = [];
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U = [];
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%--------------------------------------------------------------------------
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% FINDS A PART OF THE MATRIX U WHICH IS NOT IN THE REQIRED FORM
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%--------------------------------------------------------------------------
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A = Ubar2(I2,I5);
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%--------------------------------------------------------------------------
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% PERFORMS LUQ DECOMPOSITION OF THE MATRIX A
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%--------------------------------------------------------------------------
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[L1,U1,Q1] = luq(A,do_pivot,tol);
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%--------------------------------------------------------------------------
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% UPDATES MATRICES L, U, Q
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%--------------------------------------------------------------------------
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Lbar2 = L(:,I3)*L1;
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Qbar2 = Q1*Q(I4,:);
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L = [Lbar1 Lbar2 L(:,Jl)];
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Q = [Qbar1; Qbar2; Q(Jq,:)];
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n1 = length(I);
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n2 = length(I3);
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m2 = length(I4);
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U = [Ubar1 sparse(n1,m-n1);sparse(n2,n1) U1 sparse(n2,m-n1-m2);sparse(n-n1-n2,m)];
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