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function [R, P, I] = cholp(A, piv) %CHOLP Cholesky factorization with pivoting of a positive semidefinite matrix. % [R, P] = CHOLP(A) returns an upper triangular matrix R and a % permutation matrix P such that R'*R = P'*A*P. Only the upper % triangular part of A is used. If A is not positive semidefinite, % an error message is printed. % % [R, P, I] = CHOLP(A) never produces an error message. % If A is positive semidefinite then I = 0 and R is the Cholesky factor. % If A is not positive semidefinite then I is positive and % R is (I-1)-by-N with P'*A*P - R'*R zero in columns 1:I-1 and % rows 1:I-1. % [R, I] = CHOLP(A, 0) forces P = EYE(SIZE(A)), and therefore behaves % like [R, I] = CHOL(A). % This routine is based on the LINPACK routine CCHDC. It works % for both real and complex matrices. % % Reference: % N. J. Higham, Accuracy and Stability of Numerical Algorithms, % Second edition, Society for Industrial and Applied Mathematics, % Philadelphia, PA, 2002; sec. 10.3. if nargin == 1, piv = 1; end n = length(A); pp = 1:n; I = 0; for k = 1:n if piv d = diag(A); [big, m] = max( d(k:n) ); m = m+k-1; else big = A(k,k); m = k; end if big < 0, I = k; break, end % Symmetric row/column permutations. if m ~= k A(:, [k m]) = A(:, [m k]); A([k m], :) = A([m k], :); pp( [k m] ) = pp( [m k] ); end if big == 0 if norm(A(k+1:n,k)) ~= 0 I = k; break else continue end end A(k,k) = sqrt( A(k,k) ); if k == n, break, end A(k, k+1:n) = A(k, k+1:n) / A(k,k); % For simplicity update the whole of the remaining submatrix (rather % than just the upper triangle). j = k+1:n; A(j,j) = A(j,j) - A(k,j)'*A(k,j); end R = triu(A); if I > 0 if nargout < 3, error('Matrix must be positive semidefinite.'), end R = R(1:I-1,:); end if piv == 0 P = I; else P = eye(n); P = P(:,pp); end