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annotate toolbox/cholp.m @ 2:c124219d7bfa draft

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Re-add the 1995 toolbox after noticing the statement in the ~higham/mctoolbox/ webpage.
author Antonio Pino Robles <data.script93@gmail.com>
date Thu, 07 May 2015 18:36:24 +0200
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1 function [R, P, I] = cholp(A, piv)
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2 %CHOLP Cholesky factorization with pivoting of a pos. semidefinite matrix.
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3 % [R, P] = CHOLP(A) returns R and a permutation matrix P such that
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4 % R'*R = P'*A*P. Only the upper triangular part of A is used.
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5 % If A is not positive definite, an error message is printed.
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6 %
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7 % [R, P, I] = CHOLP(A) never produces an error message.
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8 % If A is positive definite then I = 0 and R is the Cholesky factor.
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9 % If A is not positive definite then I is positive and
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10 % R is (I-1)-by-N with P'*A*P - R'*R zeros in columns 1:I-1 and
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11 % rows 1:I-1.
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12 % [R, I] = CHOLP(A, 0) forces P = EYE(SIZE(A)), and therefore behaves
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13 % like [R, I] = CHOL(A).
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14
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15 % This routine is based on the LINPACK routine CCHDC. It works
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16 % for both real and complex matrices.
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17 %
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18 % Reference:
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19 % G.H. Golub and C.F. Van Loan, Matrix Computations, Second
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20 % Edition, Johns Hopkins University Press, Baltimore, Maryland,
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21 % 1989, sec. 4.2.9.
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22
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23 if nargin == 1, piv = 1; end
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24
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25 n = length(A);
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26 pp = 1:n;
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27 I = 0;
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28
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29 for k = 1:n
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30
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31 if piv
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32 d = diag(A);
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33 [big, m] = max( d(k:n) );
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34 m = m+k-1;
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35 else
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36 big = A(k,k); m = k;
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37 end
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38 if big <= 0, I = k; break, end
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39
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40 % Symmetric row/column permutations.
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41 if m ~= k
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42 A(:, [k m]) = A(:, [m k]);
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43 A([k m], :) = A([m k], :);
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44 pp( [k m] ) = pp( [m k] );
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45 end
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46
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47 A(k,k) = sqrt( A(k,k) );
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48 if k == n, break, end
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49 A(k, k+1:n) = A(k, k+1:n) / A(k,k);
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50
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51 % For simplicity update the whole of the remaining submatrix (rather
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52 % than just the upper triangle).
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53
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54 j = k+1:n;
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55 A(j,j) = A(j,j) - A(k,j)'*A(k,j);
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56
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57 end
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58
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59 R = triu(A);
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60 if I > 0
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61 if nargout < 3, error('Matrix must be positive definite.'), end
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62 R = R(1:I-1,:);
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63 end
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64
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65 if piv == 0
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66 P = I;
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67 else
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68 P = eye(n); P = P(:,pp);
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69 end