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annotate toolbox/cod.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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8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
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1 function [U, R, V] = cod(A, tol)
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2 %COD Complete orthogonal decomposition.
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3 % [U, R, V] = COD(A, TOL) computes a decomposition A = U*T*V,
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4 % where U and V are unitary, T = [R 0; 0 0] has the same dimensions as
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5 % A, and R is upper triangular and nonsingular of dimension rank(A).
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6 % Rank decisions are made using TOL, which defaults to approximately
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7 % MAX(SIZE(A))*NORM(A)*EPS.
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8 % By itself, COD(A, TOL) returns R.
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9
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10 % Reference:
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11 % G.H. Golub and C.F. Van Loan, Matrix Computations, Second
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12 % Edition, Johns Hopkins University Press, Baltimore, Maryland,
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13 % 1989, sec. 5.4.2.
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15 [m, n] = size(A);
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16
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17 % QR decomposition.
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18 [U, R, P] = qr(A); % AP = UR
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19 V = P'; % A = URV;
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20 if nargin == 1, tol = max(m,n)*eps*abs(R(1,1)); end % |R(1,1)| approx NORM(A).
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21
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22 % Determine r = effective rank.
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23 r = sum(abs(diag(R)) > tol);
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24 r = r(1); % Fix for case where R is vector.
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25 R = R(1:r,:); % Throw away negligible rows (incl. all zero rows, m>n).
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27 if r ~= n
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29 % Reduce nxr R' = r [L] to lower triangular form: QR' = [Lbar].
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30 % n-r [M] [0]
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31
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32 [Q, R] = trap2tri(R');
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33 V = Q*V;
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34 R = R';
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35
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36 end
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37
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38 if nargout <= 1, U = R; end