Mercurial > matrix-functions
comparison toolbox/trap2tri.m @ 0:8f23314345f4 draft
Create local repository for matrix toolboxes. Step #0 done.
| author | Antonio Pino Robles <data.script93@gmail.com> |
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| date | Wed, 06 May 2015 14:56:53 +0200 |
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| -1:000000000000 | 0:8f23314345f4 |
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| 1 function [Q, T] = trap2tri(L) | |
| 2 %TRAP2TRI Unitary reduction of trapezoidal matrix to triangular form. | |
| 3 % [Q, T] = TRAP2TRI(L), where L is an m-by-n lower trapezoidal | |
| 4 % matrix with m >= n, produces a unitary Q such that QL = [T; 0], | |
| 5 % where T is n-by-n and lower triangular. | |
| 6 % Q is a product of Householder transformations. | |
| 7 | |
| 8 % Called by RANDSVD. | |
| 9 % | |
| 10 % Reference: | |
| 11 % G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, | |
| 12 % Johns Hopkins University Press, Baltimore, Maryland, 1989. | |
| 13 % P5.2.5, p. 220. | |
| 14 | |
| 15 [n, r] = size(L); | |
| 16 | |
| 17 if r > n | norm(L-tril(L),1) | |
| 18 error('Matrix must be lower trapezoidal and m-by-n with m >= n.') | |
| 19 end | |
| 20 | |
| 21 Q = eye(n); % To hold product of H.T.s | |
| 22 | |
| 23 if r ~= n | |
| 24 | |
| 25 % Reduce nxr L = r [L1] to lower triangular form: QL = [T]. | |
| 26 % n-r [L2] [0] | |
| 27 | |
| 28 for j=r:-1:1 | |
| 29 % x is the vector to be reduced, which we overwrite with the H.T. vector. | |
| 30 x = L(j:n,j); | |
| 31 x(2:r-j+1) = zeros(r-j,1); % These elts of column left unchanged. | |
| 32 s = norm(x)*(sign(x(1)) + (x(1)==0)); % Modification for sign(1)=1. | |
| 33 | |
| 34 % Nothing to do if x is zero (or x=a*e_1, but we don't check for that). | |
| 35 if s ~= 0 | |
| 36 x(1) = x(1) + s; | |
| 37 beta = s'*x(1); | |
| 38 | |
| 39 % Implicitly apply H.T. to pivot column. | |
| 40 % L(r+1:n,j) = zeros(n-r,1); % We throw these elts away at the end. | |
| 41 L(j,j) = -s; | |
| 42 | |
| 43 % Apply H.T. to rest of matrix. | |
| 44 if j > 1 | |
| 45 y = x'*L(j:n, 1:j-1); | |
| 46 L(j:n, 1:j-1) = L(j:n, 1:j-1) - x*(y/beta); | |
| 47 end | |
| 48 | |
| 49 % Update H.T. product. | |
| 50 y = x'*Q(j:n,:); | |
| 51 Q(j:n,:) = Q(j:n,:) - x*(y/beta); | |
| 52 end | |
| 53 end | |
| 54 end | |
| 55 | |
| 56 T = L(1:r,:); % Rows r+1:n have been zeroed out. |