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comparison toolbox/augment.m @ 0:8f23314345f4 draft

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Create local repository for matrix toolboxes. Step #0 done.
author Antonio Pino Robles <data.script93@gmail.com>
date Wed, 06 May 2015 14:56:53 +0200
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1 function C = augment(A, alpha)
2 %AUGMENT Augmented system matrix.
3 % AUGMENT(A, ALPHA) is the square matrix
4 % [ALPHA*EYE(m) A; A' ZEROS(n)] of dimension m+n, where A is m-by-n.
5 % It is the symmetric and indefinite coefficient matrix of the
6 % augmented system associated with a least squares problem
7 % minimize NORM(A*x-b). ALPHA defaults to 1.
8 % Special case: if A is a scalar, n say, then AUGMENT(A) is the
9 % same as AUGMENT(RANDN(p,q)) where n = p+q and
10 % p = ROUND(n/2), that is, a random augmented matrix
11 % of dimension n is produced.
12 % The eigenvalues of AUGMENT(A) are given in terms of the singular
13 % values s(i) of A (where m>n) by
14 % 1/2 +/- SQRT( s(i)^2 + 1/4 ), i=1:n (2n eigenvalues),
15 % 1, (m-n eigenvalues).
16 % If m < n then the first expression provides 2m eigenvalues and the
17 % remaining n-m eigenvalues are zero.
18 %
19 % See also SPAUGMENT.
20
21 % Reference:
22 % G.H. Golub and C.F. Van Loan, Matrix Computations, Second
23 % Edition, Johns Hopkins University Press, Baltimore, Maryland,
24 % 1989, sec. 5.6.4.
25
26 [m, n] = size(A);
27 if nargin < 2, alpha = 1; end
28
29 if max(m,n) == 1
30 n = A;
31 p = round(n/2);
32 q = n - p;
33 A = randn(p,q);
34 m = p; n = q;
35 end
36
37 C = [alpha*eye(m) A; A' zeros(n)];