The Matrix Computation Toolbox
11 Sep 2002
11 Sep 2002)
A collection of M-files for carrying out various numerical linear algebra tasks.
function C = augment(A, alpha)
%AUGMENT Augmented system matrix.
% AUGMENT(A, ALPHA) is the square matrix
% [ALPHA*EYE(m) A; A' ZEROS(n)] of dimension m+n, where A is m-by-n.
% It is the symmetric and indefinite coefficient matrix of the
% augmented system associated with a least squares problem
% minimize NORM(A*x-b). ALPHA defaults to 1.
% Special case: if A is a scalar, n say, then AUGMENT(A) is the
% same as AUGMENT(RANDN(p,q)) where n = p+q and
% p = ROUND(n/2), that is, a random augmented matrix
% of dimension n is produced.
% The eigenvalues of AUGMENT(A,ALPHA) are given in terms of the
% singular values s(i) of A (where m>n) by
% ALPHA/2 +/- SQRT( s(i)^2*ALPHA^2 + 1/4 ), i=1:n (2n eigenvalues),
% ALPHA, (m-n eigenvalues).
% If m < n then the first expression provides 2m eigenvalues and the
% remaining n-m eigenvalues are zero.
% See also SPAUGMENT.
% G. H. Golub and C. F. Van Loan, Matrix Computations, third
% Edition, Johns Hopkins University Press, Baltimore, Maryland,
% 1996; sec. 5.6.4.
% N. J. Higham, Accuracy and Stability of Numerical Algorithms,
% Second edition, Society for Industrial and Applied Mathematics,
% Philadelphia, PA, 2002; sec. 20.5.
[m, n] = size(A);
if nargin < 2, alpha = 1; end
if max(m,n) == 1
n = A;
p = round(n/2);
q = n - p;
A = randn(p,q);
m = p; n = q;
C = [alpha*eye(m) A; A' zeros(n)];