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