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annotate toolbox/kahan.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.
Antonio Pino Robles <data.script93@gmail.com>
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1 function U = kahan(n, theta, pert)
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2 %KAHAN Kahan matrix - upper trapezoidal.
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3 % KAHAN(N, THETA) is an upper trapezoidal matrix
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4 % that has some interesting properties regarding estimation of
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5 % condition and rank.
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6 % The matrix is N-by-N unless N is a 2-vector, in which case it
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7 % is N(1)-by-N(2).
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8 % The parameter THETA defaults to 1.2.
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9 % The useful range of THETA is 0 < THETA < PI.
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10 %
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11 % To ensure that the QR factorization with column pivoting does not
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12 % interchange columns in the presence of rounding errors, the diagonal
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13 % is perturbed by PERT*EPS*diag( [N:-1:1] ).
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14 % The default is PERT = 25, which ensures no interchanges for KAHAN(N)
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15 % up to at least N = 90 in IEEE arithmetic.
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16 % KAHAN(N, THETA, PERT) uses the given value of PERT.
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17
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18 % The inverse of KAHAN(N, THETA) is known explicitly: see
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19 % Higham (1987, p. 588), for example.
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20 % The diagonal perturbation was suggested by Christian Bischof.
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21 %
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22 % References:
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23 % W. Kahan, Numerical linear algebra, Canadian Math. Bulletin,
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24 % 9 (1966), pp. 757-801.
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25 % N.J. Higham, A survey of condition number estimation for
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26 % triangular matrices, SIAM Review, 29 (1987), pp. 575-596.
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27
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28 if nargin < 3, pert = 25; end
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29 if nargin < 2, theta = 1.2; end
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31 r = n(1); % Parameter n specifies dimension: r-by-n.
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32 n = n(max(size(n)));
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33
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34 s = sin(theta);
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35 c = cos(theta);
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36
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37 U = eye(n) - c*triu(ones(n), 1);
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38 U = diag(s.^[0:n-1])*U + pert*eps*diag( [n:-1:1] );
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39 if r > n
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40 U(r,n) = 0; % Extend to an r-by-n matrix.
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41 else
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42 U = U(1:r,:); % Reduce to an r-by-n matrix.
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43 end