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annotate toolbox/triw.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 t = triw(n, alpha, k)
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2 %TRIW Upper triangular matrix discussed by Wilkinson and others.
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Antonio Pino Robles <data.script93@gmail.com>
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3 % TRIW(N, ALPHA, K) is the upper triangular matrix with ones on
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4 % the diagonal and ALPHAs on the first K >= 0 superdiagonals.
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5 % N may be a 2-vector, in which case the matrix is N(1)-by-N(2) and
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6 % upper trapezoidal.
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7 % Defaults: ALPHA = -1,
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8 % K = N - 1 (full upper triangle).
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9 % TRIW(N) is a matrix discussed by Kahan, Golub and Wilkinson.
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10 %
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11 % Ostrowski (1954) shows that
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12 % COND(TRIW(N,2)) = COT(PI/(4*N))^2,
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13 % and for large ABS(ALPHA),
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14 % COND(TRIW(N,ALPHA)) is approximately ABS(ALPHA)^N*SIN(PI/(4*N-2)).
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15 %
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16 % Adding -2^(2-N) to the (N,1) element makes TRIW(N) singular,
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17 % as does adding -2^(1-N) to all elements in the first column.
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18
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19 % References:
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20 % G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the
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21 % computation of the Jordan canonical form, SIAM Review,
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22 % 18(4), 1976, pp. 578-619.
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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 % A.M. Ostrowski, On the spectrum of a one-parametric family of
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26 % matrices, J. Reine Angew. Math., 193 (3/4), 1954, pp. 143-160.
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27 % J.H. Wilkinson, Singular-value decomposition---basic aspects,
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28 % in D.A.H. Jacobs, ed., Numerical Software---Needs and Availability,
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29 % Academic Press, London, 1978, pp. 109-135.
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30
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31 m = n(1); % Parameter n specifies dimension: m-by-n.
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32 n = n(max(size(n)));
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33
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34 if nargin < 3, k = n-1; end
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35 if nargin < 2, alpha = -1; end
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36
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37 if max(size(alpha)) ~= 1
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38 error('Second argument must be a scalar.')
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39 end
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40
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41 t = tril( eye(m,n) + alpha*triu(ones(m,n), 1), k);