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annotate toolbox/frank.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 F = frank(n, k)
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2 %FRANK Frank matrix---ill conditioned eigenvalues.
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Antonio Pino Robles <data.script93@gmail.com>
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3 % F = FRANK(N, K) is the Frank matrix of order N. It is upper
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Antonio Pino Robles <data.script93@gmail.com>
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4 % Hessenberg with determinant 1. K = 0 is the default; if K = 1 the
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Antonio Pino Robles <data.script93@gmail.com>
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5 % elements are reflected about the anti-diagonal (1,N)--(N,1).
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Antonio Pino Robles <data.script93@gmail.com>
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6 % F has all positive eigenvalues and they occur in reciprocal pairs
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Antonio Pino Robles <data.script93@gmail.com>
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7 % (so that 1 is an eigenvalue if N is odd).
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Antonio Pino Robles <data.script93@gmail.com>
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8 % The eigenvalues of F may be obtained in terms of the zeros of the
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Antonio Pino Robles <data.script93@gmail.com>
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9 % Hermite polynomials.
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Antonio Pino Robles <data.script93@gmail.com>
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10 % The FLOOR(N/2) smallest eigenvalues of F are ill conditioned,
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Antonio Pino Robles <data.script93@gmail.com>
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11 % the more so for bigger N.
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12
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13 % DET(FRANK(N)') comes out far from 1 for large N---see Frank (1958)
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14 % and Wilkinson (1960) for discussions.
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15 %
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Antonio Pino Robles <data.script93@gmail.com>
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16 % This version incorporates improvements suggested by W. Kahan.
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17 %
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18 % References:
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19 % W.L. Frank, Computing eigenvalues of complex matrices by determinant
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Antonio Pino Robles <data.script93@gmail.com>
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20 % evaluation and by methods of Danilewski and Wielandt, J. Soc.
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21 % Indust. Appl. Math., 6 (1958), pp. 378-392 (see pp. 385, 388).
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22 % G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the
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23 % computation of the Jordan canonical form, SIAM Review, 18 (1976),
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24 % pp. 578-619 (Section 13).
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25 % H. Rutishauser, On test matrices, Programmation en Mathematiques
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26 % Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165,
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27 % 1966, pp. 349-365. Section 9.
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28 % J.H. Wilkinson, Error analysis of floating-point computation,
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29 % Numer. Math., 2 (1960), pp. 319-340 (Section 8).
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30 % J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University
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31 % Press, 1965 (pp. 92-93).
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32 % The next two references give details of the eigensystem, as does
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33 % Rutishauser (see above).
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34 % P.J. Eberlein, A note on the matrices denoted by B_n, SIAM J. Appl.
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35 % Math., 20 (1971), pp. 87-92.
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36 % J.M. Varah, A generalization of the Frank matrix, SIAM J. Sci. Stat.
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37 % Comput., 7 (1986), pp. 835-839.
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38
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39 if nargin == 1, k = 0; end
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40
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41 p = n:-1:1;
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42 F = triu( p( ones(n,1), :) - diag( ones(n-1,1), -1), -1 );
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43 if k ~= 0
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44 F = F(p,p)';
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45 end