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annotate toolbox/clement.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 A = clement(n, k)
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2 %CLEMENT Clement matrix - tridiagonal with zero diagonal entries.
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3 % CLEMENT(N, K) is a tridiagonal matrix with zero diagonal entries
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4 % and known eigenvalues. It is singular if N is odd. About 64
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5 % percent of the entries of the inverse are zero. The eigenvalues
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6 % are plus and minus the numbers N-1, N-3, N-5, ..., (1 or 0).
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7 % For K = 0 (the default) the matrix is unsymmetric, while for
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
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8 % K = 1 it is symmetric.
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9 % CLEMENT(N, 1) is diagonally similar to CLEMENT(N).
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11 % Similar properties hold for TRIDIAG(X,Y,Z) where Y = ZEROS(N,1).
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12 % The eigenvalues still come in plus/minus pairs but they are not
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13 % known explicitly.
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14 %
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15 % References:
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16 % P.A. Clement, A class of triple-diagonal matrices for test
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17 % purposes, SIAM Review, 1 (1959), pp. 50-52.
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18 % A. Edelman and E. Kostlan, The road from Kac's matrix to Kac's
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19 % random polynomials. In John~G. Lewis, editor, Proceedings of
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20 % the Fifth SIAM Conference on Applied Linear Algebra Society
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21 % for Industrial and Applied Mathematics, Philadelphia, 1994,
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22 % pp. 503-507.
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23 % O. Taussky and J. Todd, Another look at a matrix of Mark Kac,
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24 % Linear Algebra and Appl., 150 (1991), pp. 341-360.
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26 if nargin == 1, k = 0; end
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27
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28 n = n-1;
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30 x = n:-1:1;
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31 z = 1:n;
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32
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33 if k == 0
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34 A = diag(x, -1) + diag(z, 1);
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35 else
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36 y = sqrt(x.*z);
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37 A = diag(y, -1) + diag(y, 1);
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38 end