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annotate toolbox/rschur.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 A = rschur(n, mu, x, y)
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2 %RSCHUR An upper quasi-triangular matrix.
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3 % A = RSCHUR(N, MU, X, Y) is an N-by-N matrix in real Schur form.
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4 % All the diagonal blocks are 2-by-2 (except for the last one, if N
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5 % is odd) and the k'th has the form [x(k) y(k); -y(k) x(k)].
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6 % Thus the eigenvalues of A are x(k) +/- i*y(k).
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7 % MU (default 1) controls the departure from normality.
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8 % Defaults: X(k) = -k^2/10, Y(k) = -k, i.e., the eigenvalues
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9 % lie on the parabola x = -y^2/10.
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10
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11 % References:
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12 % F. Chatelin, Eigenvalues of Matrices, John Wiley, Chichester, 1993;
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13 % Section 4.2.7.
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14 % F. Chatelin and V. Fraysse, Qualitative computing: Elements
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15 % of a theory for finite precision computation, Lecture notes,
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16 % CERFACS, Toulouse, France and THOMSON-CSF, Orsay, France,
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17 % June 1993.
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18
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19 m = floor(n/2)+1;
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20 alpha = 10; beta = 1;
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21
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22 if nargin < 4, y = -(1:m)/beta; end
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23 if nargin < 3, x = -(1:m).^2/alpha; end
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24 if nargin < 2, mu = 1; end
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25
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26 A = diag( mu*ones(n-1,1), 1 );
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27 for i=1:2:2*(m-1)
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28 j = (i+1)/2;
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29 A(i:i+1,i:i+1) = [x(j) y(j); -y(j) x(j)];
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30 end
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31 if 2*m ~= n,
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32 A(n,n) = x(m);
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33 end