Mercurial > matrix-functions
comparison toolbox/frank.m @ 2:c124219d7bfa draft
Re-add the 1995 toolbox after noticing the statement in the ~higham/mctoolbox/ webpage.
author | Antonio Pino Robles <data.script93@gmail.com> |
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date | Thu, 07 May 2015 18:36:24 +0200 |
parents | 8f23314345f4 |
children |
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1:e471a92d17be | 2:c124219d7bfa |
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1 function F = frank(n, k) | |
2 %FRANK Frank matrix---ill conditioned eigenvalues. | |
3 % F = FRANK(N, K) is the Frank matrix of order N. It is upper | |
4 % Hessenberg with determinant 1. K = 0 is the default; if K = 1 the | |
5 % elements are reflected about the anti-diagonal (1,N)--(N,1). | |
6 % F has all positive eigenvalues and they occur in reciprocal pairs | |
7 % (so that 1 is an eigenvalue if N is odd). | |
8 % The eigenvalues of F may be obtained in terms of the zeros of the | |
9 % Hermite polynomials. | |
10 % The FLOOR(N/2) smallest eigenvalues of F are ill conditioned, | |
11 % the more so for bigger N. | |
12 | |
13 % DET(FRANK(N)') comes out far from 1 for large N---see Frank (1958) | |
14 % and Wilkinson (1960) for discussions. | |
15 % | |
16 % This version incorporates improvements suggested by W. Kahan. | |
17 % | |
18 % References: | |
19 % W.L. Frank, Computing eigenvalues of complex matrices by determinant | |
20 % evaluation and by methods of Danilewski and Wielandt, J. Soc. | |
21 % Indust. Appl. Math., 6 (1958), pp. 378-392 (see pp. 385, 388). | |
22 % G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the | |
23 % computation of the Jordan canonical form, SIAM Review, 18 (1976), | |
24 % pp. 578-619 (Section 13). | |
25 % H. Rutishauser, On test matrices, Programmation en Mathematiques | |
26 % Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165, | |
27 % 1966, pp. 349-365. Section 9. | |
28 % J.H. Wilkinson, Error analysis of floating-point computation, | |
29 % Numer. Math., 2 (1960), pp. 319-340 (Section 8). | |
30 % J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University | |
31 % Press, 1965 (pp. 92-93). | |
32 % The next two references give details of the eigensystem, as does | |
33 % Rutishauser (see above). | |
34 % P.J. Eberlein, A note on the matrices denoted by B_n, SIAM J. Appl. | |
35 % Math., 20 (1971), pp. 87-92. | |
36 % J.M. Varah, A generalization of the Frank matrix, SIAM J. Sci. Stat. | |
37 % Comput., 7 (1986), pp. 835-839. | |
38 | |
39 if nargin == 1, k = 0; end | |
40 | |
41 p = n:-1:1; | |
42 F = triu( p( ones(n,1), :) - diag( ones(n-1,1), -1), -1 ); | |
43 if k ~= 0 | |
44 F = F(p,p)'; | |
45 end |