--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/toolbox/frank.m	Thu May 07 18:36:24 2015 +0200
@@ -0,0 +1,45 @@
+function F = frank(n, k)
+%FRANK   Frank matrix---ill conditioned eigenvalues.
+%        F = FRANK(N, K) is the Frank matrix of order N.  It is upper
+%        Hessenberg with determinant 1.  K = 0 is the default; if K = 1 the
+%        elements are reflected about the anti-diagonal (1,N)--(N,1).
+%        F has all positive eigenvalues and they occur in reciprocal pairs
+%        (so that 1 is an eigenvalue if N is odd).
+%        The eigenvalues of F may be obtained in terms of the zeros of the
+%        Hermite polynomials.
+%        The FLOOR(N/2) smallest eigenvalues of F are ill conditioned,
+%        the more so for bigger N.
+
+%        DET(FRANK(N)') comes out far from 1 for large N---see Frank (1958)
+%        and Wilkinson (1960) for discussions.
+%
+%        This version incorporates improvements suggested by W. Kahan.
+%
+%        References:
+%        W.L. Frank, Computing eigenvalues of complex matrices by determinant
+%           evaluation and by methods of Danilewski and Wielandt, J. Soc.
+%           Indust. Appl. Math., 6 (1958), pp. 378-392 (see pp. 385, 388).
+%        G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the
+%           computation of the Jordan canonical form, SIAM Review, 18 (1976),
+%             pp. 578-619 (Section 13).
+%        H. Rutishauser, On test matrices, Programmation en Mathematiques
+%           Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165,
+%           1966, pp. 349-365.  Section 9.
+%        J.H. Wilkinson, Error analysis of floating-point computation,
+%           Numer. Math., 2 (1960), pp. 319-340 (Section 8).
+%        J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University
+%           Press, 1965 (pp. 92-93).
+%        The next two references give details of the eigensystem, as does
+%        Rutishauser (see above).
+%        P.J. Eberlein, A note on the matrices denoted by B_n, SIAM J. Appl.
+%           Math., 20 (1971), pp. 87-92.
+%        J.M. Varah, A generalization of the Frank matrix, SIAM J. Sci. Stat.
+%           Comput., 7 (1986), pp. 835-839.
+
+if nargin == 1, k = 0; end
+
+p = n:-1:1;
+F = triu( p( ones(n,1), :) - diag( ones(n-1,1), -1), -1 );
+if k ~= 0
+   F = F(p,p)';
+end