Description

`Frank' is a Hessenberg, matrix with ill conditioned eigenvalues, whose determinant is 1. The elements may be optionally reflected about the anti-diagonal.

The matrix has all positive eigenvalues and they occur in reciprocal pairs (so that 1 is an eigenvalue if the order is odd). The eigenvalues may be obtained in terms of the zeros of the Hermite polynomials. The smallest half of the eigenvalues are ill conditioned, the more so for larger order.

This version incorporates improvements suggested by W. Kahan.

References

  1. This generator is adapted from Nicholas J. Higham's Test Matrix Toolbox.
  2. 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).
  3. 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).
  4. H. Rutishauser, On test matrices, Programmation en Mathematiques Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165, 1966, pp. 349-365. Section 9.
  5. J.H. Wilkinson, Error analysis of floating-point computation, Numer. Math., 2 (1960), pp. 319-340 (Section 8).
  6. J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press, 1965 (pp. 92-93).
  7. 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.
  8. J.M. Varah, A generalization of the Frank matrix, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 835-839.

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