A symmetric, positive definite, totally positive, Hankel matrix with elements Ai,j= 1/(i+j-1). It is a famous example of a badly conditioned matrix.


The condition number grows like e3.5n, for order n.

The exact inverse has (large!) integer entries. The program can compute the inverse using exact integer arithmetic through order=13. Past that point double precision approximation is used. The inverse can only be computed through order=200 due to overflow.


  1. This generator is adapted from Nicholas J. Higham's Test Matrix Toolbox.
  2. M.-D. Choi, Tricks or treats with the Hilbert matrix, Amer. Math. Monthly, 90 (1983), pp. 301-312.
  3. N.J. Higham, Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1996; sec. 26.1.
  4. M. Newman and J. Todd, The evaluation of matrix inversion programs, J. Soc. Indust. Appl. Math., 6 (1958), pp. 466-476.
  5. D.E. Knuth, The Art of Computer Programming, Volume 1, Fundamental Algorithms, second edition, Addison-Wesley, Reading, Massachusetts, 1973, p. 37.

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