A symmetric, positive definite,
totally positive, Hankel matrix with
elements A = 1/(i+j-1).
It is a famous example of a badly conditioned matrix.
The condition number grows like
e , 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.
This generator is adapted from Nicholas J. Higham's
Test Matrix Toolbox.
M.-D. Choi, Tricks or treats with the Hilbert matrix, Amer. Math.
Monthly, 90 (1983), pp. 301-312.
N.J. Higham, Accuracy and Stability of Numerical Algorithms,
Society for Industrial and Applied Mathematics, Philadelphia, PA,
USA, 1996; sec. 26.1.
M. Newman and J. Todd, The evaluation of matrix inversion
programs, J. Soc. Indust. Appl. Math., 6 (1958), pp. 466-476.
D.E. Knuth, The Art of Computer Programming,
Volume 1, Fundamental Algorithms, second edition, Addison-Wesley,
Reading, Massachusetts, 1973, p. 37.
The Matrix Market is a service of the
Mathematical and Computational Sciences Division /
Information Technology Laboratory /
National Institute of Standards and Technology
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