Inverse of Hilbert matrix

`H = invhilb(n)`

`H = invhilb(n)`

generates
the exact inverse of the exact Hilbert matrix for `n`

less
than about 15. For larger `n`

, `invhilb(n)`

generates
an approximation to the inverse Hilbert matrix.

The exact inverse of the exact Hilbert matrix is a matrix whose
elements are large integers. These integers may be represented as
floating-point numbers without roundoff error as long as the order
of the matrix, `n`

, is less than 15.

Comparing `invhilb(n)`

with `inv(hilb(n))`

involves
the effects of two or three sets of roundoff errors:

The errors caused by representing

`hilb(n)`

The errors in the matrix inversion process

The errors, if any, in representing

`invhilb(n)`

It turns out that the first of these, which involves representing fractions like 1/3 and 1/5 in floating-point, is the most significant.

`invhilb(4)`

is

16 -120 240 -140 -120 1200 -2700 1680 240 -2700 6480 -4200 -140 1680 -4200 2800

[1] Forsythe, G. E. and C. B. Moler, *Computer Solution
of Linear Algebraic Systems*, Prentice-Hall, 1967, Chapter
19.

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