Inverse of the Hilbert matrix.
H = invhilb(n)
The exact inverse of the exact Hilbert matrix is a matrix whose elements are large integers. These integers can be represented as floatingpoint numbers without roundoff error as long as the order of the matrix, n, is less than 12 or 13. invhilb(n) generates this matrix. For larger n, invhilb(n) generates an approximation to the inverse Hilbert matrix.
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
hilb
[1] G. E. Forsythe and C. B. Moler, Computer Solution of Linear Algebraic Systems, Chapter 19, Prentice-Hall, 1967.
(c) Copyright 1994 by The MathWorks, Inc.