The accurate inversion of vandermonde matrices

• Published in: Computers & mathematics with applications (1987) 2004-03, Vol.47 (6), p.921-929
• Main Author: Jog, C.S
• Format: Article
• Language: English
• Subjects: Accurate Vandermonde inverse
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/S0898-1221(04)90076-4

Two modifications are suggested in the commonly used algorithms (such as the O( n 2) Parker algorithm) for the explicit inversion of Vandermonde matrices resulting in an algorithm whose accuracy is no worse than those of the existing algorithms, but which is significantly more accurate in many pathological situations. The first modification circumvents, to some extent, the subtraction of ‘two big like-signed numbers’ which in turn reduces round-off errors, while the second modification exploits the structure of the inverse and uses two recursive formulae instead of one to bring about an increase in accuracy. Numerical results are presented to demonstrate the increase in accuracy that results from these two modifications. Although the modified algorithm is always at least as accurate as the Parker algorithm, it does, unfortunately, involve an increase in complexity from O( n 2) to O( n 3), so that use of this algorithm to increase the relative accuracy is recommended only in situations where the standard algorithms fail to yield accurate results.