Finite field computational techniques for exact solution of numerical problems

Abstract

In this dissertation, we consider the application of residue arithmetic for the exact computation of g-inverses in order to obviate the round-off errors normally associated with their computation. It turns out that for using residue arithmetic for these problems, we face the difficulty of choosing a very large prime which often goes beyond the range of integers representable in a digital computer. This difficulty can be obviated by choosing a set of reasonably small primes and computing the required result with respect to each prime; finally, these results are combined using the Chinese Remainder Theorem or other alternative methods. It is also possible to completely eliminate this multiple prime base procedure if we use the p-adic number representation. In the p-adic arithmetic system, we have the advantage of choosing a smaller prime ppp and a flexibility in the choice of the number of digits rrr depending on the problem. The application of p-adic number systems to exact computation is also dealt with in detail in this dissertation.