Efficient Algorithms and Implementation for Error-Free Computation Using P-adic

Abstract

Our research team including graduate students both in Computer Science and Mathematics has been developing P-adic Exact Scientific Computational Library (ESCL) for rational matrix operations. The effort has been focusing on converting all rational number operations to integer calculation, and fully taking the advantage of fast integer multiplication of modern computer architectures [1]. By properly selecting prime numbers as the bases and practically choosing the length r for P-adic expansion of rational numbers, we have shown some promising results for large matrix operations, such as matrix multiplication and inverse. Dixon algorithm [2] and his improved version, we call it GeneralizedDixon (or Dixon-Krishnamurthy) algorithm, which combines Dixon algorithm with Krishnamurthy algorithm [3], has been developed, which can be extended to various matrix computations.

A method for Hensel code overflow detection

“…The error rate is 0. We have developed a software package [10,15] to do matrix calculation by D-K algorithm and the Hensel code overflow detection method. This software has been well tested on matrix calculations so far, and the performance is good.…”

“…One of them is the detection of overflow and underflow [3]. Hensel code can be used to develop a new word-based computer data structure [10]. But for Hensel code, if the word length is not sufficient for the rational number it represents, the transformed rational number will be meaningless.…”

Parallel Implementation of Exact Matrix Computation Using Multiple P-adic Arithmetic

“…According to the theory, for a fixed value, the larger you 0.000% choose, the larger the bound that will result. But for computer architectures with 32 bit or 64 bit CPUs, when using the existing integer classes, the largest should be chosen with respect to 46337 or 2147483647 to assure overflow protection [4]. This means that for a 32bit CPU architecture, , while for a 64-bit CPU architecture, .…”

“…For the past several years, we have been developing Padic Exact Scientific Computational Library (ESCL) for rational matrix operations. Based on Krishnamurthy [1,2] and Dixon [3] theories, we have established a finite P-adic sequence calculation system [4][5][6][7]. But there is a problem that for certain complex matrix operations, even with small matrix sizes, the new method requires a long P-adic sequence to guarantee against overflow [6].…”