A method for Hensel code overflow detection

Abstract

Hensel code was originally defined by Krishnamurthy, Rao and Subramanian [1], which was developed from the P-adic number system first proposed by Hensel in 1900s. The purpose was to realize exact computation for rational numbers. The Hensel code arithmetic has been well developed [1, 2, 3], but the problem of detecting Hensel code overflow and underflow has not been properly addressed [3]. In this paper, we proposed a method for Hensel code overflow detection. The method can realize overflow detection by the prime p and the Hensel code itself. Using this method, a few digits of the Hensel code will be sacrificed.

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

“…One of the important issues of finite P-adic arithmetic is to choose the P-adic sequence length If the initial is not long enough, Hensel code overflow will happen [6].…”

“…One way to detect the overflow is to predict the bound, then decide the size of the prime set by Newman [12]. Another way to detect overflow is to provide some extra digits [6]. This method can detect the overflow by using the prime set and the residue number set itself.…”

“…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].…”

“…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]. The longer the P-adic sequences are, the longer the calculation will take, and computational efficiency becomes an issue.…”