Estimation of Cancellation Errors in Multivariate Hensel Construction with Floating-Point Numbers

Abstract

Multivariate Hensel construction with floating-point numbers often cause large cancellation errors which are errors due to cancellation of almost the same numbers. Sasaki and Yamaguchi [SY98] showed that multivariate Hensel construction causes large cancellation errors if the expansion point is chosen near a singular point, and Sasaki [Sas00] studied four echanisms of term cancellations near a singular point. However, an analysis of cancellation errors, if the expansion point is chosen randomly, has never been studied. Moreover, in practical computations, it is better that we employ the lowest working precision which is almost enough to that a result of the Hensel construction includes required significant digits, since a higher precision takes more computation times. In this paper, we investigate cancellation errors without an assumption that the expansion point is chosen near a singular point, and estimate a precision which is almost enough. [SY98] : T. Sasaki and S. Yamaguchi, An Analysis of Cancellation Error in Multivariate Hensel Construction with Floting-point Number Arithmetic, ISSAC'98. [Sas00] : T. Sasaki, Mechanism of cancellation errors in multivariate Hensel construction with floting-point numbers, Preprint of Univ. Tsukuba.