On Cancellation Error in Newton's Method for Power Series Roots of Multivariate Polynomial (original) (raw)

Abstract

Let F(x,u1,...,u') be a square-free polynomial which is monic w.r.t. x and let (s1,...,s') 2 C ' . If F(x,s1,...,s') is square-free then the roots of F w.r.t. x can be expanded into integral power series in u1 ¡ s1,...,u' ¡ s', and the power series roots can be calculated by well-known Newton's method. We use the floating-point number arithmetic to calculate the numerical coecients, and this paper investigates the numerical errors contained in the roots computed. We first express the power series root in terms of sub-polynomials of F(x,s1+v1,...,s'+v'), with (v1,...,v') a set of new variables, and clarify various properties of the power series roots. Then, we investigate the cancellation errors when the expansion point (s1,...,s') is close to a "singular point" and far from the origin, respectively. We find that, although the cancellation errors are not large usually, they become extremely large in some cases. In fact,...

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References (8)

  1. K.O. Geddes, S.R. Czapor and G. Labahn: Algorithms for Computer Algebra, Ch. 6, Kluwer Academic, Boston-Dordrecht-London, 1992.
  2. S.Y. Ku and R.J. Adler: Computing polynomial resultants: Bezout's determinant vs. Collins' reduced P.R.S. algorithm, Communi. ACM, 12 (1969), 23-30.
  3. H.T. Kung and J.F. Traub: All algebraic functions can be computed fast, J. ACM 25 (1978), 245-260.
  4. T. Sasaki and F. Kako: Solving multivariate algebraic equation by Hensel con- struction, Japan J. Indus. Appl. Math., 16 (1999), 257-285. (This paper was sub- mitted in March, 1993, and the authors received a referees' reports in September, 1996 and the letter of acceptance in June, 1998.)
  5. T. Sasaki and M. Sasaki: Analysis of accuracy decreasing in polynomial remainder sequence with floating-point number coefficients, J. Inf. Proces. 12 (1989), 394-403.
  6. T. Sasaki and T. Sato: Cancellation errors in multivariate resultant computation, ACM SIGSAM Bulletin, Vol. 22, pp. 13-20 (1998).
  7. T. Sasaki and S. Yamaguchi: An analysis of cancellation error in multivariate Hensel construction with floating-point number arithmetic, Proc. of ISSAC'98, pp. 1-8, ACM Press (1998).
  8. R. J. Walker: Algebraic curves, Ch. 4, Springer-Verlag, New York-Heidelberg- Berlin, 1978.