Abstract
The ISZ method (Interval-Symbol method with Zero rewriting) based on stabilization theory was proposed to reduce the amount of exact computations as much as possible but obtain the exact results by aid of floating-point computations. In this paper, we applied the ISZ method to Trager's algorithm which factors univariate polynomials over algebraic number fields. By Maple experiments, we show the efficiency of the ISZ method over the purely exact approach which uses exact computations throughout the execution of the algorithm. Furthermore, we propose a new method called the ISZ* method, which is similar to the ISZ method but beforehand excludes insufficient precisions of floating-point approximation by checking the correctness of the obtained supports. We confirmed that the ISZ* method is more effective than the ISZ method when the initially set precision is not sufficiently high.
- G. Alefeld, J. Herzberger : Introduction to Interval Computation, Academic Press, 1983.Google Scholar
- K. O. Geddes, S. R. Czapor, G. Labahn : Algorithms for Computer Algebra, 1992.Google ScholarCross Ref
- E. Kaltofen : Factorization of polynomials. In Computer algebra, Springer, Vienna, 1983, pp.95--113.Google ScholarCross Ref
- H. Nagashima, K. Shirayanagi : Effect of the Interval-Symbol Method with Correct Zero Rewriting on the Δ-LLL Algorithm, ACM Communications in Computer Algebra, Vol. 52, No. 2, Issue 204, June 2018, pp.24--31.Google ScholarDigital Library
- S. Schirra, M. Wilhelm : On Interval Methods with Zero Rewriting and Exact Geometric Computation, LNCS 10693, 2017, pp.211--226.Google Scholar
- K. Shirayanagi, H.Sekigawa : A New Groebner Basis Conversion Method Based on Stabilization Techniques, Theoretical Computer Science, Vol. 409, 2008, pp.311--317.Google ScholarDigital Library
- K. Shirayanagi, H. Sekigawa : Reducing Exact Computations to Obtain Exact Results Based on Stabilization Techniques. In Proc. International Workshop on Symbolic-Numeric Computation 2009 (SNC2009), 2009, pp.191--197.Google ScholarDigital Library
- K. Shirayanagi, M. Sweedler : A Theory of Stabilizing Algebraic Algorithms, Technical Report 95-28, Mathematical Sciences Institute, Cornell University, 1995, 92 pages.Google Scholar
- K. Shirayanagi, M. Sweedler : Remarks on automatic algorithm stabilization. J. Symbolic Computation, 26(6), 1998, pp.761--766.Google ScholarDigital Library
- B. M. Trager : Algebraic Factoring and Rational Function Integration, Proc. SYMSAC, 1976, pp.219--226.Google ScholarDigital Library
- P. J. Weinberger, L. P. Rothchild : Factoring Polynomials over Algebraic Number Fields, ACM Trans, Math. Soft 2, 1976, pp.335--350.Google ScholarDigital Library
- H. Zassenhaus : On Hensel factorization. I. J. Number Theory 1, 1969, pp.291--311.Google ScholarCross Ref
Recommendations
Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields
ISSAC '06: Proceedings of the 2006 international symposium on Symbolic and algebraic computationWe present algorithms that compute all irreducible factors of degree ≤ d of supersparse (lacunary) multivariate polynomials in n variables over an algebraic number field in deterministic polynomial-time in (l+d)n, where l is the size of the input ...
Factoring multivariate polynomials over algebraic number fields in MACSYMA
The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ...
Number of zeros of interval polynomials
In this paper, we develop a rigorous algorithm for counting the real interval zeros of polynomials with perturbed coefficients that lie within a given interval, without computing the roots of any polynomials. The result generalizes Sturm's Theorem for ...
Comments