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Computing polynomial resultants: Bezout's determinant vs. Collins' reduced P.R.S. algorithm

Computing polynomial resultants: Bezout's determinant vs. Collins' reduced P.R.S. algorithm Algorithms for computing the resultant of two polynomials in several variables, a key repetitive step of computation in solving systems of polynomial equations by elimination, are studied. Determining the best algorithm for computer implementation depends upon the extent to which extraneous factors are introduced, the extent of propagation of errors caused by truncation of real coeffcients, memory requirements, and computing speed. Preliminary considerations narrow the choice of the best algorithm to Bezout's determinant and Collins' reduced polynomial remainder sequence (p.r.s.) algorithm. Detailed tests performed on sample problems conclusively show that Bezout's determinant is superior in all respects except for univariate polynomials, in which case Collins' reduced p.r.s. algorithm is somewhat faster. In particular Bezout's determinant proves to be strikingly superior in numerical accuracy, displaying excellent stability with regard to round-off errors. Results of tests are reported in detail. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications of the ACM Association for Computing Machinery

Computing polynomial resultants: Bezout's determinant vs. Collins' reduced P.R.S. algorithm

Communications of the ACM , Volume 12 (1) – Jan 1, 1969

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References (27)

Publisher
Association for Computing Machinery
Copyright
Copyright © 1969 by ACM Inc.
ISSN
0001-0782
DOI
10.1145/362835.362839
Publisher site
See Article on Publisher Site

Abstract

Algorithms for computing the resultant of two polynomials in several variables, a key repetitive step of computation in solving systems of polynomial equations by elimination, are studied. Determining the best algorithm for computer implementation depends upon the extent to which extraneous factors are introduced, the extent of propagation of errors caused by truncation of real coeffcients, memory requirements, and computing speed. Preliminary considerations narrow the choice of the best algorithm to Bezout's determinant and Collins' reduced polynomial remainder sequence (p.r.s.) algorithm. Detailed tests performed on sample problems conclusively show that Bezout's determinant is superior in all respects except for univariate polynomials, in which case Collins' reduced p.r.s. algorithm is somewhat faster. In particular Bezout's determinant proves to be strikingly superior in numerical accuracy, displaying excellent stability with regard to round-off errors. Results of tests are reported in detail.

Journal

Communications of the ACMAssociation for Computing Machinery

Published: Jan 1, 1969

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