Abstract
In [2] a new method is presented for the computation of a greatest common divisor (gcd) of two polynomials, along with their polynomial remainder sequence (prs). This method is based on our generalization of a theorem by Van Vleck (1899)[12] and uniformly treats both normal and abnormal prs’s, making use of Bareiss’s (1968)[4] integer-preserving transformation algorithm for Gaussian elimination; moreover, for the polynomials of the prs’s, this method provides the smallest coefficients that can be expected without coefficient gcd computations. In this paper we present efficient, exact algorithms for the implementation of this new method, along with an example where bubble pivot is needed.
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Bibliography
Akritas, A.G.:A simple validity proof of the reduced prs algorithm. Computing 38, 369–372, 1987.
Akritas, A.G.:A new method for computing greatest common divisors and polynomial remainder sequences. Numerische Mathematik 52,119–127, 1988.
Akritas, A.G.:Elements of Computer Algebra with Applications. John Wiley, New York, in press.
Bareiss, E.H.:Sylvester’s identity and multistep integer-preserving Gaussian elimination. Mathematics of Computation 22,565–578, 1968.
Brown, W.S.:On Euclid’s algorithm and the computation of polynomial greatest common divisors. JACM 18, 476–504, 1971.
Brown, W.S.:The subresultant prs algorithm. ACM Transactions On Mathematical Software 4, 237–249, 1978.
Collins, G.E.:Subresultants and reduced polynomial remainder sequences. JACM 14, 128–142, 1967.
Habicht, W.:Eine Verallgemeinerung des Sturmschen Wurzelzählverfahrens. Commentarii Mathematici Helvetia 21, 99–116, 1948.
Laidacker, M.A.:Another theorem relating Sylvester’s matrix and the greatest common divisor. Mathematics Magazine 42,126–128,1969.
Loos, R.:Generalized polynomial remainder sequences. In: Computer Algebra Symbolic and Algebraic Computations. Ed. by B. Buchberger, G.E. Collins and R. Loos, Springer Verlag, Wien, New York, 1982, Computing Supplement 4,115–137.
Sylvester, J.J.:On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraical common measure. Philoshophical Transactions 143,407–548, 1853.
Van Vleck, E. B.:On the determination of a series of Sturm’s functions by the calculation of a single determinant. Annals of Mathematics, Second Series, Vol. 1,1–13, 1899–1900.
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© 1989 Springer-Verlag New York Inc.
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Akritas, A.G. (1989). Exact Algorithms for the Matrix-Triangularization Subresultant PRS Method. In: Kaltofen, E., Watt, S.M. (eds) Computers and Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9647-5_19
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DOI: https://doi.org/10.1007/978-1-4613-9647-5_19
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