Journal of Symbolic Computation - Special issue on computational algebraic complexity
Algorithms for computer algebra
Algorithms for computer algebra
Algorithmic algebra
On computing greatest common divisors with polynomials given by black boxes for their evaluations
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Modern computer algebra
Subresultants and Reduced Polynomial Remainder Sequences
Journal of the ACM (JACM)
On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors
Journal of the ACM (JACM)
On Euclid's Algorithm and the Theory of Subresultants
Journal of the ACM (JACM)
Polynomials and Linear Control Systems
Polynomials and Linear Control Systems
Fraction-Free Computation of Matrix Rational Interpolants and Matrix GCDs
SIAM Journal on Matrix Analysis and Applications
Journal of Symbolic Computation
Exact solution of linear equations
SYMSAC '71 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation
Algorithms for normal forms for matrices of polynomials and ore polynomials
Algorithms for normal forms for matrices of polynomials and ore polynomials
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Output-sensitive modular algorithms for polynomial matrix normal forms
Journal of Symbolic Computation
Computing the Greatest Common Divisor of Polynomials Using the Comrade Matrix
Computer Mathematics
Computing polynomial LCM and GCD in lagrange basis
ACM Communications in Computer Algebra
Multivariate resultants in Bernstein basis
ADG'08 Proceedings of the 7th international conference on Automated deduction in geometry
Ruppert matrix as subresultant mapping
CASC'07 Proceedings of the 10th international conference on Computer Algebra in Scientific Computing
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In this paper, we examine the problem of computing the greatest common divisor (GCD) of univariate polynomials represented in different bases. When the polynomials are represented in Newton basis or a basis of orthogonal polynomials, we show that the well-known Sylvester matrix can be generalized. We give fraction-free and modular algorithms to directly compute the GCD in the alternate basis. These algorithms are suitable for computation in domains where growth of coefficients in intermediate computations are a central concern. In the cases of Newton basis and bases using certain orthogonal polynomials, we also show that the standard subresultant algorithm can be applied easily. If the degrees of the input polynomials is at most n and the degree of the GCD is at least n/2, our algorithms outperform the corresponding algorithms using the standard power basis.