Computing the Greatest Common Divisor of Polynomials Using the Comrade Matrix
- 1.1k Downloads
The comrade matrix of a polynomial is an analogue of the companion matrix when the matrix is expressed in terms of a general basis such that the basis is a set of orthogonal polynomials satisfying the three-term recurrence relation. We present the algorithms for computing the comrade matrix, and the coefficient matrix of the corresponding linear systems derived from the recurrence relation. The computing times of these algorithms are analyzed. The computing time bounds, which dominate these times, are obtained as functions of the degree and length of the integers that represent the rational number coefficients of the input polynomials. The ultimate aim is to apply these computing time bounds in the analysis of the performance of the generalized polynomial greatest common divisor algorithms.
Keywordscomrade matrix orthogonal polynomials three-term recurrence relation greatest common divisor of generalized polynomials
Unable to display preview. Download preview PDF.
- 1.Rahman, A.A., Aris, N.: The State of the art in Exact polynomial GCD computations. In: Proceedings Malaysian Science and Technology Conference (2002)Google Scholar
- 2.Rahman, A.A.: The use of GCD computation to remove repeated zeroes from a floating point polynomial. In: Proceedings SCAN 1992, Oldedenberg, Germany (1992)Google Scholar
- 5.Barnett, S.: Polynomial and Linear Control Systems. Marcel Dekker, New York (1983)Google Scholar
- 7.Brown, W.S.: On Euclid’s algorithm and polynomial greatest common divisors 18(1), 478–504 (1971)Google Scholar
- 9.Collins, G.E., Mignotte, M., Winkler, F.: Arithmetic in basic algebraic domains: Computing Suppl. 4, 189–220 (1982)Google Scholar
- 10.Labahn, G., Cheng, H.: On computing polynomial GCDs in alternate bases. In: Proceedings ISSAC 2006, pp. 47–54 (2006)Google Scholar
- 13.Aris, N., Rahman, A.A.: On the division of generalized polynomials. Lecture Notes Series On Computing, vol. 10, pp. 40–51. World Scientific Computing, Singapore (2003)Google Scholar
- 14.Rubald, C.M.: Algorithms for Polynomials over a Real Algebraic Number Field. University of Wisconsin: Ph.D. Thesis (1973)Google Scholar