Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Rahman, A.A., Aris, N.: The State of the art in Exact polynomial GCD computations. In: Proceedings Malaysian Science and Technology Conference (2002)
Rahman, A.A.: The use of GCD computation to remove repeated zeroes from a floating point polynomial. In: Proceedings SCAN 1992, Oldedenberg, Germany (1992)
Barnett, S.: A companion matrix analogue for orthogonal polynomials. Linear Algebra and its Applications 12, 197–208 (1975)
Barnett, S., Maroulas, J.: Greatest common divisor of generalized polynomial and polynomial matrices. Linear Algebra and its Applications 22, 195–210 (1978)
Barnett, S.: Polynomial and Linear Control Systems. Marcel Dekker, New York (1983)
Barnett, S.: Division of generalized polynomials using the comrade matrix. Linear Algebra and its Applications 60, 159–175 (1984)
Brown, W.S.: On Euclid’s algorithm and polynomial greatest common divisors  18(1), 478–504 (1971)
Collins, G.E.: The computing time of the Euclidean algorithm. SIAM Journal on Computing 3(1), 1–10 (1974)
Collins, G.E., Mignotte, M., Winkler, F.: Arithmetic in basic algebraic domains: Computing Suppl.  4, 189–220 (1982)
Labahn, G., Cheng, H.: On computing polynomial GCDs in alternate bases. In: Proceedings ISSAC 2006, pp. 47–54 (2006)
McClellan, M.T.: The exact solution of systems of linear equations with polynomial coefficients. J. ACM 20(4), 563–588 (1973)
McClellan, M.T.: A comparison of algorithms for the exact solutions of linear equations. ACM Trans. Math. Software 3(2), 147–158 (1977)
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)
Rubald, C.M.: Algorithms for Polynomials over a Real Algebraic Number Field. University of Wisconsin: Ph.D. Thesis (1973)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Aris, N., Nahar Ahmad, S. (2008). Computing the Greatest Common Divisor of Polynomials Using the Comrade Matrix. In: Kapur, D. (eds) Computer Mathematics. ASCM 2007. Lecture Notes in Computer Science(), vol 5081. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87827-8_7
Download citation
DOI: https://doi.org/10.1007/978-3-540-87827-8_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87826-1
Online ISBN: 978-3-540-87827-8
eBook Packages: Computer ScienceComputer Science (R0)