Analysis and comparison of numerical algorithms for finding the GCD of certain types of polynomials in the Chebyshev basis

Abstract

This research investigates on the numerical methods for computing the greatest common divisors (GCD) of two polynomials in the orthogonal basis without having to convert to the power series form. Previous implementations were conducted using the Gauss Elimination with partial pivoting (GEPP) and QR Householder algorithms, respectively. This work proceeds to seek for a better approximate solution by comparing the results of the implementations with the QR with column pivoting (QRCP) algorithm. The results reveal that QRCP is as competent as the GEPP algorithm, up to a certain degree, giving a reasonably good approximate solution. It is also found that normalizing the columns of the associated coefficient matrix slightly reduces the condition number of the matrix but has no significant effect on the GCD solutions when applying the GEPP and QR Householder algorithms. However equilibration of the columns by computing its ∞-norm is capable to improve the solution when QRCP is applied. Comparing the three algorithms on some test problems, QR Householder outperforms the rest and is able to give a good approximate solution in the worst case condition when the smallest element of the matrix is 1, the entries ranging up to 15 digits integers.

The Mechanization of the Comrade Matrix Approach in Determining the GCD of Orthogonal Polynomials

“…On the other hand, in the floating point environment the coefficient matrix can be ill conditioned. Solving the problem numerically leads to new challenges not only for the purpose of solving the GCD problem but also to investigate on the extensiveness of applying appropriate numerical methods for solving ill-conditionedsystems of linear equations, see in [7,8].…”