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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.

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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.

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