Solution of ax + by = d for generalized polynomials

Abstract

A simple matrix method is described for solving the equation ax + by = d, where a and b are given, and x, y and the greatest common divisor (g.c.d.) d are to be determined. All the polynomials are expressed relative to an arbitrary given basis of orthogonal polynomials, and there are no conversions to power form. The procedure utilizes the comrade matrix, a generalization of the companion matrix, and involves elementary row operations performed on easily constructed matrices.