Biorthogonal polynomials and the bordering method for linear systems

Abstract

First, the problem of solving a system of linear equations is shown to be equivalent to the computation of biorthogonal polynomials. The bordering method is a procedure for solving recursively a sequence of linear systems with increasing dimensions and it gave rise to a recurrence relationship between two adjacent families of biorthogonal polynomials. Of course, one relation is not sufficient for computing two families. However, in some particular cases, a second recurrence relationship exists between these biorthogonal polynomials thus leading to procedures for solving recursively such linear systems with increasing dimensions. The cases of Hankel and Toeplitz matrices are treated in details.