Tridiagonal Companion Matrices and their use for Computing Orthogonal and Nonorthogonal Polynomial Zeros

In this paper, we show that poor conditioning can arise in trivial polynomials, making it difficult to compute zeros with standard companion matrix approaches. Furthermore, using these methods, we show that although the conditioning of an orthogonal polynomial sequence may degrade gracefully, its numerically-computed zeros generally do not. Hence we introduce a tridiagonal form for the companion matrices of orthogonal polynomials whose eigenvalues can be computed more accurately than the standard form. We then introduce a method of obtaining tridiagonal companion matrices for arbitrary polynomials in an attempt to exploit this advantage in the general case.