A method of order 1 + √ 3 for computing the smallest eigenvalue of a symmetric Toeplitz matrix

Abstract: In this note we discuss a method of order 1 + 3 for computing the smallest eigenvalue λ1 of a symmetric and positive definite Toeplitz matrix. It generalizes and improves a method introduced in [7] which is based on rational Hermitean interpolation of the secular equation. Taking advantage of a further rational approximation of the secular equation which is essentially for free and which yields lower bounds of λ1 we obtain an improved stopping criterion