The Minimum Eigenvalue of a Symmetric Positive De nite Toeplitz Matrix and Rational Hermitian Interpolation

A novel method for computing the minimal eigenvalue of a symmetric positive deenite Toeplitz matrix is presented. Similarly to the algorithm of Cybenko and Van Loan it is a combination of bisection and a root nding method. Both phases of the method are accelerated considerably by rational Hermite interpolation of the secular equation. For randomly generated test problems of dimension 800 the average number of linear systems which have to be solved to determine the smallest eigenvalue is 6.6 which reduces the computational cost of the method of Cybenko and Van Loan to approximately 35%. The method includes a rigorous error bound.