The Minimum Eigenvalue of a Symmetric Positive-Definite Toeplitz Matrix and Rational Hermitian Interpolation
Wolfgang Mackens, Heinrich Voß · SIAM Journal on Matrix Analysis and Applications · 1997
A novel method for computing the minimal eigenvalue of a symmetric positive-definite Toeplitz matrix is presented. Similar to the algorithm of Cybenko and Van Loan, it is a combination of bisection and a root finding method. Both phases of the method are accelerated considerably by rational Hermitian interpolation of the secular equation. For randomly generated test problems of dimension 800 the average number of linear systems which must 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.