Application of the Kato-Temple Inequality for Eigenvalues of Symmetric Matrices to Numerical Algorithms with Shift for Singular Values

Abstract

The Kato-Temple inequality for eigenvalues of symmetric matrices gives a lower bound of the minimal eigenvalue lambdam. Let A be a symmetric positive definite tridiagonal matrix defined by A = BT B, where B is bidiagonal. Then the so-called Kato-Temple bound gives a lower bound of the minimal singular value sigmam of B. In this paper we discuss how to apply the Kato-Temple inequality to shift of origin which appears in the mdLVs algorithm, for example, for computing all singular values of B. To make use of the Kato-Temple inequality a Rayleigh quotient for the matrix A = BT B and a right endpoint of interval where lambdam = sigmam 2 belongs are necessary. Then it is shown that the execution time of mdLVs with the standard shifts can be shorten by a possible choice of the generalized Newton bound or the Kato-Temple bound.