Estimating the Quadratic Form <i>x</i>^T<i>A</i>^−m<i>x</i> for Symmetric Matrices: Further Progress and Numerical Computations

Abstract

In this paper, we study estimates for quadratic forms of the type xTA−mx, m∈N, for symmetric matrices. We derive a general approach for estimating this type of quadratic form and we present some upper bounds for the corresponding absolute error. Specifically, we consider three different approaches for estimating the quadratic form xTA−mx. The first approach is based on a projection method, the second is a minimization procedure, and the last approach is heuristic. Numerical examples showing the effectiveness of the estimates are presented. Furthermore, we compare the behavior of the proposed estimates with other methods that are derived in the literature.

Keywords

• quadratic form
• estimates
• upper bounds