Efficient estimates for matrix-inverse quadratic forms

In this paper we present two approaches for estimating matrix-inverse quadratic forms $x^T{A}^{-1}x$, where A is a symmetric positive definite matrix of order n, and $x\in R^n$. Using the first, analytic approach, we establish two families of estimates which are convenient for matrices with small condition number. Based on the second, heuristic approach, we derive two families of estimates which are suitable for matrices when vector x is close enough to an eigenvector. The low complexity and stability of the estimates is proved. Several numerical results illustrating the effectiveness of the methods are presented.