## Estimates for the bilinear form $x^T A^{-1} y$ with applications to linear algebra problems

### Abstract

Let $A \in \mathbb{R}^{p\times p}$ be a nonsingular matrix and $x,y$ vectors in $\mathbb{R}^p$. The task of this paper is to develop efficient estimation methods for the bilinear form $x^TA^{-1}y$ based on the extrapolation of moments of the matrix $A$ at the point $-1.$ The extrapolation method and estimates for the trace of $A^{-1}$ presented in Brezinski et al. [Numer. Linear Algebra Appl., 19 (2012), pp. 937–953] are extended, and families of estimates efficiently approximating the bilinear form requiring only few matrix vector products are derived. Numerical approximations of the entries and the trace of the inverse of any real nonsingular matrix are presented and several numerical results, discussions, and comparisons are given.

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### Key words

extrapolation, matrix moments, matrix inverse, trace

### AMS subject classifications

65F15, 65F30, 65B05, 65C05, 65J10, 15A18, 15A45

### Links to the cited ETNA articles

 [8] Vol. 39 (2012), pp. 144-155 Claude Brezinski, Paraskevi Fika, and Marilena Mitrouli : Estimations of the trace of powers of positive self-adjoint operators by extrapolation of the moments [15] Vol. 31 (2008), pp. 178-203 Gene H. Golub, Martin Stoll, and Andy Wathen: Approximation of the scattering amplitude and linear systems