ESTIMATES FOR THE BILINEAR FORM xTA-1y WITH APPLICATIONS TO LINEAR ALGEBRA PROBLEMS

Let A is an element of R-p x p be a nonsingular matrix and x, y vectors in R-p. The task of this paper is to develop efficient estimation methods for the bilinear form x(T)A(-1)y based on the extrapolation of moments of the matrix Lambda at the point -1. The extrapolation method and estimates for the trace of Lambda(-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.