Estimates in quadratic formulas

Abstract

LetA be a real symmetric positive definite matrix. We consider three particular questions, namely estimates for the error in linear systemsAx=b, minimizing quadratic functional minx(x T Ax−2b T x) subject to the constraint ‖x‖=α, α<‖A −1 b‖, and estimates for the entries of the matrix inverseA −1. All of these questions can be formulated as a problem of finding an estimate or an upper and lower bound onu T F(A)u, whereF(A)=A −1 resp.F(A)=A −2,u is a real vector. This problem can be considered in terms of estimates in the Gauss-type quadrature formulas which can be effectively computed exploiting the underlying Lanczos process. Using this approach, we first recall the exact arithmetic solution of the questions formulated above and then analyze the effect of rounding errors in the quadrature calculations. It is proved that the basic relation between the accuracy of Gauss quadrature forf(λ)=λ−1 and the rate of convergence of the corresponding conjugate gradient process holds true even for finite precision computation. This allows us to explain experimental results observed in quadrature calculations and in physical chemistry and solid state physics computations which are based on continued fraction recurrences.