Estimation of the bilinear form y⁎f(A)x for Hermitian matrices

Abstract

For a Hermitian matrix A∈Cp×p, given vectors x, y∈Cp and for suitable functions f, the bilinear form y⁎f(A)x is estimated by extending the extrapolation method proposed by C. Brezinski in 1999. Families of one term and two term estimates ef,ν, ν∈C and eˆf,n,k, n,k∈Z, respectively, are derived by extrapolation of the moments of the matrix A. For the positive definite case, bounds for the optimal value of ν, which leads to an efficient one term estimate in only one matrix vector product, are derived. For f(A)=A−1, a formula approximating this optimal value of ν is specified. Numerical results for several matrix functions and comparisons are provided to demonstrate the effectiveness of the extrapolation method.