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

For a Hermitian matrix A∈Cp×pA∈Cp×p, given vectors x  , y∈Cpy∈Cp and for suitable functions f  , the bilinear form y⁎f(A)xy⁎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,νef,ν, ν∈Cν∈C and eˆf,n,k, n,k∈Zn,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−1f(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.