Abstract
The approximation of matrix functionals appears in many applications arising from the fields of mathematics, statistics, mechanics, networks, machine learning and physics. In this paper, we estimate matrix functionals of the form XTf(A)Y, where A∈Rp×p is a given diagonalizable matrix, X, Y∈Rp×k are skinny “block vectors” with k≪p columns and f is a smooth function defined on the spectrum of the matrix A. We apply a direct approach based on the extrapolation of the moments of the given matrix, for estimating this kind of matrix functionals. This approach avoids the application of the polarization identity, is fairly inexpensive and leads to a stable procedure. Moreover, we develop a detailed backward error analysis for the derived estimates. Several numerical results illustrating the effectiveness of the direct method are presented and concrete classes of matrices, suitable for the extrapolation estimates, are proposed.
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