Fast estimates for the diagonal of the inverse of large scale matrices appearing in applications

Abstract

For large scale problems, the explicit computation of the inverse of a given matrix has high computational complexity and therefore a crucial problem is its efficient approximation. In this work, we present a readily implementable procedure for approximating individual diagonal elements and the entire diagonal of the inverse of large-scale diagonalizable matrices. In particular, based on extrapolation procedures, backward stable families of low cost estimates approximating efficiently the theoretical values are proposed. Several applications involving the precision matrix in Statistics, the matrix resolvent in Network Analysis, matrices coming from economic problems and from the discretization of physical problems, require the diagonal elements of the inverse of the associated matrix. For these classes of problems, the effectiveness of the derived estimates is validated through several numerical examples implemented in serial and parallel forms (OpenMP) on the high-performance computing system ARIS.