Approximating functions of a large sparse positive definite matrix using a spectral splitting method

Abstract

The computation of functions of large sparse matrices f(A) is an important topic in numerical linear algebra and finds application in many fields of applied mathematics and statistics. In previous research we considered ? matrices with compact spectrum ?( A ) ? [a,b] and proposed low degree matrix polynomial approximations p( A ) such that e = ?f( A ) ? p( A ) ? was small on the spectral interval, where the extreme eigenvalues a and b were calculated using Krylov subspace approximation. For the class of matrices examined, the thick restarted Lanczos scheme enabled rapid convergence to the extreme eigenvalues and these Ritz values were used to construct cubic near-minimax Chebyshev least squares approximations of the desired matrix functions. There is a good balance between accuracy and efficiency for this approximation method. The aim of the present study is to extend the previously developed matrix function approximation technique to enable ? matrices with a wider spectrum to be treated using a novel splitting of ?( A ). In this case, the decomposition of f( A ) as a sum of a 'singular' part and a 'regular' part is investigated. To perform the split a projector onto the singular part is here constructed using Krylov subspace approximation. Numerical results for a representative large sparse positive definite matrix appear promising.