Deflation approach for matrix function calculations based on double exponential-type numerical integral formula

This paper focuses on integration-type methods that employ the double-exponential (DE) integral formula, DE-type methods , for computing matrix functions. DE-type methods exhibit high convergence rate thanks to the highly efficient DE integral formula; however, it is known that existence of eigenvalues near a specific region on the complex plane adversely affects the convergence. This paper proposes a deflation approach that improves the convergence of the DE-type methods by extracting and separating the eigenspaces corresponding to the eigenvalues within such a region. To extract these interior eigenspaces, the proposed approach uses a complex moment-based eigensolver. The components of the matrix function corresponding to the interior eigenspace are directly obtained from the eigenpairs, whereas those corresponding to the exterior eigenspace are computed through the DE-type method with deflation. Error analysis and numerical experiments demonstrate the effectiveness of the proposed approach.