Differentiable implicitly restarted Lanczos method for computing eigenpairs and their derivatives of large symmetric matrix-valued functions

Abstract

A differentiable implicitly restarted Lanczos method is presented in this paper to compute the smallest eigenpairs and their derivatives of large matrix-valued function. The convergence of the proposed method is also established. This method improves the efficiency of computing the eigenpair derivatives greatly and there are three advantages of it. First, eigenpairs and their derivatives are computed simultaneously. Second, the equation system for eigenvector derivatives can be greatly reduced from the original matrix size. Third, the left eigenvector and its derivative are not required. Numerical experiments show that the method is efficient.