Shifted Lanczos method for quadratic forms with Hermitian matrix resolvents

Abstract: Quadratic forms of Hermitian matrix resolvents involve the solutions of shifted linear systems. Efficient iterative solutions use the shift-invariance property of Krylov subspaces The Hermitian Lanczos method reduces a given vector and matrix to a Jacobi matrix (real symmetric tridiagonal matrix with positive super and sub-diagonal entries) and approximates the quadratic form using the Jacobi matrix. This study develops a shifted Lanczos method that deals directly with the Hermitian matrix resolvent. We derive a matrix representation of a linear operator that approximates the resolvent by solving a Vorobyev moment problem associated with the shifted Lanczos method. We show that an entry of the Jacobi matrix resolvent can approximate the quadratic form, matching the moments. We give a sufficient condition such that the method does not break down, an error bound, and error estimates. Numerical experiments on matrices drawn from real-world applications compare the proposed method with previous methods and show that the proposed method outperforms well-established methods in solving some problems.