Shifted Lanczos method for quadratic forms with Hermitian matrix resolvents

Quadratic forms of Hermitian matrix resolvents involve the solutions of shifted linear systems. Efficient solutions use the shift-invariance property of Krylov subspaces. The Hermitian Lanczos method reduces a given vector and matrix to a Jacobi matrix (a real symmetric tridiagonal matrix with positive super and sub-diagonal entries) and approximates the quadratic form with the Jacobi matrix. This study develops a shifted Lanczos method that deals directly with the Hermitian matrix resolvent to extend the scope of problems that the Lanczos method can solve. 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. We show the moment-matching property of the shifted Lanczos method and give a sufficient condition such that the method does not break down. Numerical experiments on matrices drawn from real-world applications compare the proposed method with previous methods and show that the proposed method outperforms a well-established method in solving some problems.

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