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Quadratic forms of Hermitian matrix resolvents involve the solutions ofshifted linear systems. Efficient iterative solutions use the shift-invarianceproperty of Krylov subspaces The Hermitian Lanczos method reduces a givenvector and matrix to a Jacobi matrix (real symmetric tridiagonal matrix withpositive super and sub-diagonal entries) and approximates the quadratic formusing the Jacobi matrix. This study develops a shifted Lanczos method thatdeals directly with the Hermitian matrix resolvent. We derive a matrixrepresentation of a linear operator that approximates the resolvent by solvinga Vorobyev moment problem associated with the shifted Lanczos method. We showthat an entry of the Jacobi matrix resolvent can approximate the quadraticform, matching the moments. We give a sufficient condition such that the methoddoes not break down, an error bound, and error estimates. Numerical experimentson matrices drawn from real-world applications compare the proposed method withprevious methods and show that the proposed method outperforms well-establishedmethods in solving some problems.

Shifted Lanczos method for quadratic forms with Hermitian matrix resolvents