A RATIONAL FILTERING ALGORITHM FOR SEQUENCES OF SHIFTED SYMMETRIC LINEAR SYSTEMS WITH APPLICATIONS TO FREQUENCY RESPONSE ANALYSIS

Abstract

We present a numerical algorithm for the solution of a large number of shifted linear systems for which the system pencil is symmetric and definite and the shifts lie inside a given real interval. Extending an earlier method due to Meerbergen and Bai [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1642–1662], the algorithm uses a rational filter with poles at Chebyshev points to compute and deflate the components of the solution in the direction of eigenvectors of the system pencil corresponding to eigenvalues within the interval. It then solves the deflated systems for the remaining components using a Krylov subspace method with a preconditioner constructed by interpolating the factorizations at the filter poles. The algorithm parallelizes naturally. We demonstrate its effectiveness using matrix pencils from both model and real-world problems and discuss applications to frequency response analysis.