Extending the eigCG algorithm to nonsymmetric Lanczos for linear systems with multiple right-hand sides

TL;DR

This paper extends the eigCG algorithm to nonsymmetric matrices using BiCG, enabling efficient computation of eigenvalues and eigenvectors while solving multiple linear systems, with applications demonstrated in Lattice QCD.

Contribution

The authors develop a novel eigBiCG algorithm for nonsymmetric matrices that computes eigenvalues and eigenvectors during linear system solves, and introduce an incremental approach for multiple right-hand sides.

Findings

EigBiCG accurately computes spectral approximations comparable to nonsymmetric Lanczos.

The incremental eigBiCG effectively computes eigenvalues during initial solves.

The combined approach with deflated BiCGStab is competitive for multiple right-hand side systems.

Abstract

The technique that was used to build the EigCG algorithm for sparse symmetric linear systems is extended to the nonsymmetric case using the BiCG algorithm. We show that, similarly to the symmetric case, we can build an algorithm that is capable of computing a few smallest magnitude eigenvalues and their corresponding left and right eigenvectors of a nonsymmetric matrix using only a small window of the BiCG residuals while simultaneously solving a linear system with that matrix. For a system with multiple right-hand sides, we give an algorithm that computes incrementally more eigenvalues while solving the first few systems and then uses the computed eigenvectors to deflate BiCGStab for the remaining systems. Our experiments on various test problems, including Lattice QCD, show the remarkable ability of EigBiCG to compute spectral approximations with accuracy comparable to that of the unrestarted, nonsymmetric Lanczos. Furthermore, our incremental EigBiCG followed by appropriately restarted and deflated BiCGStab provides a competitive method for systems with multiple right-hand sides.