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SUMMARY  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, similar 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 with 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. Copyright © 2013 John Wiley & Sons, Ltd.

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