  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.
