2013/02/17 by Abdou Abdel-Rehim, A. M. Abdel-Rehim, Andreas Stathopoulos +1 · 1 citation
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Algorithm #Applied mathematics #Biconjugate gradient stabilized method #Eigenvalues and eigenvectors #Electromagnetic Scattering and Analysis #Lanczos algorithm #Lanczos resampling #Linear system #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Matrix Theory and Algorithms #cs.NA #hep-lat #math.NA
paper · pdf · doi:10.1002/nla.1893
published as Numer. Linear Algebra Appl., 21: 473-493 (2014) · 25 pages, 18 figures
arxiv created 2013/02/17 · openalex publication_date 2013/07/15 · arxiv updated 2014/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.