Extending the eigCG algorithm to non-symmetric linear systems with multiple right-hand sides

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For Hermitian positive definite linear systems and eigenvalue problems, the eigCG algorithm is a memory efficient algorithm that solves the linear system and simultaneously computes some of its eigenvalues. The algorithm is based on the Conjugate-Gradient (CG) algorithm, however, it uses only a window of the vectors generated by the CG algorithm to compute approximate eigenvalues. The number and accuracy of the eigenvectors can be increased by solving more right-hand sides. For Hermitian systems with multiple right-hand sides, the computed eigenvectors can be used to speed up the solution of subsequent systems. The algorithm was tested on Lattice QCD problems ... continued below

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Abdou Abdel-Rehim, Kostas Orginos, Andreas Stathopoulos November 1, 2009.

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For Hermitian positive definite linear systems and eigenvalue problems, the eigCG algorithm is a memory efficient algorithm that solves the linear system and simultaneously computes some of its eigenvalues. The algorithm is based on the Conjugate-Gradient (CG) algorithm, however, it uses only a window of the vectors generated by the CG algorithm to compute approximate eigenvalues. The number and accuracy of the eigenvectors can be increased by solving more right-hand sides. For Hermitian systems with multiple right-hand sides, the computed eigenvectors can be used to speed up the solution of subsequent systems. The algorithm was tested on Lattice QCD problems by solving the normal equations and was shown to give large speed up factors and to remove the critical slowing down as we approach light quark masses. Here, an extension to the non-symmetric case based on the two-sided Lanczos algorithm is given. The new algorithm is tested on Lattice QCD problems and is shown to give promising results. We also study the removal of the critical slowing down and compare results with those of the eigCG algorithm. We also discuss the case when the system is gamma5-Hermitian.

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  • 27th International Symposium on Lattice Field Theory (Lattice 2009), Beijing, China, 25-31 Jul 2009

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  • Report No.: JLAB-THY-09-978
  • Report No.: DOE/OR/23177-1778
  • Report No.: arXiv:0911.2285
  • Grant Number: AC05-06OR23177
  • Office of Scientific & Technical Information Report Number: 1023440
  • Archival Resource Key: ark:/67531/metadc840593

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  • November 1, 2009

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  • May 19, 2016, 3:16 p.m.

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  • Aug. 3, 2016, 6:27 p.m.

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Abdou Abdel-Rehim, Kostas Orginos, Andreas Stathopoulos. Extending the eigCG algorithm to non-symmetric linear systems with multiple right-hand sides, article, November 1, 2009; Newport News, Virginia. (digital.library.unt.edu/ark:/67531/metadc840593/: accessed November 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.