Block symplectic Gram–Schmidt method

Authors

  • Yoichi Matsuo School of Fundamental Science and Technology, Graduate School of Science and Technology, Keio University
  • Takashi Nodera Keio University

DOI:

https://doi.org/10.21914/anziamj.v56i0.9380

Keywords:

symplectic block Gram-Schmidt, optimal block size, J-orthogonality

Abstract

For large scale linear problems, it is common to use the symplectic Lanczos method which uses the symplectic Gram–Schmidt method to compute symplectic vectors. However, previous studies showed that the selection process of the parameter in the symplectic Gram–Schmidt method is flawed, as it results in a partially destroyed \(J\)-orthogonality of the \(J\)-orthogonal matrix. We exploree a block type symplectic Gram–Schmidt method and a new condition for the reorthogonalization to maintain \(J\)-orthogonality and to more accurately compute symplectic factorization. Applying the block size scheme to this method, we develop a new procedure for computing symplectic vectors. References
  • G. Ammar and V. Mehrmann. On Hamiltonian and symplectic Hessenberg forms. Linear Algebra Appl. 149:55–72, 1991. doi:10.1016/0024-3795(91)90325-Q
  • P. Benner and H. Fassbender. An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem. Linear Algebra Appl. 263:75–111, 1997. doi:10.1016/S0024-3795(96)00524-1
  • A. Bunse-Gerstner and V. Mehrmann. A symplectic QR like algorithm for the solution of the real algebraic Riccati equation. IEEE T. Automat. Contr. 31:1104–1113, 1986. doi:10.1109/TAC.1986.1104186
  • G. Runger and M. Schwind. Comparison of different parallel modified Gram–Schmidt algorithms, Euro-Par 2005, Lecture Notes in Computer Science 3648:826–836, 2005. doi:10.1007/11549468_90
  • H. Kwakernaak and R. Sivan. Linear Optimal Control Systems. Wiley, 1972. http://dl.acm.org/citation.cfm?id=578807
  • C. Van Loan. A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix. Linear Algebra Appl., 61:233–251, 1984. doi:10.1016/0024-3795(84)90034-X
  • Y. Matsuo and T. Nodera. The optimal block-size for the block Gram–Schmidt orthogonalization. J. Sci. Tech., 49:569–584, 2011.
  • Y. Matsuo and T. Nodera. An Efficient Implementation of the Block Gram–Schmidt Method. CTAC2012, ANZIAM J., 54:C394–409, 2013. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6327
  • A. Salam. On theoretical and numerical aspects of symplectic Gram–Schmidt-like algorithms, Numer. Algorithms 39:437–462, 2005. doi:10.1007/s11075-005-0963-2
  • A. Salam and E. Al-Aidarous. Equivalence between modified symplectic Gram–Schmidt and Householder SR algorithms. BIT Numer. Math., 54:283–302, 2014. doi:10.1007/s10543-013-0441-5
  • S. J. Leon, A. Bjorck and W. Gander. Gram–Schmidt orthogonalization: 100 years and more. Numer. Linear Algebra Appl. 20:492–532, 2013. doi:10.1002/nla.1839
  • G. W. Stewart. Block Gram–Schmidt orthogonalization, SIAM J. Sci. Comput., 31:761–775, 2008. doi:10.1137/070682563

Published

2016-02-22

Issue

Section

Proceedings Computational Techniques and Applications Conference