On the preconditioned biconjugate gradients for solving linear complex equations arising from finite elements

Michal Křížek; Jaroslav Mlýnek

Banach Center Publications (1994)

  • Volume: 29, Issue: 1, page 195-205
  • ISSN: 0137-6934

Abstract

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The paper analyses the biconjugate gradient algorithm and its preconditioned version for solving large systems of linear algebraic equations with nonsingular sparse complex matrices. Special emphasis is laid on symmetric matrices arising from discretization of complex partial differential equations by the finite element method.

How to cite

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Křížek, Michal, and Mlýnek, Jaroslav. "On the preconditioned biconjugate gradients for solving linear complex equations arising from finite elements." Banach Center Publications 29.1 (1994): 195-205. <http://eudml.org/doc/262752>.

@article{Křížek1994,
abstract = {The paper analyses the biconjugate gradient algorithm and its preconditioned version for solving large systems of linear algebraic equations with nonsingular sparse complex matrices. Special emphasis is laid on symmetric matrices arising from discretization of complex partial differential equations by the finite element method.},
author = {Křížek, Michal, Mlýnek, Jaroslav},
journal = {Banach Center Publications},
keywords = {conjugate gradient method; large linear system; finite element; iterative method; preconditioning},
language = {eng},
number = {1},
pages = {195-205},
title = {On the preconditioned biconjugate gradients for solving linear complex equations arising from finite elements},
url = {http://eudml.org/doc/262752},
volume = {29},
year = {1994},
}

TY - JOUR
AU - Křížek, Michal
AU - Mlýnek, Jaroslav
TI - On the preconditioned biconjugate gradients for solving linear complex equations arising from finite elements
JO - Banach Center Publications
PY - 1994
VL - 29
IS - 1
SP - 195
EP - 205
AB - The paper analyses the biconjugate gradient algorithm and its preconditioned version for solving large systems of linear algebraic equations with nonsingular sparse complex matrices. Special emphasis is laid on symmetric matrices arising from discretization of complex partial differential equations by the finite element method.
LA - eng
KW - conjugate gradient method; large linear system; finite element; iterative method; preconditioning
UR - http://eudml.org/doc/262752
ER -

References

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  2. [2] F. S. Beckman, The solution of linear equations by the conjugate gradient method, in: Mathematical Methods for Digital Computers, A. Ralston and H. S. Wilf (eds.), Wiley, New York 1960, 62-72. 
  3. [3] R. Fletcher, Conjugate gradient methods for indefinite systems, in: Proc. Dundee Conf. on Numerical Analysis, A. Dold and B.Eckmann (eds.), Lecture Notes in Math. 506, Springer, New York 1975, 73-89. 
  4. [4] R. W. Freund, Conjugate gradient type methods for linear systems with complex symmetric coefficient matrices, SIAM J. Sci. Statist. Comput. 13 (1992), 1-23. 
  5. [5] R. W. Freund and N. M. Nachtigal, QMR: a quasi-minimal residual method for non-Hermitian linear systems, RIACS Technical Report 90.51, NASA, Columbia 1990, 33 pp. 
  6. [6] D. A. H. Jacobs, A generalization of the conjugate-gradient method to solve complex systems, IMA J. Numer. Anal. 6 (1986), 447-452. Zbl0614.65028
  7. [7] A. Kiełbasiński, Catalogue of linear algebra algorithms of the journal Numerische Mathematik, Mat. Stos. 2 (1974), 5-13 (in Polish). 
  8. [8] M. Křížek and P. Neittaanmäki, Finite Element Approximation of Variational Problems and Applications, Longman, Harlow 1990. 
  9. [9] D. G. Luenberger, Hyperbolic pairs in the conjugate gradients, SIAM J. Appl. Math. 17 (1969), 1263-1267. Zbl0187.09704
  10. [10] Z. Mikić and E. C. Morse, The use of a preconditioned bi-conjugate gradient method for hybrid plasma stability analysis, J. Comput. Phys. 61 (1985), 154-185. Zbl0637.76129
  11. [11] B. N. Parlett, D. R. Taylor and Z. A. Liu, A look-ahead Lanczos algorithm for symmetric matrices, Math. Comp. 44 (1985), 105-124. Zbl0564.65022
  12. [12] Y. Saad, The Lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems, SIAM J. Numer. Anal. 19 (1982), 485-506. Zbl0483.65022
  13. [13] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1962. 
  14. [14] H. A. van der Vorst and K. Dekker, Conjugate gradient type methods and preconditioning, J. Comput. Appl. Math. 24 (1988), 73-87. Zbl0659.65033
  15. [15] O. Widlund, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal. 15 (1978), 801-812. Zbl0398.65030

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