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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 408167, 16 pages
Exploiting the Composite Step Strategy to the Biconjugate -Orthogonal Residual Method for Non-Hermitian Linear Systems
1School of Mathematical Sciences, Institute of Computational Science, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
2Institute of Mathematics and Computing Science, University of Groningen, Nijenborgh 9, P.O. Box 407, 9700 AK Groningen, The Netherlands
Received 15 October 2012; Accepted 19 December 2012
Academic Editor: Zhongxiao Jia
Copyright © 2013 Yan-Fei Jing et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Y. Saad and H. A. van der Vorst, “Iterative solution of linear systems in the 20th century,” Journal of Computational and Applied Mathematics, vol. 43, pp. 1155–1174, 2005.
- V. Simoncini and D. B. Szyld, “Recent computational developments in Krylov subspace methods for linear systems,” Numerical Linear Algebra with Applications, vol. 14, no. 1, pp. 1–59, 2007.
- J. Dongarra and F. Sullivan, “Guest editors’introduction to the top 10 algorithms,” Computer Science and Engineering, vol. 2, pp. 22–23, 2000.
- B. Philippe and L. Reichel, “On the generation of Krylov subspace bases,” Applied Numerical Mathematics, vol. 62, no. 9, pp. 1171–1186, 2012.
- A. Greenbaum, Iterative Methods for Solving Linear Systems, vol. 17 of Frontiers in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1997.
- Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, Pa, USA, 2nd edition, 2003.
- Y.-F. Jing, T.-Z. Huang, Y. Zhang et al., “Lanczos-type variants of the COCR method for complex nonsymmetric linear systems,” Journal of Computational Physics, vol. 228, no. 17, pp. 6376–6394, 2009.
- Y.-F. Jing, B. Carpentieri, and T.-Z. Huang, “Experiments with Lanczos biconjugate -orthonormalization methods for MoM discretizations of Maxwell’s equations,” Progress in Electromagnetics Research, vol. 99, pp. 427–451, 2009.
- Y.-F. Jing, T.-Z. Huang, Y. Duan, and B. Carpentieri, “A comparative studyof iterative solutions to linear systems arising in quantum mechanics,” Journal of Computational Physics, vol. 229, no. 22, pp. 8511–8520, 2010.
- B. Carpentieri, Y.-F. Jing, and T.-Z. Huang, “The BICOR and CORS iterative algorithms for solving nonsymmetric linear systems,” SIAM Journal on Scientific Computing, vol. 33, no. 5, pp. 3020–3036, 2011.
- R. Fletcher, “Conjugate gradient methods for indefinite systems,” in Numerical Analysis (Proc 6th Biennial Dundee Conf., Univ. Dundee, Dundee, 1975), vol. 506 of Lecture Notes in Mathematics, pp. 73–89, Springer, Berlin, Germany, 1976.
- W. Joubert, Generalized conjugate gradient and lanczos methods for the solution of nonsymmetric systems of linear equations [Ph.D. thesis], University of Texas, Austin, Tex, USA, 1990.
- W. Joubert, “Lanczos methods for the solution of nonsymmetric systems of linear equations,” SIAM Journal on Matrix Analysis and Applications, vol. 13, no. 3, pp. 926–943, 1992.
- M. H. Gutknecht, “A completed theory of the unsymmetric Lanczos process and related algorithms. I,” SIAM Journal on Matrix Analysis and Applications, vol. 13, no. 2, pp. 594–639, 1992.
- M. H. Gutknecht, “A completed theory of the unsymmetric Lanczos process and related algorithms. II,” SIAM Journal on Matrix Analysis and Applications, vol. 15, no. 1, pp. 15–58, 1994.
- M. H. Gutknecht, “Block Krylov space methods for linear systems withmultiple right-hand sides: an introduction,” in Modern Mathematical Models, Methods and Algorithms for Real World Systems, A. H. Siddiqi, I. S. Duff, and O. Christensen, Eds., Anamaya Publishers, New Delhi, India, 2006.
- D. G. Luenberger, “Hyperbolic pairs in the method of conjugate gradients,” SIAM Journal on Applied Mathematics, vol. 17, pp. 1263–1267, 1969.
- R. E. Bank and T. F. Chan, “An analysis of the composite step biconjugate gradient method,” Numerische Mathematik, vol. 66, no. 3, pp. 295–319, 1993.
- R. E. Bank and T. F. Chan, “A composite step bi-conjugate gradient algorithm for nonsymmetric linear systems,” Numerical Algorithms, vol. 7, no. 1, pp. 1–16, 1994.
- R. W. Freund and N. M. Nachtigal, “QMR: a quasi-minimal residual method for non-Hermitian linear systems,” Numerische Mathematik, vol. 60, no. 3, pp. 315–339, 1991.
- M. H. Gutknecht, “The unsymmetric Lanczos algorithms and their relations to Pade approximation, continued fraction and the QD algorithm,” in Proc. Copper Mountain Conference on Iterative Methods, Book 2, Breckenridge Co., Breckenridge, Colo, USA, 1990.
- B. N. Parlett, D. R. Taylor, and Z. A. Liu, “A look-ahead Lánczos algorithm for unsymmetric matrices,” Mathematics of Computation, vol. 44, no. 169, pp. 105–124, 1985.
- B. N. Parlett, “Reduction to tridiagonal form and minimal realizations,” SIAM Journal on Matrix Analysis and Applications, vol. 13, no. 2, pp. 567–593, 1992.
- C. Brezinski, M. Redivo Zaglia, and H. Sadok, “Avoiding breakdown and near-breakdown in Lanczos type algorithms,” Numerical Algorithms, vol. 1, no. 3, pp. 261–284, 1991.
- C. Brezinski, M. Redivo Zaglia, and H. Sadok, “A breakdown-free Lanczos type algorithm for solving linear systems,” Numerische Mathematik, vol. 63, no. 1, pp. 29–38, 1992.
- C. Brezinski, M. Redivo-Zaglia, and H. Sadok, “Breakdowns in the implementation of the Lánczos method for solving linear systems,” Computers & Mathematics with Applications, vol. 33, no. 1-2, pp. 31–44, 1997.
- C. Brezinski, M. R. Zaglia, and H. Sadok, “New look-ahead Lanczos-type algorithms for linear systems,” Numerische Mathematik, vol. 83, no. 1, pp. 53–85, 1999.
- N. M. Nachtigal, A look-ahead variant of the lanczos algorithm and its application to the quasi-minimal residual method for non-Hermitian linear systems [Ph.D. thesis], Massachusettes Institute of Technology, Cambridge, Mass, USA, 1991.
- R. W. Freund, M. H. Gutknecht, and N. M. Nachtigal, “An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices,” SIAM Journal on Scientific Computing, vol. 14, no. 1, pp. 137–158, 1993.
- M. H. Gutknecht, “Lanczos-type solvers for nonsymmetric linear systems of equations,” in Acta Numerica, 1997, vol. 6 of Acta Numerica, pp. 271–397, Cambridge University Press, Cambridge, UK, 1997.
- T. Sogabe, M. Sugihara, and S.-L. Zhang, “An extension of the conjugate residual method to nonsymmetric linear systems,” Journal of Computational and Applied Mathematics, vol. 226, no. 1, pp. 103–113, 2009.
- P. Concus, G. H. Golub, and D. P. O'Leary, “A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations,” in Sparse Matrix Computations (Proc. Sympos., Argonne Nat. Lab., Lemont, Ill., 1975), pp. 309–332, Academic Press, New York, NY, USA, 1976.
- A. B. J. Kuijlaars, “Convergence analysis of Krylov subspace iterations with methods from potential theory,” SIAM Review, vol. 48, no. 1, pp. 3–40, 2006.
- T. Davis, “The university of Florida sparse matrix collection,” NA Digest, vol. 97, no. 23, 1997.
- T. Huckle, “Approximate sparsity patterns for the inverse of a matrix and preconditioning,” Applied Numerical Mathematics, vol. 30, no. 2-3, pp. 291–303, 1999.
- G. A. Gravvanis, “Explicit approximate inverse preconditioning techniques,” Archives of Computational Methods in Engineering, vol. 9, no. 4, pp. 371–402, 2002.
- B. Carpentieri, I. S. Duff, L. Giraud, and G. Sylvand, “Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations,” SIAM Journal on Scientific Computing, vol. 27, no. 3, pp. 774–792, 2005.
- M. Benzi, “Preconditioning techniques for large linear systems: a survey,” Journal of Computational Physics, vol. 182, no. 2, pp. 418–477, 2002.