A parallel algorithm of ICSYM for complex symmetric linear systems in quantum chemistry

Yingchun Zhang, Quanyi Lv, Manyu Xiao, Gongnan Xie, Piotr Breitkopf


Computational effort is a common issue for solving large-scale complex symmetric linear systems, particularly in quantum chemistry applications. In order to alleviate this problem, we propose a parallel algorithm of improved conjugate gradient-type iterative (CSYM). Using three-term recurrence relation and orthogonal properties of residual vectors to replace the tridiagonalization process of classical CSYM, which allows to decrease the degree of the reduce-operator from two to one communication at each iteration and to reduce the amount of vector updates and vector multiplications. Several numerical examples are implemented to show that high performance of proposed improved version is obtained both in convergent rate and in parallel efficiency.

Full Text:



Bai, Z. J., Day, D., Demmel, J. and Dongarra, J. A test matrix collection for non-Hermitian eigenvalue problems. Technical Report CS-97-355, Department of Computer Science, University of Tennessee, Knoxville, TN, (1997)

Li, C. L. and Qiao, Z. H. A fast preconditioned iterative algorithm for the electromagnetic scattering from a large cavity. Journal of Scientific Computing, 53(2), 435–450 (2012)

Xiang, T. M. and Liang, C. H. Iterative solution for dense linear systems arising in computation electromagnetics. Journal of Xidian University, 30(6), 748–751 (2003)

Hu, Q. Y. and Yuan, L. A plane-wave least-squares method for time-harmonic Maxwell’s equations in absorbing media. SIAM Journal on Scientific Computing, 36(4), A1937–A1959 (2014)

Huttunen, T., Malinen, M. and Monk, P. Solving Maxwell’s equations using the ultra weak variational Formulation. Journal of Computational Physics, 223(2), 731–758 (2007)

Hu, Q. Y. and Yuan, L. A weighted variational formulation based on plane wave basis for discretization of Helmholtz equations. International Journal of Numerical Analysis & Modeling, 11(3), 587–607 (2014)

Elman, H. C. and OLeary, D. P. Eigenanalysis of some preconditioned Helmholtz problems. Numerische Mathematik, 83(2), 231–257 (1999)

Van der Vorst, H. A. Iterative Krylov Methods for Large Linear Systems, Cambridge University Press, New York (2003)

Gu, X. M., Clemens, M., Huang, T. Z. and Li, L. The SCBiCG class of algorithms for symmetric linear systems with

applications in several electromagnetic model problems. Computer Physics Communication, 191, 52–64 (2015)

Clemens Markus, Thomas Weiland, and Ursula Van Rienen. Comparison of Krylov-Type Methods for Complex Linear Systems Applied to High-Voltage Problems. IEEE Transactions of Magnetics, 34(5), 3335–3338 (1998)

Van der Vorst, H. A. and Melissen, J. A Petrov-Galerkin type method for solving Ax = b, where A is symmetric complex.

IEEE Transactions on Magnetics, 26(2), 706–708 (1990)

Sogabe, T. and Zhang, S. L. A COCR method for solving complex symmetric linear systems. Journal of Computational and

Applied Mathematics, 199, 297–303 (2007)

Bunse-Gerstner,A.andSt ̈over,R.Onaconjugategradient-typemethodforsolvingcomplexsymmetriclinearsystems.Linear

Algebra and its Applications, 287, 105–123 (1999)

Bu ̈cker, H. M. and Sauren, M. A parallel version of the quasi-minimal residual method based on coupled two-term recurrences.

International Workshop on Applied Parallel Computing, 1184, 157–165 (1996)

Zhao, L. T., Zuo, X. Y., Gu, T. X., Huang, T. Z. and Yue, J. H. Conjugate residual squared method and its improvement for

non-symmetric linear systems. International Journal of Computer Mathematics, 87(7), 1578–1590 (2010)

Yang, T. R. and Brent, R. P. The improved BiCG method for large and sparse nonsymmetric linear systems on parallel

distributed memory architectures. International Parallel & Distributed Processing Symposium, 3, 1–7 (2002)

Yang, T. R. and Brent, R. P. The improved BiCGSTAB method for large and sparse nonsymmetric linear systems on parallel distributed memory architectures. International Conference on Algorithms & Architectures for Parallel Processing, 3, 324–328 (2002)

Zuo, X. Y., Liu, Y., Zhang, L. T., and Meng, H. A parallel version of COCR method for solving complex symmetric linear systems. Parallel and Cloud Computing Research, 2, 12-18 (2014)

Wu, J. P., Wang, Z. H. and Li, X. M. Efficient solution and parallel computation of sparse linear equations, Hunan Science and Technology Press, Changsha (2004)

Chen, G. L., Hong, A., Chen, J., Zheng, Q. L. and Shan, J. L. The Parallel Algorithm, Higher Education Press, Beijing (2004)

DOI: https://doi.org/10.7494/csci.2018.19.4.2868


  • There are currently no refbacks.