Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Doubling algorithm for the discretized Bethe-Salpeter eigenvalue problem

Authors: Zhen-Chen Guo, Eric King-Wah Chu and Wen-Wei Lin
Journal: Math. Comp. 88 (2019), 2325-2350
MSC (2010): Primary 15A18, 65F15
Published electronically: January 9, 2019
MathSciNet review: 3957895
Full-text PDF
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

Abstract: The discretized Bethe-Salpeter eigenvalue problem arises in the Green’s function evaluation in many body physics and quantum chemistry. Discretization leads to a matrix eigenvalue problem for $H \in \mathbb {C}^{2n \times 2n}$ with a Hamiltonian-like structure. After an appropriate transformation of $H$ to a standard symplectic form, the structure-preserving doubling algorithm, originally for algebraic Riccati equations, is extended for the discretized Bethe-Salpeter eigenvalue problem. Potential breakdowns of the algorithm, due to the ill condition or singularity of certain matrices, can be avoided with a double-Cayley transform or a three-recursion remedy. A detailed convergence analysis is conducted for the proposed algorithm, especially on the benign effects of the double-Cayley transform. Numerical results are presented to demonstrate the efficiency and the structure-preserving nature of the algorithm.

References [Enhancements On Off] (What's this?)

  • Zhaojun Bai and Ren-Cang Li, Minimization principles for the linear response eigenvalue problem I: Theory, SIAM J. Matrix Anal. Appl. 33 (2012), no. 4, 1075–1100. MR 3023465, DOI
  • Zhaojun Bai and Ren-Cang Li, Minimization principles for the linear response eigenvalue problem II: Computation, SIAM J. Matrix Anal. Appl. 34 (2013), no. 2, 392–416. MR 3046810, DOI
  • Peter Benner, Sergey Dolgov, Venera Khoromskaia, and Boris N. Khoromskij, Fast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation, J. Comput. Phys. 334 (2017), 221–239. MR 3606226, DOI
  • P. Benner, H. Fassbender, and C. Yang, Some remarks on the complex $J$-symmetric eigenvalue problem, Preprint, MPIMD/15-12, Max Planck Institute Magdeburg, 2015 (available at
  • P. Benner, V. Khoromskaia, and B. N. Khoromskij, A reduced basis approach for calculation of the Bethe-Salpeter excitation energies using low-rank tensor factorizations, Molecular Phys., 114 (2016) 1148–1161.
  • M. E. Casida, Time-dependent density-functional response theory for molecules, Recent Advances in Density Functional Methods, Part I, D.P. Chong (Ed.), World Scientific, Singapore, 155 (1995) 1207–1216.
  • E. K.-W. Chu, H.-Y. Fan, and W.-W. Lin, A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations, Linear Algebra Appl. 396 (2005), 55–80. MR 2112199, DOI
  • E. K.-W. Chu, H.-Y. Fan, W.-W. Lin, and C.-S. Wang, Structure-preserving algorithms for periodic discrete-time algebraic Riccati equations, Internat. J. Control 77 (2004), no. 8, 767–788. MR 2072208, DOI
  • Eric King-Wah Chu, Tsung-Min Hwang, Wen-Wei Lin, and Chin-Tien Wu, Vibration of fast trains, palindromic eigenvalue problems and structure-preserving doubling algorithms, J. Comput. Appl. Math. 219 (2008), no. 1, 237–252. MR 2437709, DOI
  • Gene H. Golub and Charles F. Van Loan, Matrix computations, 3rd ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996. MR 1417720
  • T.-M. Huang, R.-C. Li, W.-W. Lin, and L. Lu., Optimal parameters for doubling algorithms, Technical Report 2017-03, Department of Mathematics, University of Texas at Arlington, May 2017. Available at
  • Tsung-Ming Huang and Wen-Wei Lin, Structured doubling algorithms for weakly stabilizing Hermitian solutions of algebraic Riccati equations, Linear Algebra Appl. 430 (2009), no. 5-6, 1452–1478. MR 2490689, DOI
  • V. Khoromskaia, B. N. Khoromskij, and R. Schneider, Tensor-structured factorized calculation of two-electron integrals in a general basis, SIAM J. Sci. Comput. 35 (2013), no. 2, A987–A1010. MR 3040965, DOI
  • S. KΓΆrbel, P. Boulanger, I. Duchemin, X. Blase, M. AL Marques, and S. Botti, Benchmark many-body GW and Bethe-Salpeter calculations for small transition metal molecules, J. Chemical Theory Comp., 10 (2014) 3934–3943.
  • X. Leng, F. Jin, M. Wei, and Y. Ma, GW method and Bethe-Salpeter equation for calculating electronic excitations, Wiley Interdisciplinary Reviews: Computation Molecular Science, Wiley Online Library, 2016.
  • Tiexiang Li, Chun-Yueh Chiang, Eric King-wah Chu, and Wen-Wei Lin, The palindromic generalized eigenvalue problem $A^*x=\lambda Ax$: numerical solution and applications, Linear Algebra Appl. 434 (2011), no. 11, 2269–2284. MR 2776795, DOI
  • Tiexiang Li, Eric King-wah Chu, Jong Juang, and Wen-Wei Lin, Solution of a nonsymmetric algebraic Riccati equation from a one-dimensional multistate transport model, IMA J. Numer. Anal. 31 (2011), no. 4, 1453–1467. MR 2846762, DOI
  • Tiexiang Li, Eric King-wah Chu, Jong Juang, and Wen-Wei Lin, Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model, Linear Algebra Appl. 434 (2011), no. 1, 201–214. MR 2737242, DOI
  • Wen-Wei Lin and Shu-Fang Xu, Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations, SIAM J. Matrix Anal. Appl. 28 (2006), no. 1, 26–39. MR 2218940, DOI
  • G. Onida, L. Reining, and A. Rubio, Electronic excitations: density-functional versus many-body Green’s-function approaches, Rev. Mod. Phys., 74 (2002) 601–659.
  • R.M. Parrish, E.G. Hohenstein, N. Schunck, C. Sherrill, and T. J. Martinez, Exact tensor hypercontraction: A universal technique for the resolution of matrix elements of local finite-range N-body potentials in many-body quantum problems, Phys. Rev. Lett., 111 (2013) 132505.
  • Y. Ping, D. Rocca, and G. Galli, Electronic excitations in light absorbers for photo-electrochemical energy conversion: First principles calculations based on many body perturbation theory, Chem. Soc. Rev., 42 (2013) 2437–2469.
  • S. Reine, T. Helgaker, and R. Lindh, Multi-electron integrals, WIREs Comput. Mol. Sci., 2 (2012) 290–303.
  • L. Reining, V. Olevano, A. Rubio, and G. Onida, Excitonic effects in solids described by time-dependent density functional theory, Phys. Rev. Lett., 88 (2002) 066404.
  • E. Ribolini, J. Toulouse, and A. Savin, Electronic excitation energies of molecular systems from the Bethe-Salpeter equation: Example of H2 molecule, Concepts and Methods in Modern Theoretical Chemistry, S. Ghosh and P. Chattaraj (eds), Vol. 1: Electronic Structure and Reactivity, 367 (2013) 367–390.
  • E. Ribolini, J. Toulouse, and A. Savin, Electronic excitations from a linear-response range-separated hybrid scheme, Molecular Phys., 111 (2013) 1219–1234.
  • M. Rohlfing and S. G. Louie, Electron-hole excitations and optical spectra from first principles. Phys. Rev. B, 62 (2000) 4927–4944.
  • E. Runge and E. Gross, Density-function theory for time-dependent systems, Phys. Rev. Lett., 52 (1984) 997–1000.
  • E. E. Salpeter and H. A. Bethe, A relativistic equation for bound-state problems, Phys. Rev. (2) 84 (1951), 1232–1242. MR 52996
  • Meiyue Shao, Felipe H. da Jornada, Chao Yang, Jack Deslippe, and Steven G. Louie, Structure preserving parallel algorithms for solving the Bethe-Salpeter eigenvalue problem, Linear Algebra Appl. 488 (2016), 148–167. MR 3419779, DOI
  • G. W. Stewart and Ji Guang Sun, Matrix perturbation theory, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1990. MR 1061154
  • R. E. Stratmann, G. E. Scuseria, and M. J. Frisch, An efficient implementation of time-dependent density-functional theory for the calculation of excitation energies of large molecules, J. Chem. Phys., 109 (1998) 8218–8224.
  • S. Wilson, Universal basis sets and Cholesky decomposition of the two-electron integral matrix, Comput. Phys. Commun., 58 (1990) 71–81.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 15A18, 65F15

Retrieve articles in all journals with MSC (2010): 15A18, 65F15

Additional Information

Zhen-Chen Guo
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
MR Author ID: 1095300

Eric King-Wah Chu
Affiliation: School of Mathematics, Monash University, 9 Rainforest Walk, Victoria 3800, Australia
MR Author ID: 49125

Wen-Wei Lin
Affiliation: Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan
MR Author ID: 232126

Keywords: Bethe-Salpeter eigenvalue problem, Cayley transform, doubling algorithm
Received by editor(s): November 20, 2017
Received by editor(s) in revised form: May 6, 2018, July 25, 2018, and August 21, 2018
Published electronically: January 9, 2019
Article copyright: © Copyright 2019 American Mathematical Society