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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Doubling algorithm for the discretized Bethe-Salpeter eigenvalue problem
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by Zhen-Chen Guo, Eric King-Wah Chu and Wen-Wei Lin HTML | PDF
Math. Comp. 88 (2019), 2325-2350 Request permission


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.
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Additional Information
  • Zhen-Chen Guo
  • Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
  • MR Author ID: 1095300
  • Email:
  • Eric King-Wah Chu
  • Affiliation: School of Mathematics, Monash University, 9 Rainforest Walk, Victoria 3800, Australia
  • MR Author ID: 49125
  • Email:
  • Wen-Wei Lin
  • Affiliation: Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan
  • MR Author ID: 232126
  • Email:
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Math. Comp. 88 (2019), 2325-2350
  • MSC (2010): Primary 15A18, 65F15
  • DOI:
  • MathSciNet review: 3957895