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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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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