## Doubling algorithm for the discretized Bethe-Salpeter eigenvalue problem

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Zhen-Chen Guo, Eric King-Wah Chu and Wen-Wei Lin
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## 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

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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: guozhch06@gmail.com
**Eric King-Wah Chu**- Affiliation: School of Mathematics, Monash University, 9 Rainforest Walk, Victoria 3800, Australia
- MR Author ID: 49125
- Email: eric.chu@monash.edu
**Wen-Wei Lin**- Affiliation: Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan
- MR Author ID: 232126
- Email: wwlin@math.nctu.edu.tw
- 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: https://doi.org/10.1090/mcom/3398
- MathSciNet review: 3957895