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A parallel structured banded DC algorithm for symmetric eigenvalue problems

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Abstract

In this paper, a novel parallel structured divide-and-conquer (DC) algorithm is proposed for symmetric banded eigenvalue problems, denoted by PBSDC, which modifies the classical parallel banded DC (PBDC) algorithm by reducing its computational cost. The main tool that PBSDC uses is a parallel structured matrix multiplication algorithm (PSMMA), which can be much faster than the general dense matrix multiplication ScaLAPACK routine PDGEMM. Numerous experiments have been performed on Tianhe-2 supercomputer to compare PBSDC with PBDC and ELPA. For matrices with few deflations, PBSDC can be much faster than PBDC since computations are saved. For matrices with many deflations and/or small bandwidths, PBSDC can be faster than the tridiagonalization-based DC implemented in LAPACK and ELPA. However, PBSDC would become slower than ELPA for matrices with relatively large bandwidths.

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Acknowledgements

The authors would like to thank the referees for their valuable comments. This work is supported in part by NSFC (No. 2021YFB0300101, 62073333, 61902411, 62032023, 12002382, 11275269, 42104078), 173 Program of China (2020-JCJQ-ZD-029), Open Research Fund from State Key Laboratory of High Performance Computing of China (HPCL) (No. 202101-01), Guangdong Natural Science Foundation (2018B030312002), and the Program for Guangdong Introducing Innovative and Entrepreneurial Teams under Grant (No. 2016ZT06D211). Jose E. Roman is supported by the Spanish Agencia Estatal de Investigación (AEI) under project SLEPc-DA (PID2019-107379RB-I00). On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Authors and Affiliations

  1. College of Computer Science, National University of Defense Technology, Changsha, 410073, China

    Shengguo Li & Xia Liao

  2. National Supercomputer Center in Guangzhou, and the School of Data and Computer Science, Sun Yatsen University, Guangzhou, 510006, China

    Yutong Lu

  3. D. Sistemes Informàtics i Computació, Universitat Politècnica de València, Camí de Vera s/n, València, 46022, Spain

    Jose E. Roman

  4. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan, 411105, China

    Xiaoqiang Yue

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Correspondence to Xia Liao.

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Li, S., Liao, X., Lu, Y. et al. A parallel structured banded DC algorithm for symmetric eigenvalue problems. CCF Trans. HPC 5, 116–128 (2023). https://doi.org/10.1007/s42514-022-00117-9

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