A parallel structured divide-and-conquer algorithm for symmetric tridiagonal eigenvalue problems

In this paper, a parallel structured divide-and-conquer (PSDC) eigensolver is proposed for symmetric tridiagonal matrices based on ScaLAPACK and a parallel structured matrix multiplication algorithm, called PSMMA. Computing the eigenvectors via matrix-matrix multiplications is the most computational...

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Vydané v:IEEE transactions on parallel and distributed systems Ročník 32; číslo 2; s. 1
Hlavní autori: Liao, Xia, Li, Shengguo, Lu, Yutong, Roman, Jose E. E.
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York IEEE 01.02.2021
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Predmet:
Algorithms
Cauchy-like matrix
Computation
Divide-and-conquer
Eigenvalues
Eigenvectors
Mathematical analysis
Matrix methods
Multiplication
PSMMA
PUMMA Algorithm
Rank-structured matrix
ScaLAPACK
Structured matrices
ISSN:1045-9219, 1558-2183
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Shrnutí:In this paper, a parallel structured divide-and-conquer (PSDC) eigensolver is proposed for symmetric tridiagonal matrices based on ScaLAPACK and a parallel structured matrix multiplication algorithm, called PSMMA. Computing the eigenvectors via matrix-matrix multiplications is the most computationally expensive part of the divide-and-conquer algorithm, and one of the matrices involved in such multiplications is a rank-structured Cauchy-like matrix. By exploiting this particular property, PSMMA constructs the local matrices by using generators of Cauchy-like matrices without any communication, and further reduces the computation costs by using a structured low-rank approximation algorithm. Thus, both the communication and computation costs are reduced. Experimental results show that both PSMMA and PSDC are highly scalable and scale to 4096 processes at least. PSDC has better scalability than PHDC that was proposed in [J. Comput. Appl. Math. 344 (2018) 512--520] and only scaled to 300 processes for the same matrices. Comparing with PDSTEDC in ScaLAPACK, PSDC is always faster and achieves 1.4 x--1.6x speedup for some matrices with few deflations. PSDC is also comparable with ELPA, with PSDC being faster than ELPA when using few processes and a little slower when using many processes.
Bibliografia:ObjectType-Article-1
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ISSN:1045-9219
1558-2183
DOI:10.1109/TPDS.2020.3019471