Abstract
In this paper, a scalable parallel eigen-solver called parallel absorbing diagonal algorithm (parallel ADA) is proposed. This algorithm is of significantly improved parallel complexity when compared to traditional parallel symmetric eigen-solver algorithms. The scalability-bottleneck of the traditional eigen-solvers is the tri-diagonalization of a matrix via Householder/Givens transforms. The basic idea of ADA is to avoid the tri-diagonalization completely by iteratively and alternatingly applying two kind of operations in multi-scales: diagonal attaction operations and diagonal absorption operations. In a diagonal attraction operation, it attracts the off-diagonal entries to make the entries near to the diagonal larger in magnitude than the entries far away from the diagonal. In a diagonal absoprtion operation, it absorbs the nearer nonzero entries into the diagonal. Theories of ADA has been established in another paper of ours that for any \(\epsilon >0\), there exists a constant \(C=C\left( \epsilon \right) \), such that within C rounds of iterations, the relative error of the algorithm will be reduced to below \(\epsilon \). Parallel complexity of ADA is analyzed in this paper to reveal its qualitative improvement of scalability.
Supported by the National Natural Science Foundation of P. R. China (No. 61672296, No. 61602261), Major Natural Science Research Projects in Colleges and Universities of Jiangsu Province (No. 18KJA520008).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bientinesi, P., Dhillon, I.S., Van de Geijn, R.A.: A parallel eigensolver for dense symmetric matrices based on multiple relatively robust representations. SIAM J. Sci. Comput. 27(1), 43–66 (2003)
Cuppen, J.J.M.: A divide and conquer method for the symmetric tridiagonal eigenproblem. Numer. Math. 36(2), 177–195 (1980)
Dhillon, I., Fann, G., Parlett, B.: Application of a new algorithm for the symmetric eigenproblem to computational quantum chemistry. In: Eighth SIAM Conference on Parallel Processing for Scientific Computing SIAM (1999)
Dhillon, I.S.: A new \(O(n^2)\) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem. Ph.D. thesis, University of California at Berkeley, October 1997
Godunov, S.K., Antonov, A.G., Kiriljuk, O.P., Kostin, V.I.: Guaranteed Accuracy in Numerical Linear Algebra. Kluwer Academic Publishers Group, Dordrecht (1993)
Golub, G.H., Loan, C.V.: Matrix Computations, 2nd edn. The Johns Hopkins University Press, Baltimore (1989)
Hegland, M., Kahn, M., Osborne, M.: A parallel algorithm for the reduction to tridiagonal form for eigendecomposition. SIAM J. Sci. Comput. 21(3), 987–1005 (1970)
Luszczek, P., Ltaief, H., Dongarra, J.: Two-stage tridiagonal reduction for dense symmetric matrices using tile algorithms on multicore architectures, pp. 944–955 (2011)
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Wu, J., Zheng, H., Li, P. (2020). Parallel Absorbing Diagonal Algorithm: A Scalable Iterative Parallel Fast Eigen-Solver for Symmetric Matrices. In: He, J., et al. Data Science. ICDS 2019. Communications in Computer and Information Science, vol 1179. Springer, Singapore. https://doi.org/10.1007/978-981-15-2810-1_61
Download citation
DOI: https://doi.org/10.1007/978-981-15-2810-1_61
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-2809-5
Online ISBN: 978-981-15-2810-1
eBook Packages: Computer ScienceComputer Science (R0)