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High Performance Solution of Skew-symmetric Eigenvalue Problems with Applications in Solving the Bethe-Salpeter Eigenvalue Problem
[article]

2020
*
arXiv
*
pre-print

We present

arXiv:1912.04062v2
fatcat:2mf6l6c7ffeltgy3x35rpjqj7a
*a*high-performance solver for dense skew-*symmetric*matrix*eigenvalue**problems*. ...*eigenvalue**problem*. ... It is always possible to solve*a*complex Hermitian*eigenvalue**problem*instead*of**a*skew-*symmetric*one. ...##
###
Solution of large, dense symmetric generalized eigenvalue problems using secondary storage

1988
*
ACM Transactions on Mathematical Software
*

*A*combination

*of*block Cholesky and block Householder transformations are used to reduce the

*problem*to

*a*

*symmetric*

*banded*eigenproblem whose

*eigenvalues*can be computed in central memory. ... Inverse iteration is applied to the

*banded*matrix to compute selected eigenvectors, which are then transformed back to eigenvectors

*of*the original

*problem*. ... COMPUTING THE

*EIGENVALUES*

*OF*THE

*BAND*MATRIX We have thus reduced the large out-

*of*-core

*problem*to

*a*standard in-core

*problem*: the computation

*of*the

*eigenvalues*

*of*

*a*

*symmetric*

*band*matrix. ...

##
###
On Tridiagonalizing and Diagonalizing Symmetric Matrices with Repeated Eigenvalues

1996
*
SIAM Journal on Matrix Analysis and Applications
*

*A*slight modi cation

*of*the usual orthogonal

*band*-

*reduction*algorithm allows us to reveal this structure, which then leads to potential parallelism in the form

*of*independent diagonal blocks. ... Such matrices arise, for example, in invariant subspace decomposition approaches to the

*symmetric*

*eigenvalue*

*problem*. ... The algorithm progresses through

*a*series

*of*

*band*

*reductions*, each

*band*

*reduction*stage forcing

*a*decoupling

*of*the

*band*matrix into independent subblocks. ...

##
###
A fast band–Krylov eigensolver for macromolecular functional motion simulation on multicore architectures and graphics processors

2016
*
Journal of Computational Physics
*

The method consists

doi:10.1016/j.jcp.2016.01.007
fatcat:6tac5detiram3lsboxetp6euka
*of*two stages, with the original*problem*first reduced into*a*simpler*band*-structured form by means*of**a*high-performance computeintensive procedure. ... The method consists*of*two stages, with the original*problem*first reduced into*a*simpler*band*-structured form by means*of**a*high-performance computeintensive procedure. ... Acknowledgements This work was supported by the CICYT projects TIN2011-23283 and TIN2014-53495-R*of*the MINECO and FEDER, and project P1-1B2013-20*of*the Fundació Caixa Castelló-Bancaixa and UJI. ...##
###
Page 687 of Mathematical Reviews Vol. , Issue 81B
[page]

1981
*
Mathematical Reviews
*

It starts with

*a*p-dimensional subspace, and computes an ortho- normal basis for the Krylov spaces*of**A*,*generated*from this starting subspace, in which*A*is represented by*a*2p+1*band*matrix, whose*eigenvalues*... Author’s summary: “*A**band*Lanczos algorithm for the iterative computation*of**eigenvalues*and eigenvectors*of**a*large sparse*symmetric*matrix is described and tested on numerical examples. ...##
###
Page 6901 of Mathematical Reviews Vol. , Issue 93m
[page]

1993
*
Mathematical Reviews
*

, Linda (1-BELL)
An algorithm for the

*banded**symmetric**generalized*matrix*eigenvalue**problem*. ... Summary: “This paper derives an algorithm for finding the eigen- values*of*the*symmetric**banded**generalized**eigenvalue**problem*Ax =ABx where*A*and B are nx n*symmetric*positive definite matrices*of*bandwidth ...##
###
Toward a High Performance Tile Divide and Conquer Algorithm for the Dense Symmetric Eigenvalue Problem

2012
*
SIAM Journal on Scientific Computing
*

Classical solvers for the dense

doi:10.1137/110823699
fatcat:celpm22sabhz7h3gtuptbq7jpy
*symmetric**eigenvalue**problem*suffer from the first step, which involves*a**reduction*to tridiagonal form that is dominated by the cost*of*accessing memory during the panel ... The standard*symmetric**eigenvalue*solver approach. The common way*of*stating the*eigenvalue**problem*for*a**symmetric*dense matrix is λ an*eigenvalue*, and x the corresponding eigenvector. ... The authors would like to thank the two anonymous reviewers for their insightful comments, which greatly helped to improve the quality*of*this article. ...##
###
List of Publications of James H. Wilkinson

1987
*
Linear Algebra and its Applications
*

Peters)

doi:10.1016/0024-3795(87)90099-1
fatcat:ekk6py45ijayzazmdktvtfox3a
*Eigenvalues**of*Ax = XBx with*band**symmetric**A*and I?, 68. (with G. ... Martin) Similarity*reduction**of**a**general*matrix to Hessenberg form, Numerische Math. 12:349 (1968). (with R. S. ...##
###
$O( n^2 )$ Reduction Algorithms for the Construction of a Band Matrix from Spectral Data

1991
*
SIAM Journal on Matrix Analysis and Applications
*

These methods can also be used in the second phase

doi:10.1137/0612030
fatcat:ndqb2cmx3jaufipvmdnz7v37ne
*of*the construction*of**a**band*matrix from the interlacing*eigenvalues*as described in Linear Algebra Appl., 40 1981 ), pp. 79-87 ]. ... Efficient rotation patterns are presented that provide stable O(n2) algorithms for the construction ofa real*symmetric**band*matrix having specified*eigenvalues*and first p components ofits normalized eigenvectors ... Let*A*be*a*real*symmetric*(2p + )-*band*matrix*of*order n, and let Ak denote the trailing principal submatrix ofA*A*,*of*order k. ...##
###
Comparison of eigensolvers for symmetric band matrices

2014
*
Science of Computer Programming
*

the BTF method is fastest.

doi:10.1016/j.scico.2014.01.005
pmid:26594079
pmcid:PMC4617464
fatcat:es3hwc34dzbqzbosird5p65k4u
*a*r t i c l e i n f o*a*b s t r*a*c t We compare different algorithms for computing*eigenvalues*and eigenvectors*of**a**symmetric**band*matrix across*a*wide range*of*synthetic ... h i g h l i g h t s • Single core performance*of**symmetric**band*eigensolvers is compared experimentally. • Tridiagonalizing*a**band*matrix produces subnormal numbers and degrades performance. • Methods ... Conclusions We performed*a*detailed experimental runtime comparison*of*five different methods for computing eigenpairs*of*randomly*generated**symmetric**band*matrices with different*eigenvalue*distributions ...##
###
A hybrid Hermitian general eigenvalue solver
[article]

2012
*
arXiv
*
pre-print

medium sized Hermitian

arXiv:1207.1773v1
fatcat:2m7iyy4fhjf6rlipewwiartbnm
*generalized**eigenvalue**problems*must be solved many times. ... Addressing these demands, we implemented*a*novel Hermitian*general*eigensolver algorithm. This algorithm is based on*a*standard*eigenvalue*solver, and existing algorithms can be used. ... The authors would also like to thank the National Science Foundation, the Department*of*Energy, NVIDIA, and the MathWorks for this research effort. ...##
###
Page 244 of Mathematical Reviews Vol. 57, Issue 1
[page]

1979
*
Mathematical Reviews
*

Simultaneous determination

*of*dominant*eigenvalues*and initia tanh ey ° » * eigenvectors*of**a**symmetric*sparse matrix; 6. The*generalized*| “(Fn otish translation: Siberian Math. ...*Reduction**of*the*symmetric*eigenproblem Ax=ABx and related applica Springer-Verlag, New York- Heidelberg, 1971. ix +439 pp.*problems*to standard form (pp. 303-314); B. N. ...##
###
A novel hybrid CPU–GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks

2013
*
The international journal of high performance computing applications
*

applications, where mediumsized

doi:10.1177/1094342013502097
fatcat:ytkh7zbpfzhjrms4xif3vqso2u
*generalized**eigenvalue**problems*must be solved many times. ... These*eigenvalue**problems*are too small to effectively solve on distributed systems, but can benefit from the massive computing power concentrated on*a*single node, hybrid CPU-GPU system. ... ACKNOWLEDGMENTS The authors would like to thank the National Science Foundation, the Department*of*Energy, NVIDIA, and MathWorks ...##
###
Condensed forms for the symmetric eigenvalue problem on multi-threaded architectures

2010
*
Concurrency and Computation
*

*of*the

*symmetric*

*eigenvalue*

*problem*, on generalpurpose multi-core processors. ... We investigate the performance

*of*the routines in LAPACK and the Successive

*Band*

*Reduction*(SBR) toolbox for the

*reduction*

*of*

*a*dense matrix to tridiagonal form,

*a*crucial preprocessing stage in the solution ... in the solution

*of*large

*symmetric*

*eigenvalue*

*problems*. ...

##
###
Efficient Reduction Algorithms for Banded Symmetric Generalized Eigenproblems via Sequentially Semiseparable (SSS) Matrices

2022
*
Mathematics
*

In this paper,

doi:10.3390/math10101676
fatcat:ggwvgymgwbcu3ev2e64w64jj6a
*a*novel algorithm is proposed for reducing*a**banded**symmetric**generalized**eigenvalue**problem*to*a**banded**symmetric*standard*eigenvalue**problem*, based on the sequentially semiseparable (SSS ... It is the first time that the SSS matrix techniques are used in such*eigenvalue**problems*. ... Conflicts*of*Interest: The authors declare no conflict*of*interest. ...
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