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

Carolin Penke, Andreas Marek, Christian Vorwerk, Claudia Draxl, Peter Benner
2020 arXiv   pre-print
We present 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.  ... 
arXiv:1912.04062v2 fatcat:2mf6l6c7ffeltgy3x35rpjqj7a

Solution of large, dense symmetric generalized eigenvalue problems using secondary storage

Roger G. Grimes, Horst D. Simon
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.  ... 
doi:10.1145/44128.44130 fatcat:x3mmxvycpjgejf3rvdmqirdtpm

On Tridiagonalizing and Diagonalizing Symmetric Matrices with Repeated Eigenvalues

Christian H. Bischof, Xiaobai Sun
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.  ... 
doi:10.1137/s0895479892227608 fatcat:j53hrhmbhbb73ahsaqjomflrty

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

José I. Aliaga, Pedro Alonso, José M. Badía, Pablo Chacón, Davor Davidović, José R. López-Blanco, Enrique S. Quintana-Ortí
2016 Journal of Computational Physics  
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.  ...  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.  ... 
doi:10.1016/j.jcp.2016.01.007 fatcat:6tac5detiram3lsboxetp6euka

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

Azzam Haidar, Hatem Ltaief, Jack Dongarra
2012 SIAM Journal on Scientific Computing  
Classical solvers for the dense 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.  ... 
doi:10.1137/110823699 fatcat:celpm22sabhz7h3gtuptbq7jpy

List of Publications of James H. Wilkinson

1987 Linear Algebra and its Applications  
Peters) 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.  ... 
doi:10.1016/0024-3795(87)90099-1 fatcat:ekk6py45ijayzazmdktvtfox3a

$O( n^2 )$ Reduction Algorithms for the Construction of a Band Matrix from Spectral Data

Gregory S. Ammar, William B. Gragg
1991 SIAM Journal on Matrix Analysis and Applications  
These methods can also be used in the second phase 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.  ... 
doi:10.1137/0612030 fatcat:ndqb2cmx3jaufipvmdnz7v37ne

Comparison of eigensolvers for symmetric band matrices

Michael Moldaschl, Wilfried N. Gansterer
2014 Science of Computer Programming  
the BTF method is fastest. 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  ... 
doi:10.1016/j.scico.2014.01.005 pmid:26594079 pmcid:PMC4617464 fatcat:es3hwc34dzbqzbosird5p65k4u

A hybrid Hermitian general eigenvalue solver [article]

Raffaele Solcà, Thomas C. Schulthess, Azzam Haidar, Stanimire Tomov, Ichitaro Yamazaki, Jack Dongarra
2012 arXiv   pre-print
medium sized Hermitian 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.  ... 
arXiv:1207.1773v1 fatcat:2m7iyy4fhjf6rlipewwiartbnm

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

Azzam Haidar, Stanimire Tomov, Jack Dongarra, Raffaele Solcà, Thomas Schulthess
2013 The international journal of high performance computing applications  
applications, where mediumsized 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  ... 
doi:10.1177/1094342013502097 fatcat:ytkh7zbpfzhjrms4xif3vqso2u

Condensed forms for the symmetric eigenvalue problem on multi-threaded architectures

Paolo Bientinesi, Francisco D. Igual, Daniel Kressner, Matthias Petschow, Enrique S. Quintana-Ortí
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.  ... 
doi:10.1002/cpe.1680 fatcat:37iivnxs3jberfrflansxtv7s4

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

Fan Yuan, Shengguo Li, Hao Jiang, Hongxia Wang, Cheng Chen, Lei Du, Bo Yang
2022 Mathematics  
In this paper, 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.  ... 
doi:10.3390/math10101676 fatcat:ggwvgymgwbcu3ev2e64w64jj6a
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