Hybrid Algorithms for Solving the Algebraic Eigenvalue Problem with Sparse Matrices
Hybrid algorithms for solving the partial generalized eigenvalue problem for symmetric positive definite sparse matrices of different structures by hybrid computers with graphic processors are proposed, coefficients for the efficiency of the algorithms are obtained, and approbation of the developed...
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| Vydáno v: | Cybernetics and systems analysis Ročník 53; číslo 6; s. 937 - 949 |
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| Jazyk: | angličtina |
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01.11.2017
Springer Springer Nature B.V |
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| ISSN: | 1060-0396, 1573-8337 |
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| Abstract | Hybrid algorithms for solving the partial generalized eigenvalue problem for symmetric positive definite sparse matrices of different structures by hybrid computers with graphic processors are proposed, coefficients for the efficiency of the algorithms are obtained, and approbation of the developed algorithms for test and practical problems is carried out. |
|---|---|
| AbstractList | Hybrid algorithms for solving the partial generalized eigenvalue problem for symmetric positive definite sparse matrices of different structures by hybrid computers with graphic processors are proposed, coefficients for the efficiency of the algorithms are obtained, and approbation of the developed algorithms for test and practical problems is carried out. Hybrid algorithms for solving the partial generalized eigenvalue problem for symmetric positive definite sparse matrices of different structures by hybrid computers with graphic processors are proposed, coefficients for the efficiency of the algorithms are obtained, and approbation of the developed algorithms for test and practical problems is carried out. Keywords: algebraic eigenvalue problem, computer of hybrid architecture, hybrid algorithm, subspace iteration method, conjugate gradient methods, efficiency of parallel algorithms. |
| Audience | Academic |
| Author | Khimich, A. N. Chistyakov, O. V. Popov, A. V. |
| Author_xml | – sequence: 1 givenname: A. N. surname: Khimich fullname: Khimich, A. N. email: khimich505@gmail.com organization: V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine – sequence: 2 givenname: A. V. surname: Popov fullname: Popov, A. V. organization: V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine – sequence: 3 givenname: O. V. surname: Chistyakov fullname: Chistyakov, O. V. organization: V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine |
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| Keywords | computer of hybrid architecture conjugate gradient methods algebraic eigenvalue problem efficiency of parallel algorithms hybrid algorithm subspace iteration method |
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| References | B. N. Parlett, The Symmetric Eigenvalue Problem (Classics in Applied Mathematics), Book 20, Society for Industrial and Applied Mathematics (1987). The SuiteSparse Matrix Collection (2015), https://cise.ufl.edu/research/sparse/matrices/. CUDA library (2015), https://nvidia.com/. O. M. Khimich and O. V. Chistyakov, “Parallel one-step iterative methods to solve the algebraic eigenvalue problem for sparse matrices,” Komp. Matem., No. 2, 81–88 (2014). LIS (2015), http://www.ssisc.org/lis/. PrikazchikovVGPrototypes of iterative processes in the eigenvalue problemDiff. Uravneniya1980169168816970443.65021 SLEPc (2015), http://slepc.upv.es/. A. N. Khimich, A. V. Popov, A. Yu. Baranov, and O. V. Chistyakov, “Hybrid algorithm to solve eigenvalue problems for band matrices,” Teoriya Optym. Rishen’, 86–94 (2016). O. V. Chistyakov, “Special features of software development to solve eigenvalue problems with sparse matrices by hybrid computers,” Komp. Matematika, No. 1, 75–84 (2015). O. V. Popov, “Efficiency analysis of parallel algorithms for computers of hybrid architecture,” in: Trans. Intern. Sci. School-Seminar “Optimization of Computations (POO XLII)” (2015). O. M. Khimich and V. A. Sidoruk, “Hybrid algorithm to solve linear systems with sparse matrices based on the block LLT method,” Komp. Matematika, No. 1, 67–74 (2015). NetLib (2015), http://www.netlib.org/. MikhalevichVSMolchanovINSergienkoIVNumerical methods for the ES Multiprocessor Computing Complex [in Russian]1986MoscowVVIA im. N. E. Zhukovskogo MATLAB (2017), https://mathworks.com/. S. Pissanetzky, Sparse Matrix Tehnology, Elsevier, Academic Press (1984). MAGMA (2015), http://icl.cs.utk.edu/magma/. G. V. Savinov, “Convergence analysis of one generalized conjugate gradient method to find extremum eigenvalues of the matrix,” Trans. Sci. Seminar LOMI, No. 111, 145–150 (1981). DekretVAZelenskiiVSBystrovVMNumerical Stability Analysis of a Layered Composite Material under Monoaxial Compression of the Filler [in Russian]2008KyivNaukova Dumka Intel Maths Kernel Library (2015), https://software.intel.com/en-us/mkl. KhimichANMolchanovINPopovAVChistyakovaTVYakovlevMFParallel Algorithms to Solve Problems in Calculus Mathematics [in Russian]2008KyivNaukova Dumka BaranovAYPopovAVSlobodyanYEKhimichANMathematical modeling of building constructions using hybrid computing systemsJ. Autom. Inform. Sci.2017497183210.1615/JAutomatInfScien.v49.i7.20 O. M. Khimich, 2. Ì. Molchanov, O. V. Popov, et al., “Smart personal supercomputer to solve scientific and technical problems,” Nauka ta Innovatsii, Vol. 12, No. 5, 17–31 (2016). 9996_CR9 9996_CR8 9996_CR7 9996_CR6 9996_CR5 9996_CR4 9996_CR3 VS Mikhalevich (9996_CR12) 1986 VA Dekret (9996_CR18) 2008 9996_CR20 9996_CR1 AY Baranov (9996_CR13) 2017; 49 9996_CR21 9996_CR11 9996_CR22 9996_CR16 9996_CR17 9996_CR14 9996_CR15 9996_CR19 VG Prikazchikov (9996_CR2) 1980; 16 AN Khimich (9996_CR10) 2008 |
| References_xml | – reference: S. Pissanetzky, Sparse Matrix Tehnology, Elsevier, Academic Press (1984). – reference: KhimichANMolchanovINPopovAVChistyakovaTVYakovlevMFParallel Algorithms to Solve Problems in Calculus Mathematics [in Russian]2008KyivNaukova Dumka – reference: O. V. Chistyakov, “Special features of software development to solve eigenvalue problems with sparse matrices by hybrid computers,” Komp. Matematika, No. 1, 75–84 (2015). – reference: O. M. Khimich and V. A. Sidoruk, “Hybrid algorithm to solve linear systems with sparse matrices based on the block LLT method,” Komp. Matematika, No. 1, 67–74 (2015). – reference: PrikazchikovVGPrototypes of iterative processes in the eigenvalue problemDiff. Uravneniya1980169168816970443.65021 – reference: SLEPc (2015), http://slepc.upv.es/. – reference: A. N. Khimich, A. V. Popov, A. Yu. Baranov, and O. V. Chistyakov, “Hybrid algorithm to solve eigenvalue problems for band matrices,” Teoriya Optym. Rishen’, 86–94 (2016). – reference: DekretVAZelenskiiVSBystrovVMNumerical Stability Analysis of a Layered Composite Material under Monoaxial Compression of the Filler [in Russian]2008KyivNaukova Dumka – reference: Intel Maths Kernel Library (2015), https://software.intel.com/en-us/mkl. – reference: BaranovAYPopovAVSlobodyanYEKhimichANMathematical modeling of building constructions using hybrid computing systemsJ. Autom. Inform. Sci.2017497183210.1615/JAutomatInfScien.v49.i7.20 – reference: NetLib (2015), http://www.netlib.org/. – reference: O. M. Khimich and O. V. Chistyakov, “Parallel one-step iterative methods to solve the algebraic eigenvalue problem for sparse matrices,” Komp. Matem., No. 2, 81–88 (2014). – reference: LIS (2015), http://www.ssisc.org/lis/. – reference: The SuiteSparse Matrix Collection (2015), https://cise.ufl.edu/research/sparse/matrices/. – reference: MAGMA (2015), http://icl.cs.utk.edu/magma/. – reference: CUDA library (2015), https://nvidia.com/. – reference: O. M. Khimich, 2. Ì. Molchanov, O. V. Popov, et al., “Smart personal supercomputer to solve scientific and technical problems,” Nauka ta Innovatsii, Vol. 12, No. 5, 17–31 (2016). – reference: G. V. Savinov, “Convergence analysis of one generalized conjugate gradient method to find extremum eigenvalues of the matrix,” Trans. Sci. Seminar LOMI, No. 111, 145–150 (1981). – reference: MikhalevichVSMolchanovINSergienkoIVNumerical methods for the ES Multiprocessor Computing Complex [in Russian]1986MoscowVVIA im. N. E. Zhukovskogo – reference: O. V. Popov, “Efficiency analysis of parallel algorithms for computers of hybrid architecture,” in: Trans. Intern. Sci. School-Seminar “Optimization of Computations (POO XLII)” (2015). – reference: MATLAB (2017), https://mathworks.com/. – reference: B. N. Parlett, The Symmetric Eigenvalue Problem (Classics in Applied Mathematics), Book 20, Society for Industrial and Applied Mathematics (1987). – ident: 9996_CR5 – volume-title: Numerical Stability Analysis of a Layered Composite Material under Monoaxial Compression of the Filler [in Russian] year: 2008 ident: 9996_CR18 – ident: 9996_CR6 – ident: 9996_CR3 – ident: 9996_CR4 – ident: 9996_CR8 – ident: 9996_CR7 – ident: 9996_CR9 – ident: 9996_CR17 – ident: 9996_CR16 – ident: 9996_CR19 – ident: 9996_CR14 – ident: 9996_CR15 – ident: 9996_CR1 doi: 10.1016/B978-0-12-557580-5.50012-0 – volume: 16 start-page: 1688 issue: 9 year: 1980 ident: 9996_CR2 publication-title: Diff. Uravneniya – ident: 9996_CR11 – volume: 49 start-page: 18 issue: 7 year: 2017 ident: 9996_CR13 publication-title: J. Autom. Inform. Sci. doi: 10.1615/JAutomatInfScien.v49.i7.20 – ident: 9996_CR22 – ident: 9996_CR20 – ident: 9996_CR21 – volume-title: Parallel Algorithms to Solve Problems in Calculus Mathematics [in Russian] year: 2008 ident: 9996_CR10 – volume-title: Numerical methods for the ES Multiprocessor Computing Complex [in Russian] year: 1986 ident: 9996_CR12 |
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| SubjectTerms | Algorithms Analysis Artificial Intelligence Control Eigenvalues Hybrid computers Mathematical analysis Mathematics Mathematics and Statistics Matrix methods Processor Architectures Processors Software Engineering/Programming and Operating Systems Software–Hardware Systems Sparse matrices Sparsity Systems Theory |
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