메뉴 건너뛰기




Volumn 123, Issue 1-2, 2000, Pages 515-530

The impact of high-performance computing in the solution of linear systems: Trends and problems

Author keywords

65F05; 65F50; Direct methods; Matrix factorization; Multifrontal methods; Parallelism; Sparse matrices

Indexed keywords

ALGORITHMS; COMPUTER SIMULATION; COMPUTER SOFTWARE; DIGITAL ARITHMETIC; ITERATIVE METHODS; LEAST SQUARES APPROXIMATIONS; MATRIX ALGEBRA; PARALLEL PROCESSING SYSTEMS;

EID: 0034319806     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(00)00401-5     Document Type: Article
Times cited : (23)

References (89)
  • 2
    • 0029375720 scopus 로고
    • Parallel sparse matrix solution and performance
    • Alaghband G. Parallel sparse matrix solution and performance. Parallel Comput. 21(9):1995;1407-1430.
    • (1995) Parallel Comput. , vol.21 , Issue.9 , pp. 1407-1430
    • Alaghband, G.1
  • 4
    • 0001877684 scopus 로고
    • Optimal parallel solution of sparse triangular systems
    • Alvarado F.L., Schreiber R. Optimal parallel solution of sparse triangular systems. SIAM J. Sci. Comput. 14:1993;446-460.
    • (1993) SIAM J. Sci. Comput. , vol.14 , pp. 446-460
    • Alvarado, F.L.1    Schreiber, R.2
  • 10
    • 22444456364 scopus 로고    scopus 로고
    • Multifrontal solvers within the PARASOL environment
    • B. Kagstrom, J. Dongarra, E. Elmroth, & J. Wasniewski. Applied Parallel Computing, PARA'98. Berlin: Springer
    • Amestoy P.R., Duff I.S., L'Excellent J.-Y. Multifrontal solvers within the PARASOL environment. Kagstrom B., Dongarra J., Elmroth E., Wasniewski J. Applied Parallel Computing, PARA'98. Lecture Notes in Computer Science. Vol. 1541:1998;7-11 Springer, Berlin.
    • (1998) Lecture Notes in Computer Science , vol.1541 , pp. 7-11
    • Amestoy, P.R.1    Duff, I.S.2    L'Excellent, J.-Y.3
  • 12
    • 21344466930 scopus 로고    scopus 로고
    • Multifrontal QR factorization in a multiprocessor environment
    • Amestoy P.R., Duff I.S., Puglisi C. Multifrontal QR factorization in a multiprocessor environment. Numer. Linear Algebra Appl. 3(4):1996;275-300.
    • (1996) Numer. Linear Algebra Appl. , vol.3 , Issue.4 , pp. 275-300
    • Amestoy, P.R.1    Duff, I.S.2    Puglisi, C.3
  • 15
    • 0001750069 scopus 로고
    • Solving sparse triangular systems on parallel computers
    • Anderson E.C., Saad Y. Solving sparse triangular systems on parallel computers. Internat. J. High Speed Comput. 1:1989;73-95.
    • (1989) Internat. J. High Speed Comput. , vol.1 , pp. 73-95
    • Anderson, E.C.1    Saad, Y.2
  • 17
    • 0032397943 scopus 로고    scopus 로고
    • Robust ordering of sparse matrices using multisection
    • Ashcraft C., Liu J.W.H. Robust ordering of sparse matrices using multisection. SIAM J. Matrix Anal. Appl. 19:1998;816-832.
    • (1998) SIAM J. Matrix Anal. Appl. , vol.19 , pp. 816-832
    • Ashcraft, C.1    Liu, J.W.H.2
  • 24
    • 0031541111 scopus 로고    scopus 로고
    • An unsymmetric-pattern multifrontal method for sparse LU factorization
    • Davis T.A., Duff I.S. An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM J. Matrix Anal. Appl. 18(1):1997;140-158.
    • (1997) SIAM J. Matrix Anal. Appl. , vol.18 , Issue.1 , pp. 140-158
    • Davis, T.A.1    Duff, I.S.2
  • 25
    • 0009170244 scopus 로고
    • A nondeterministic parallel algorithm for general unsymmetric sparse LU factorization
    • Davis T.A., Yew P.C. A nondeterministic parallel algorithm for general unsymmetric sparse LU factorization. SIAM J. Matrix Anal. Appl. 11:1990;383-402.
    • (1990) SIAM J. Matrix Anal. Appl. , vol.11 , pp. 383-402
    • Davis, T.A.1    Yew, P.C.2
  • 26
    • 0004295534 scopus 로고
    • Technical Report, Computer Science Division, U.C. Berkeley, Berkeley, California, February (available from netlib)
    • J.W. Demmel, J.R. Gilbert, X.S. Li, SuperLU users' guide, Technical Report, Computer Science Division, U.C. Berkeley, Berkeley, California, February 1995. (available from netlib).
    • (1995) SuperLU Users' Guide
    • Demmel, J.W.1    Gilbert, J.R.2    Li, X.S.3
  • 28
    • 0040352754 scopus 로고    scopus 로고
    • Technical Report 99-2, Institute for Computer Applications in Science and Engineering (ICASE), MS 132C, NASA Langley Research Center, Hampton, VA 23681-0001, USA
    • F. Dobrian, G. Kumfert, A. Pothen, Object-oriented design for sparse direct solvers, Technical Report 99-2, Institute for Computer Applications in Science and Engineering (ICASE), MS 132C, NASA Langley Research Center, Hampton, VA 23681-0001, USA, 1999.
    • (1999) Object-oriented Design for Sparse Direct Solvers
    • Dobrian, F.1    Kumfert, G.2    Pothen, A.3
  • 29
    • 0003533609 scopus 로고    scopus 로고
    • Technical Report CS-89-85, University of Tennessee, Knoxville, Tennessee, Updated version at Web address
    • J.J. Dongarra, Performance of various computers using standard linear algebra software, Technical Report CS-89-85, University of Tennessee, Knoxville, Tennessee, 1999. Updated version at Web address http: // www.netlib.org / benchmark / performance.ps .
    • (1999) Performance of Various Computers Using Standard Linear Algebra Software
    • Dongarra, J.J.1
  • 32
    • 0034320008 scopus 로고    scopus 로고
    • Numerical linear algebra algorithms and software, this issue
    • J.J. Dongarra, V. Eijkhout, Numerical linear algebra algorithms and software, this issue, J. Comput. Appl. Math. 123 (2000) 489-514.
    • (2000) J. Comput. Appl. Math. , vol.123 , pp. 489-514
    • Dongarra, J.J.1    Eijkhout, V.2
  • 35
    • 0013136934 scopus 로고
    • Full matrix techniques in sparse Gaussian elimination
    • G.A. Watson. Numerical Analysis Proceedings, Dundee 1981. Berlin: Springer
    • Duff I.S. Full matrix techniques in sparse Gaussian elimination. Watson G.A. Numerical Analysis Proceedings, Dundee 1981. Lecture Notes in Mathematics. Vol. 912:1981;71-84 Springer, Berlin.
    • (1981) Lecture Notes in Mathematics , vol.912 , pp. 71-84
    • Duff, I.S.1
  • 36
    • 0001191910 scopus 로고
    • Design features of a frontal code for solving sparse unsymmetric linear systems out-of-core
    • Duff I.S. Design features of a frontal code for solving sparse unsymmetric linear systems out-of-core. SIAM J. Sci. Statist. Comput. 5:1984;270-280.
    • (1984) SIAM J. Sci. Statist. Comput. , vol.5 , pp. 270-280
    • Duff, I.S.1
  • 37
    • 0022754738 scopus 로고
    • Parallel implementation of multifrontal schemes
    • Duff I.S. Parallel implementation of multifrontal schemes. Parallel Comput. 3:1986;193-204.
    • (1986) Parallel Comput. , vol.3 , pp. 193-204
    • Duff, I.S.1
  • 38
    • 0024720802 scopus 로고
    • Multiprocessing a sparse matrix code on the Alliant FX / 8
    • Duff I.S. Multiprocessing a sparse matrix code on the Alliant FX. / 8 J. Comput. Appl. Math. 27:1989;229-239.
    • (1989) J. Comput. Appl. Math. , vol.27 , pp. 229-239
    • Duff, I.S.1
  • 40
    • 0003452796 scopus 로고    scopus 로고
    • Technical Report RAL-TR-97-031, Rutherford Appleton Laboratory, Also Technical Report ISSTECH-97-017 from Boeing Information & Support Services, Seattle and Report TR / PA / 97 / 36 from CERFACS, Toulouse
    • I.S. Duff, R.G. Grimes, J.G. Lewis, The Rutherford-Boeing sparse matrix collection, Technical Report RAL-TR-97-031, Rutherford Appleton Laboratory, 1997. Also Technical Report ISSTECH-97-017 from Boeing Information & Support Services, Seattle and Report TR / PA / 97 / 36 from CERFACS, Toulouse.
    • (1997) The Rutherford-Boeing Sparse Matrix Collection
    • Duff, I.S.1    Grimes, R.G.2    Lewis, J.G.3
  • 41
    • 0033242220 scopus 로고    scopus 로고
    • The design and use of algorithms for permuting large entries to the diagonal of sparse matrices
    • Duff I.S., Koster J. The design and use of algorithms for permuting large entries to the diagonal of sparse matrices. SIAM J. Matrix Anal. Appl. 20(4):1999;889-901.
    • (1999) SIAM J. Matrix Anal. Appl. , vol.20 , Issue.4 , pp. 889-901
    • Duff, I.S.1    Koster, J.2
  • 42
    • 0004171315 scopus 로고    scopus 로고
    • Technical Report RAL-TR-1999-030, Rutherford Appleton Laboratory, Also appeared as Report TR / PA / 99 / 13, CERFACS, Toulouse, France
    • I.S. Duff, J. Koster, On algorithms for permuting large entries to the diagonal of a sparse matrix, Technical Report RAL-TR-1999-030, Rutherford Appleton Laboratory, 1999. Also appeared as Report TR / PA / 99 / 13, CERFACS, Toulouse, France.
    • (1999) On Algorithms for Permuting Large Entries to the Diagonal of a Sparse Matrix
    • Duff, I.S.1    Koster, J.2
  • 43
    • 0020822138 scopus 로고
    • The multifrontal solution of indefinite sparse symmetric linear systems
    • Duff I.S., Reid J.K. The multifrontal solution of indefinite sparse symmetric linear systems. ACM Trans. Math. Software. 9:1983;302-325.
    • (1983) ACM Trans. Math. Software , vol.9 , pp. 302-325
    • Duff, I.S.1    Reid, J.K.2
  • 44
    • 0033285109 scopus 로고    scopus 로고
    • Developments and trends in the parallel solution of linear systems
    • I.S. Duff, H.A. van der Vorst, Developments and trends in the parallel solution of linear systems, Parallel Comput. 25 (1999) 1931-1970.
    • (1999) Parallel Comput. , vol.25 , pp. 1931-1970
    • Duff, I.S.1    Van Der Vorst, H.A.2
  • 45
    • 0041731470 scopus 로고    scopus 로고
    • Two-dimensional block partitionings for the parallel sparse Cholesky factorization
    • Dumitrescu B., Doreille M., Roch J.-L., Trystram D. Two-dimensional block partitionings for the parallel sparse Cholesky factorization. Numer. Algorithms. 16(1):1997;17-38.
    • (1997) Numer. Algorithms , vol.16 , Issue.1 , pp. 17-38
    • Dumitrescu, B.1    Doreille, M.2    Roch, J.-L.3    Trystram, D.4
  • 46
    • 0040946753 scopus 로고    scopus 로고
    • Efficient sparse LU factorization with partial pivoting on distributed memory architectures
    • Technical report, University of California at Santa Barbara, submitted for publication
    • C. Fu, X. Jiao, T. Yang, Efficient sparse LU factorization with partial pivoting on distributed memory architectures, Technical report, University of California at Santa Barbara, 1997; IEEE Trans. Parallel Distrib. Systems, submitted for publication.
    • (1997) IEEE Trans. Parallel Distrib. Systems
    • Fu, C.1    Jiao, X.2    Yang, T.3
  • 47
    • 0030421529 scopus 로고    scopus 로고
    • Solving large nonsymmetric sparse linear systems using MCSPARSE
    • Gallivan K.A., Marsolf B.A., Wijshoff H.A.G. Solving large nonsymmetric sparse linear systems using MCSPARSE. Parallel Comput. 22:1996;1291-1333.
    • (1996) Parallel Comput. , vol.22 , pp. 1291-1333
    • Gallivan, K.A.1    Marsolf, B.A.2    Wijshoff, H.A.G.3
  • 48
    • 0000264382 scopus 로고
    • Nested dissection of a regular finite element mesh
    • George A. Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10:1973;345-363.
    • (1973) SIAM J. Numer. Anal. , vol.10 , pp. 345-363
    • George, A.1
  • 49
    • 0018478506 scopus 로고
    • The design of a user interface for a sparse matrix package
    • George A., Liu J.W.H. The design of a user interface for a sparse matrix package. ACM Trans. Math. Software. 5(2):1979;139-162.
    • (1979) ACM Trans. Math. Software , vol.5 , Issue.2 , pp. 139-162
    • George, A.1    Liu, J.W.H.2
  • 52
    • 0040946757 scopus 로고
    • Incomplete nested dissection for solving n × n grid problems
    • George A., Poole J.W., Voigt R. Incomplete nested dissection for solving. n × n grid problems SIAM J. Numer. Anal. 15:1978;663-673.
    • (1978) SIAM J. Numer. Anal. , vol.15 , pp. 663-673
    • George, A.1    Poole, J.W.2    Voigt, R.3
  • 53
    • 0029358116 scopus 로고
    • Exploiting large grain parallelism in a sparse direct linear system solver
    • Geschiere J.P., Wijshoff H.A.G. Exploiting large grain parallelism in a sparse direct linear system solver. Parallel Comput. 21(8):1995;1339-1364.
    • (1995) Parallel Comput. , vol.21 , Issue.8 , pp. 1339-1364
    • Geschiere, J.P.1    Wijshoff, H.A.G.2
  • 54
    • 21144481596 scopus 로고
    • Elimination structures for unsymmetric sparse LU factors
    • Gilbert J.R., W.H. Liu J. Elimination structures for unsymmetric sparse LU factors. SIAM J. Matrix Anal. Appl. 14:1993;334-354.
    • (1993) SIAM J. Matrix Anal. Appl. , vol.14 , pp. 334-354
    • Gilbert, J.R.1    W.H. Liu, J.2
  • 57
  • 58
    • 0031273280 scopus 로고    scopus 로고
    • Recursion leads to automatic variable blocking for dense linear algebra algorithms
    • Gustavson F.G. Recursion leads to automatic variable blocking for dense linear algebra algorithms. IBM J. Res. Develop. 41:1997;737-755.
    • (1997) IBM J. Res. Develop. , vol.41 , pp. 737-755
    • Gustavson, F.G.1
  • 60
    • 0014701614 scopus 로고
    • Symbolic generation of an optimal Crout algorithm for sparse systems of linear equations
    • Gustavson F.G., Liniger W.M., Willoughby R.A. Symbolic generation of an optimal Crout algorithm for sparse systems of linear equations. J. ACM. 17(1):1970;87-109.
    • (1970) J. ACM , vol.17 , Issue.1 , pp. 87-109
    • Gustavson, F.G.1    Liniger, W.M.2    Willoughby, R.A.3
  • 61
    • 0028586358 scopus 로고
    • Performance of a fully parallel sparse solver
    • May 23-25, 1994, Knoxville, Tennessee, Los Alamitos, California, IEEE Computer Society Press, Silver Spring, MD
    • M.T. Heath, P. Raghavan, Performance of a fully parallel sparse solver, Proceedings of SHPCC '94, Scalable High-Performance Computing Conference, May 23-25, 1994, Knoxville, Tennessee, Los Alamitos, California, IEEE Computer Society Press, Silver Spring, MD, 1994, pp. 334-341.
    • (1994) Proceedings of SHPCC '94, Scalable High-Performance Computing Conference , pp. 334-341
    • Heath, M.T.1    Raghavan, P.2
  • 62
    • 0003573801 scopus 로고
    • Technical Report SAND94-2692, Sandia National Laboratories, Albuquerque, October
    • B. Hendrickson, R. Leland, The CHACO User's Guide, Version 2.0, Technical Report SAND94-2692, Sandia National Laboratories, Albuquerque, October 1994.
    • (1994) The CHACO User's Guide, Version 2.0
    • Hendrickson, B.1    Leland, R.2
  • 63
    • 0032228620 scopus 로고    scopus 로고
    • Improving the runtime and quality of nested dissection ordering
    • Hendrickson B., Rothberg E. Improving the runtime and quality of nested dissection ordering. SIAM J. Sci. Comput. 20:1998;468-489.
    • (1998) SIAM J. Sci. Comput. , vol.20 , pp. 468-489
    • Hendrickson, B.1    Rothberg, E.2
  • 64
    • 0040690069 scopus 로고    scopus 로고
    • HSL, Harwell Subroutine Library, AEA Technology, Harwell Laboratory, Oxfordshire, England, For information concerning HSL contact: Dr Nick Brealey, HSL Manager, AEA Technology Products & Systems, Culham Science Centre, Oxon OX14 3ED, England (tel: +44-1235-463404, fax: +44-1235-463480, email: hsl@aeat.co.uk)
    • HSL, Harwell Subroutine Library, A Catalogue of Subroutines (Release 12). AEA Technology, Harwell Laboratory, Oxfordshire, England, 1996. For information concerning HSL contact: Dr Nick Brealey, HSL Manager, AEA Technology Products & Systems, Culham Science Centre, Oxon OX14 3ED, England (tel: +44-1235-463404, fax: +44-1235-463480, email: hsl@aeat.co.uk).
    • (1996) A Catalogue of Subroutines (Release 12)
  • 65
    • 0003735970 scopus 로고
    • Technical Report TR-95-036, Department of Computer Science, University of Minnesota, May
    • G. Karypis, V. Kumar, Parallel multilevel graph partitioning, Technical Report TR-95-036, Department of Computer Science, University of Minnesota, May 1995.
    • (1995) Parallel Multilevel Graph Partitioning
    • Karypis, G.1    Kumar, V.2
  • 69
    • 0007956992 scopus 로고    scopus 로고
    • Making sparse Gaussian elimination scalable by static pivoting
    • Orlando, Florida, November
    • X.S. Li, J.W. Demmel, Making sparse Gaussian elimination scalable by static pivoting. Proceedings of Supercomputing, Orlando, Florida, November 1998.
    • (1998) Proceedings of Supercomputing
    • Li, X.S.1    Demmel, J.W.2
  • 70
    • 0022785798 scopus 로고
    • On the storage requirement in the out-of-core multifrontal method for sparse factorization
    • Liu J.W.H. On the storage requirement in the out-of-core multifrontal method for sparse factorization. ACM Trans. Math. Software. 12:1987;249-264.
    • (1987) ACM Trans. Math. Software , vol.12 , pp. 249-264
    • Liu, J.W.H.1
  • 71
    • 0024698976 scopus 로고
    • Reordering sparse matrices for parallel elimination
    • Liu J.W.H. Reordering sparse matrices for parallel elimination. Parallel Comput. 11:1989;73-91.
    • (1989) Parallel Comput. , vol.11 , pp. 73-91
    • Liu, J.W.H.1
  • 72
    • 0000444373 scopus 로고
    • The elimination form of the inverse and its application to linear programming
    • Markowitz H.M. The elimination form of the inverse and its application to linear programming. Management Sci. 3:1957;255-269.
    • (1957) Management Sci. , vol.3 , pp. 255-269
    • Markowitz, H.M.1
  • 73
    • 0029267316 scopus 로고
    • Parallel sparse QR factorization on shared memory architectures
    • Matstoms P. Parallel sparse QR factorization on shared memory architectures. Parallel Comput. 21:1995;473-486.
    • (1995) Parallel Comput. , vol.21 , pp. 473-486
    • Matstoms, P.1
  • 74
    • 0000018801 scopus 로고
    • Block sparse Cholesky algorithms on advanced uniprocessor computers
    • Ng E.G., Peyton B.W. Block sparse Cholesky algorithms on advanced uniprocessor computers. SIAM J. Sci. Comput. 14:1993;1034-1056.
    • (1993) SIAM J. Sci. Comput. , vol.14 , pp. 1034-1056
    • Ng, E.G.1    Peyton, B.W.2
  • 76
    • 0000812783 scopus 로고
    • Partitioning sparse matrices with eigenvectors of graphs
    • Pothen A., Simon H.D., Liou K.P. Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl. 11(3):1990;430-452.
    • (1990) SIAM J. Matrix Anal. Appl. , vol.11 , Issue.3 , pp. 430-452
    • Pothen, A.1    Simon, H.D.2    Liou, K.P.3
  • 78
    • 0031209254 scopus 로고    scopus 로고
    • Parallel ordering using edge contraction
    • Raghavan P. Parallel ordering using edge contraction. Parallel Comput. 23(8):1997;1045-1067.
    • (1997) Parallel Comput. , vol.23 , Issue.8 , pp. 1045-1067
    • Raghavan, P.1
  • 80
    • 0001102965 scopus 로고    scopus 로고
    • Performance of panel and block approaches to sparse Cholesky factorization on the iPSC / 860 and Paragon multicomputers
    • Rothberg E. Performance of panel and block approaches to sparse Cholesky factorization on the iPSC. / 860 and Paragon multicomputers SIAM J. Sci. Comput. 17(3):1996;699-713.
    • (1996) SIAM J. Sci. Comput. , vol.17 , Issue.3 , pp. 699-713
    • Rothberg, E.1
  • 83
    • 0039179703 scopus 로고    scopus 로고
    • Iterative solution of linear systems in the 20th century, this issue
    • Y. Saad, H.A. van der Vorst, Iterative solution of linear systems in the 20th century, this issue, J. Comput. Appl. Math. 123 (2000) 1-33.
    • (2000) J. Comput. Appl. Math. , vol.123 , pp. 1-33
    • Saad, Y.1    Van Der Vorst, H.A.2
  • 84
    • 84937995839 scopus 로고
    • Direct solutions of sparse network equations by optimally ordered triangular factorization
    • Tinney W.F., Walker J.W. Direct solutions of sparse network equations by optimally ordered triangular factorization. Proc. IEEE. 55:1967;1801-1809.
    • (1967) Proc. IEEE , vol.55 , pp. 1801-1809
    • Tinney, W.F.1    Walker, J.W.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.