메뉴 건너뛰기




Volumn 35, Issue 1, 2013, Pages

An optimized sparse approximate matrix multiply for matrices with decay

Author keywords

Matrices with decay; N body; Quantum chemistry; Reduced complexity algorithm; SpAMM; Sparse approximate matrix multiply; Sparse linear algebra

Indexed keywords

COMPUTATIONAL COMPLEXITY; QUANTUM CHEMISTRY;

EID: 84876211167     PISSN: 10648275     EISSN: 10957200     Source Type: Journal    
DOI: 10.1137/120870761     Document Type: Article
Times cited : (31)

References (154)
  • 1
    • 9344227702 scopus 로고
    • Experiments with quadtree representation of matrices
    • P. Gianni, ed., Lecture Notes in Comput. Sci., Springer, Berlin
    • S. K. Abdali and D. S. Wise, Experiments with quadtree representation of matrices, in Symbolic and Algebraic Computation, P. Gianni, ed., Lecture Notes in Comput. Sci. 358, Springer, Berlin, 1989, pp. 96-108.
    • (1989) Symbolic and Algebraic Computation , vol.358 , pp. 96-108
    • Abdali, S.K.1    Wise, D.S.2
  • 5
    • 84876235136 scopus 로고    scopus 로고
    • New trends in collision detection performance
    • Laval, France, S. Richir and A. Shirai, ed.
    • Q. Avril, V. Gouranton, and B. Arnaldi, New trends in collision detection performance, in VRIC'09 Proceedings, vol. 11, Laval, France, S. Richir and A. Shirai, ed., 2009, pp. 53-62.
    • (2009) VRIC'09 Proceedings , vol.11 , pp. 53-62
    • Avril, Q.1    Gouranton, V.2    Arnaldi, B.3
  • 6
    • 33747416213 scopus 로고    scopus 로고
    • The efficacy of software prefetching and locality optimizations on future memory systems
    • A. H. Badawy, A. Aggarwal, D. Yeung, and C. W. Tseng, The efficacy of software prefetching and locality optimizations on future memory systems, J. Instruction-Level Parallelism, 6 (2004).
    • (2004) J. Instruction-Level Parallelism , vol.6
    • Badawy, A.H.1    Aggarwal, A.2    Yeung, D.3    Tseng, C.W.4
  • 7
    • 45449120592 scopus 로고    scopus 로고
    • Hardware-oriented implementation of cache oblivious matrix operations based on space-filling curves
    • R. Wyrzykowski, J. Dongarra, K. Karczewski, and J. Wasniewski, eds., Lecture Notes in Comput. Sci., Springer, Berlin
    • M. Bader, R. Franz, S. Guenther, and A. Heinecke, Hardware-oriented implementation of cache oblivious matrix operations based on space-filling curves, in Parallel Processing and Applied Mathematics, 7th International Conference, PPAM 2007, R. Wyrzykowski, J. Dongarra, K. Karczewski, and J. Wasniewski, eds., Lecture Notes in Comput. Sci. 4967, Springer, Berlin, 2008, p. 628.
    • (2008) Parallel Processing and Applied Mathematics, 7th International Conference, PPAM 2007 , vol.4967 , pp. 628
    • Bader, M.1    Franz, R.2    Guenther, S.3    Heinecke, A.4
  • 9
    • 0037252817 scopus 로고    scopus 로고
    • A new error estimate of the fast Gauss transform
    • B. J. C. Baxter and G. Roussos, A new error estimate of the fast Gauss transform, SIAM J. Sci. Comput., 24 (2002), pp. 257-259.
    • (2002) SIAM J. Sci. Comput. , vol.24 , pp. 257-259
    • Baxter, B.J.C.1    Roussos, G.2
  • 11
    • 0000843403 scopus 로고
    • Three-dimensional adaptive mesh refinement for hyperbolic conservation laws
    • J. Bell, M. J. Berger, J. Saltzman, and M. Welcome, Three-dimensional adaptive mesh refinement for hyperbolic conservation laws, SIAM J. Sci. Comput., 15 (1994), pp. 127- 138.
    • (1994) SIAM J. Sci. Comput. , vol.15 , pp. 127-138
    • Bell, J.1    Berger, M.J.2    Saltzman, J.3    Welcome, M.4
  • 13
    • 0043044851 scopus 로고    scopus 로고
    • Bounds for the entries of matrix functions with applications to preconditioning
    • M. Benzi and G. H. Golub, Bounds for the entries of matrix functions with applications to preconditioning, BIT, 39 (1999), pp. 417-438.
    • (1999) BIT , vol.39 , pp. 417-438
    • Benzi, M.1    Golub, G.H.2
  • 14
    • 54549085632 scopus 로고    scopus 로고
    • Decay bounds and O(n) algorithms for approximating functions of sparse matrices
    • M. Benzi and N. Razouk, Decay bounds and O(n) algorithms for approximating functions of sparse matrices, Electron. Trans. Numer. Anal., 28 (2007), pp. 16-39.
    • (2007) Electron. Trans. Numer. Anal. , vol.28 , pp. 16-39
    • Benzi, M.1    Razouk, N.2
  • 15
    • 18844471650 scopus 로고    scopus 로고
    • Orderings for factorized sparse approximate inverse preconditioners
    • M. Benzi and M. Tuma, Orderings for factorized sparse approximate inverse preconditioners, SIAM J. Sci. Comput., 21 (2000), pp. 1851-1868.
    • (2000) SIAM J. Sci. Comput. , vol.21 , pp. 1851-1868
    • Benzi, M.1    Tuma, M.2
  • 16
    • 11744289966 scopus 로고
    • Local adaptive mesh refinement for shock hydrodynamics
    • M. J. Berger and P. Colella, Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82 (1989), pp. 64-84.
    • (1989) J. Comput. Phys. , vol.82 , pp. 64-84
    • Berger, M.J.1    Colella, P.2
  • 17
    • 48749141209 scopus 로고
    • Adaptive mesh refinement for hyperbolic partial differential equations
    • M. J. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53 (1984), pp. 484-512.
    • (1984) J. Comput. Phys. , vol.53 , pp. 484-512
    • Berger, M.J.1    Oliger, J.2
  • 18
    • 0003018688 scopus 로고
    • Stability of fast algorithms for matrix multiplication
    • D. Bini and G. Lotti, Stability of fast algorithms for matrix multiplication, Numer. Math., 36 (1980), pp. 63-72.
    • (1980) Numer. Math. , vol.36 , pp. 63-72
    • Bini, D.1    Lotti, G.2
  • 21
    • 0037171094 scopus 로고    scopus 로고
    • Recent progress in linear scaling ab initio electronic structure techniques
    • D. R. Bowler, T. Miyazaki, and M. J. Gillan, Recent progress in linear scaling ab initio electronic structure techniques, J. Phys. Condensed Matter, 14 (2002), pp. 2781-2798.
    • (2002) J. Phys. Condensed Matter , vol.14 , pp. 2781-2798
    • Bowler, D.R.1    Miyazaki, T.2    Gillan, M.J.3
  • 26
    • 0001222111 scopus 로고    scopus 로고
    • A linear scaling method for Hartree-Fock exchange calculations of large molecules
    • J. C. Burant, G. E. Scuseria, and M. J. Frisch, A linear scaling method for Hartree-Fock exchange calculations of large molecules, J. Chem. Phys., 105 (1996), pp. 8969-8972.
    • (1996) J. Chem. Phys. , vol.105 , pp. 8969-8972
    • Burant, J.C.1    Scuseria, G.E.2    Frisch, M.J.3
  • 31
    • 0001027297 scopus 로고    scopus 로고
    • Fast assembly of the Coulomb matrix: A quantum chemical tree code
    • M. Challacombe, E. Schwegler, and J. Almlöf, Fast assembly of the Coulomb matrix: A quantum chemical tree code, J. Chem. Phys., 104 (1996), pp. 4685-4698.
    • (1996) J. Chem. Phys. , vol.104 , pp. 4685-4698
    • Challacombe, M.1    Schwegler, E.2    Almlöf, J.3
  • 32
    • 0039758422 scopus 로고    scopus 로고
    • Modern developments in hartree-fock theory: Fast methods for computing the coulomb matrix
    • J. Leszczynski, ed., Computational Chemistry: Reviews of Current Trends, World Scientific, Singapore
    • M. Challacombe, E. Schwegler, and J. Almlöf, Modern Developments in Hartree-Fock Theory: Fast Methods for Computing the Coulomb Matrix, in Computational Chemistry: Reviews of Current Trends, J. Leszczynski, ed., Computational Chemistry: Reviews of Current Trends, vol. 1, World Scientific, Singapore, 1996, pp. 53-107.
    • (1996) Computational Chemistry: Reviews of Current Trends , vol.1 , pp. 53-107
    • Challacombe, M.1    Schwegler, E.2    Almlöf, J.3
  • 33
    • 0010259404 scopus 로고    scopus 로고
    • Linear scaling computation of the Fock matrix
    • M. Challacombe and E. Schwegler, Linear scaling computation of the Fock matrix, J. Chem. Phys., 106 (1997), pp. 5526-5536.
    • (1997) J. Chem. Phys. , vol.106 , pp. 5526-5536
    • Challacombe, M.1    Schwegler, E.2
  • 34
    • 0001514496 scopus 로고    scopus 로고
    • A simplified density matrix minimization for linear scaling self-consistent field theory
    • M. Challacombe, A simplified density matrix minimization for linear scaling self-consistent field theory, J. Chem. Phys., 110 (1999), pp. 2332-2342.
    • (1999) J. Chem. Phys. , vol.110 , pp. 2332-2342
    • Challacombe, M.1
  • 35
    • 0034625292 scopus 로고    scopus 로고
    • A general parallel sparse-blocked matrix multiply for linear scaling SCF theory
    • M. Challacombe, A general parallel sparse-blocked matrix multiply for linear scaling SCF theory, Comput. Phys. Comm., 128 (2000), pp. 93-107.
    • (2000) Comput. Phys. Comm. , vol.128 , pp. 93-107
    • Challacombe, M.1
  • 36
    • 0034506257 scopus 로고    scopus 로고
    • Linear scaling computation of the Fock matrix. V. Hierarchical cubature for numerical integration of the exchange-correlation matrix
    • M. Challacombe, Linear scaling computation of the Fock matrix. V. Hierarchical cubature for numerical integration of the exchange-correlation matrix, J. Chem. Phys., 113 (2000), p. 10037.
    • (2000) J. Chem. Phys. , vol.113 , pp. 10037
    • Challacombe, M.1
  • 37
    • 33845283552 scopus 로고    scopus 로고
    • A fast adaptive solver for hierarchically semiseparable representations
    • S. Chandrasekaran, M. Gu, and W. Lyons, A fast adaptive solver for hierarchically semiseparable representations, Calcolo, 42 (2005), pp. 171-185.
    • (2005) Calcolo , vol.42 , pp. 171-185
    • Chandrasekaran, S.1    Gu, M.2    Lyons, W.3
  • 38
    • 34547225006 scopus 로고    scopus 로고
    • A fast ULV decomposition solver for hierarchically semiseparable representations
    • S. Chandrasekaran, M. Gu, and T. Pals, A Fast ULV Decomposition Solver for Hierarchically Semiseparable Representations, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 603- 622.
    • (2006) SIAM J. Matrix Anal. Appl. , vol.28 , pp. 603-622
    • Chandrasekaran, S.1    Gu, M.2    Pals, T.3
  • 39
    • 0000362580 scopus 로고    scopus 로고
    • A fast adaptive multipole algorithm in three dimensions
    • H. Cheng, L. Greengard, and V. Rokhlin, A fast adaptive multipole algorithm in three dimensions, J. Comput. Phys., 155 (1999), pp. 468-498.
    • (1999) J. Comput. Phys. , vol.155 , pp. 468-498
    • Cheng, H.1    Greengard, L.2    Rokhlin, V.3
  • 41
    • 0034501522 scopus 로고    scopus 로고
    • Making pointer-based data structures cache conscious
    • T. M. Chilimbi, M. D. Hill, and J. R. Larus, Making pointer-based data structures cache conscious, Computer, 33 (2000), pp. 67-74.
    • (2000) Computer , vol.33 , pp. 67-74
    • Chilimbi, T.M.1    Hill, M.D.2    Larus, J.R.3
  • 43
    • 85023205150 scopus 로고
    • Matrix multiplication via arithmetic progressions
    • D. Coppersmith and S. Winograd, Matrix multiplication via arithmetic progressions, J. Symbolic Comput., 9 (1990), pp. 251-280.
    • (1990) J. Symbolic Comput. , vol.9 , pp. 251-280
    • Coppersmith, D.1    Winograd, S.2
  • 44
    • 0000876466 scopus 로고    scopus 로고
    • Semiempirical methods with conjugate gradient density matrix search to replace diagonalization for molecular systems containing thousands of atoms
    • A. D. Daniels, J. M. Millam, and G. E. Scuseria, Semiempirical methods with conjugate gradient density matrix search to replace diagonalization for molecular systems containing thousands of atoms, J. Chem. Phys., 107 (1997), pp. 425-431.
    • (1997) J. Chem. Phys. , vol.107 , pp. 425-431
    • Daniels, A.D.1    Millam, J.M.2    Scuseria, G.E.3
  • 45
    • 84966252352 scopus 로고
    • Decay rates for inverses of band matrices
    • S. Demko, W. F. Moss, and P. W. Smith, Decay rates for inverses of band matrices, Math. Comp., 43 (1984), pp. 491-499.
    • (1984) Math. Comp. , vol.43 , pp. 491-499
    • Demko, S.1    Moss, W.F.2    Smith, P.W.3
  • 47
    • 0026913668 scopus 로고
    • Stability of block algorithms with fast level-3 blas
    • J. W. Demmel and N. J. Higham, Stability of block algorithms with fast level-3 blas, ACM Trans. Math. Software, 18 (1992), pp. 274-291.
    • (1992) ACM Trans. Math. Software , vol.18 , pp. 274-291
    • Demmel, J.W.1    Higham, N.J.2
  • 49
    • 0016353777 scopus 로고
    • Quad trees a data structure for retrieval on composite keys
    • J. L. Finkel and R. A. Bentley, Quad trees a data structure for retrieval on composite keys, Acta Inform., 4 (1974), pp. 1-9.
    • (1974) Acta Inform. , vol.4 , pp. 1-9
    • Finkel, J.L.1    Bentley, R.A.2
  • 50
    • 0347771690 scopus 로고    scopus 로고
    • Auto-blocking matrix-multiplication or tracking blas3 performance from source code
    • J. D. Frens and D. S. Wise, Auto-blocking matrix-multiplication or tracking blas3 performance from source code, SIGPLAN Not., 32 (1997), pp. 206-216.
    • (1997) SIGPLAN Not. , vol.32 , pp. 206-216
    • Frens, J.D.1    Wise, D.S.2
  • 53
    • 0000639195 scopus 로고    scopus 로고
    • Linear scaling methods for electronic structure calculations and quantum molecular-dynamics simulations
    • G. Galli, Linear scaling methods for electronic structure calculations and quantum molecular-dynamics simulations, Current Opinion in Solid State & Materials Science, 1 (1996), pp. 864-874.
    • (1996) Current Opinion in Solid State & Materials Science , vol.1 , pp. 864-874
    • Galli, G.1
  • 54
    • 0037603314 scopus 로고    scopus 로고
    • Linear scaling computation of the Fock matrix. VI. Data parallel computation of the exchange-correlation matrix
    • C. K. Gan and M. Challacombe, Linear scaling computation of the Fock matrix. VI. Data parallel computation of the exchange-correlation matrix, J. Comput. Phys., 118 (2003), pp. 9128-9135.
    • (2003) J. Comput. Phys. , vol.118 , pp. 9128-9135
    • Gan, C.K.1    Challacombe, M.2
  • 55
    • 0020249952 scopus 로고
    • An effective way to represent quadtrees
    • I. Gargantini, An effective way to represent quadtrees, Commun. ACM, 25 (1982), pp. 905- 910.
    • (1982) Commun. ACM , vol.25 , pp. 905-910
    • Gargantini, I.1
  • 57
    • 84876265888 scopus 로고    scopus 로고
    • GNU compiler collection, http://gcc.gnu.org/.
  • 59
    • 0038450200 scopus 로고    scopus 로고
    • Linear scaling electronic structure methods in chemistry and physics
    • S. Goedecker and G. Scuseria, Linear scaling electronic structure methods in chemistry and physics, Comput. Sci. Engrg., 5 (2003), pp. 14-21.
    • (2003) Comput. Sci. Engrg. , vol.5 , pp. 14-21
    • Goedecker, S.1    Scuseria, G.2
  • 61
    • 0033246389 scopus 로고    scopus 로고
    • Linear scaling electronic structure methods
    • S. Goedecker, Linear scaling electronic structure methods, Rev. Mod. Phys., 71 (1999), pp. 1085-1123.
    • (1999) Rev. Mod. Phys. , vol.71 , pp. 1085-1123
    • Goedecker, S.1
  • 63
    • 44249094647 scopus 로고    scopus 로고
    • Anatomy of high-performance matrix multiplication
    • K. Goto and R. A. van de Geijn, Anatomy of high-performance matrix multiplication, ACM Trans. Math. Software, 34 (2008), pp. 12:1-12:25.
    • (2008) ACM Trans. Math. Software , vol.34 , pp. 121-1225
    • Goto, K.1    Geijn De Van, R.A.2
  • 64
    • 48849089104 scopus 로고    scopus 로고
    • High-performance implementation of the level-3 blas
    • K. Goto and R. A. van de Geijn, High-performance implementation of the level-3 blas, ACM Trans. Math. Software, 35 (2008), pp. 4:1-4:14.
    • (2008) ACM Trans. Math. Software , vol.35 , pp. 41-414
    • Goto, K.1    Geijn De Van, R.A.2
  • 66
    • 0141829814 scopus 로고    scopus 로고
    • Construction and arithmetics of H-matrices
    • L. Grasedyck and W. Hackbusch, Construction and arithmetics of H-matrices, Computing, 70 (2003), pp. 295-334.
    • (2003) Computing , vol.70 , pp. 295-334
    • Grasedyck, L.1    Hackbusch, W.2
  • 67
    • 84899027721 scopus 로고    scopus 로고
    • N-body' Problems in statistical learning
    • MIT Press, Cambridge, MA
    • A. G. Gray and A. W. Moore, N-Body' Problems in Statistical Learning, in Advances in Neural Information Processing Systems, vol. 4, MIT Press, Cambridge, MA, 2001, pp. 521-527.
    • (2001) Advances in Neural Information Processing Systems , vol.4 , pp. 521-527
    • Gray, A.G.1    Moore, A.W.2
  • 69
    • 0000396658 scopus 로고
    • A fast algorithm for particle simulations
    • L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys., 73 (1987), pp. 325-348.
    • (1987) J. Comput. Phys. , vol.73 , pp. 325-348
    • Greengard, L.1    Rokhlin, V.2
  • 70
    • 0000468568 scopus 로고
    • The rapid evaluation of potential fields in three dimensions
    • C. Anderson and C. Greengard, eds., Lecture Notes in Math., Springer, Berlin
    • L. Greengard and V. Rokhlin, The rapid evaluation of potential fields in three dimensions, in Vortex Methods, C. Anderson and C. Greengard, eds., Lecture Notes in Math. 1360, Springer, Berlin, 1988, pp. 121-141.
    • (1988) Vortex Methods , vol.1360 , pp. 121-141
    • Greengard, L.1    Rokhlin, V.2
  • 71
    • 85011484616 scopus 로고    scopus 로고
    • A new version of the fast multipole method for the Laplace equation in three dimensions
    • L. Greengard and V. Rokhlin, A new version of the fast multipole method for the Laplace equation in three dimensions, Acta Numer., (1997), pp. 229-269.
    • (1997) Acta Numer. , pp. 229-269
    • Greengard, L.1    Rokhlin, V.2
  • 74
    • 84862107202 scopus 로고    scopus 로고
    • Parallel and cache-efficient in-place matrix storage format conversion
    • F. G. Gustavson, L. Karlsson, and B. Ka°gström, Parallel and cache-efficient in-place matrix storage format conversion, ACM Trans. Math. Software, 38 (2012), pp. 1-32.
    • (2012) ACM Trans. Math. Software , vol.38 , pp. 1-32
    • Gustavson, F.G.1    Karlsson, L.2    Kagström, B.3
  • 75
    • 0036387334 scopus 로고    scopus 로고
    • Data-sparse approximation by adaptive H2-matrices
    • W. Hackbusch and S. Börm, Data-sparse approximation by adaptive H2-matrices, Computing, 69 (2002), pp. 1-35.
    • (2002) Computing , vol.69 , pp. 1-35
    • Hackbusch, W.1    Börm, S.2
  • 77
    • 0023165505 scopus 로고
    • Hierarchical n-body methods
    • L. Hernquist, Hierarchical n-body methods, Comput. Phys. Commun., 48 (1988), pp. 107- 115.
    • (1988) Comput. Phys. Commun. , vol.48 , pp. 107-115
    • Hernquist, L.1
  • 78
    • 33846316938 scopus 로고
    • Über die stetige Abbildung einer Linie auf ein Flächenstü ck
    • D. Hilbert, Über die stetige Abbildung einer Linie auf ein Flächenstück, Math. Ann. (1891), p. 459.
    • (1891) Math. Ann. , pp. 459
    • Hilbert, D.1
  • 81
    • 70449098063 scopus 로고    scopus 로고
    • available online from Intel
    • Intel 64 and IA-32 Architectures Optimization Reference Manual, 2009; available online from Intel, http://www.intel.com/content/www/us/en/ architecture-and-technology/64- ia-32-architectures-optimization-manual.html.
    • (2009) Intel 64 and IA-32 Architectures Optimization Reference Manual
  • 82
    • 0007295634 scopus 로고    scopus 로고
    • How large is the exponential of a banded matrix?
    • A. Iserles, How large is the exponential of a banded matrix?, J. New Zealand Math. Soc., 29 (2000), pp. 177-192.
    • (2000) J. New Zealand Math. Soc. , vol.29 , pp. 177-192
    • Iserles, A.1
  • 86
    • 0002479236 scopus 로고    scopus 로고
    • Charm++: Parallel programming with message-driven objects
    • G. V. Wilson and P. Lu, eds., MIT Press, Cambridge, MA
    • L. V. Kale and S. Krishnan, Charm++: Parallel programming with message-driven objects, in Parallel Programming Using C++, G. V. Wilson and P. Lu, eds., MIT Press, Cambridge, MA, 1996, pp. 175-213.
    • (1996) Parallel Programming Using C++ , pp. 175-213
    • Kale, L.V.1    Krishnan, S.2
  • 89
    • 0000990502 scopus 로고    scopus 로고
    • Linear-scaling quantum mechanical calculations of biological molecules: The divide-and-conquer approach
    • T. S. Lee, J. P. Lewis, and W. Yang, Linear-scaling quantum mechanical calculations of biological molecules: The divide-and-conquer approach, Comput. Materials Sci., 12 (1998), pp. 259-277.
    • (1998) Comput. Materials Sci. , vol.12 , pp. 259-277
    • Lee, T.S.1    Lewis, J.P.2    Yang, W.3
  • 90
    • 50949096788 scopus 로고    scopus 로고
    • Perfect spatial hashing
    • S. Lefebvre and H. Hoppe, Perfect spatial hashing, ACM Trans. Graph., 25 (2006), pp. 579- 588.
    • (2006) ACM Trans. Graph. , vol.25 , pp. 579-588
    • Lefebvre, S.1    Hoppe, H.2
  • 98
    • 16244396071 scopus 로고    scopus 로고
    • Recent progress in density functional theory and its numerical methods
    • Z. Y. Li, W. He, and J. L. Yang, Recent progress in density functional theory and its numerical methods, Progress in Chemistry, 17 (2005), pp. 192-202.
    • (2005) Progress in Chemistry , vol.17 , pp. 192-202
    • Li, Z.Y.1    He, W.2    Yang, J.L.3
  • 101
    • 0001332894 scopus 로고
    • The density matrix in self-consistent field theory. I. Iterative construction of the density matrix
    • R. McWeeny, The density matrix in self-consistent field theory. I. Iterative construction of the density matrix, Proc. Roy. Soc. London A Mat., 235 (1956), pp. 496-509.
    • (1956) Proc. Roy. Soc. London A Mat. , vol.235 , pp. 496-509
    • McWeeny, R.1
  • 102
    • 1542601822 scopus 로고    scopus 로고
    • Improving memory hierarchy performance for irregular applications using data and computation reorderings
    • J. Mellor-Crummey, D. Whalley, and K. Kennedy, Improving memory hierarchy performance for irregular applications using data and computation reorderings, Internat. J. Parallel Programming, 29 (2001), pp. 217-247.
    • (2001) Internat. J. Parallel Programming , vol.29 , pp. 217-247
    • Mellor-Crummey, J.1    Whalley, D.2    Kennedy, K.3
  • 103
    • 5244240756 scopus 로고    scopus 로고
    • Linear scaling conjugate gradient density matrix search as an alternative to diagonalization for first principles electronic structure calculations
    • J. M. Millam and G. E. Scuseria, Linear scaling conjugate gradient density matrix search as an alternative to diagonalization for first principles electronic structure calculations, J. Chem. Phys., 106 (1997), pp. 5569-5577.
    • (1997) J. Chem. Phys. , vol.106 , pp. 5569-5577
    • Millam, J.M.1    Scuseria, G.E.2
  • 104
    • 0026826969 scopus 로고
    • Join processing in relational databases
    • P. Mishra and M. H. Eich, Join processing in relational databases, ACM Comput. Surv., 24 (1992), pp. 63-113.
    • (1992) ACM Comput. Surv. , vol.24 , pp. 63-113
    • Mishra, P.1    Eich, M.H.2
  • 109
    • 0037863744 scopus 로고    scopus 로고
    • Trace resetting density matrix purification in O(N) self-consistent-field theory
    • A. M. N. Niklasson, C. J. Tymczak, and M. Challacombe, Trace resetting density matrix purification in O(N) self-consistent-field theory, J. Chem. Phys., 118 (2003), pp. 8611- 8620.
    • (2003) J. Chem. Phys. , vol.118 , pp. 8611-8620
    • Niklasson, A.M.N.1    Tymczak, C.J.2    Challacombe, M.3
  • 110
    • 0000247202 scopus 로고    scopus 로고
    • Linear and sublinear scaling formation of Hartree-Fock-type exchange matrices
    • C. Ochsenfeld, C. A. White, and M. Head-Gordon, Linear and sublinear scaling formation of Hartree-Fock-type exchange matrices, J. Chem. Phys., 109 (1998), pp. 1663-1669.
    • (1998) J. Chem. Phys. , vol.109 , pp. 1663-1669
    • Ochsenfeld, C.1    White, C.A.2    Head-Gordon, M.3
  • 111
    • 25944474315 scopus 로고    scopus 로고
    • Master's thesis, Department of Computer Science, University of Copenhagen, Denmark
    • J. H. Olsen and S. C. Skov, Cache-Oblivious Algorithms in Practice, Master's thesis, Department of Computer Science, University of Copenhagen, Denmark, 2002.
    • (2002) Cache-Oblivious Algorithms in Practice
    • Olsen, J.H.1    Skov, S.C.2
  • 112
    • 0001241557 scopus 로고    scopus 로고
    • Canonical purification of the density matrix in electronic-structure theory
    • A. H. R. Palser and D. E. Manolopoulos, Canonical purification of the density matrix in electronic-structure theory, Phys. Rev. B, 58 (1998), pp. 12704-12711.
    • (1998) Phys. Rev. B , vol.58 , pp. 12704-12711
    • Palser, A.H.R.1    Manolopoulos, D.E.2
  • 115
    • 0005083863 scopus 로고
    • Sur une courbe, qui remplit toute une aire plane
    • G. Peano, Sur une courbe, qui remplit toute une aire plane, Math. Ann., 36 (1890), pp. 157- 160.
    • (1890) Math. Ann. , vol.36 , pp. 157-160
    • Peano, G.1
  • 118
    • 0000343198 scopus 로고
    • Tradeoffs in processing complex join queries via hashing in multiprocessor database machines
    • San Francisco, Morgan Kaufmann
    • D. A. Schneider and D. J. DeWitt, Tradeoffs in processing complex join queries via hashing in multiprocessor database machines, in Proceedings of the 16th International Conference on Very Large Databases, San Francisco, Morgan Kaufmann, 1990, pp. 469-480.
    • (1990) Proceedings of the 16th International Conference on Very Large Databases , pp. 469-480
    • Schneider, D.A.1    Dewitt, D.J.2
  • 119
    • 0000484587 scopus 로고    scopus 로고
    • Linear scaling computation of the Fock matrix. II. Rigorous bounds on exchange integrals and incremental Fock build
    • E. Schwegler, M. Challacombe, and M. Head-Gordon, Linear scaling computation of the Fock matrix. II. Rigorous bounds on exchange integrals and incremental Fock build, J. Chem. Phys., 106 (1997), pp. 9708-9717.
    • (1997) J. Chem. Phys. , vol.106 , pp. 9708-9717
    • Schwegler, E.1    Challacombe, M.2    Head-Gordon, M.3
  • 120
    • 0007132664 scopus 로고    scopus 로고
    • A multipole acceptability criterion for electronic structure theory
    • E. Schwegler, M. Challacombe, and M. Head-Gordon, A multipole acceptability criterion for electronic structure theory, J. Chem. Phys., 109 (1998), pp. 8764-8769.
    • (1998) J. Chem. Phys. , vol.109 , pp. 8764-8769
    • Schwegler, E.1    Challacombe, M.2    Head-Gordon, M.3
  • 123
    • 0007898394 scopus 로고
    • The fast Gauss transform with variable scales
    • J. Strain, The fast Gauss transform with variable scales, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 1131-1139.
    • (1991) SIAM J. Sci. Statist. Comput. , vol.12 , pp. 1131-1139
    • Strain, J.1
  • 124
    • 34250487811 scopus 로고
    • Gaussian elimination is not optimal
    • V. Strassen, Gaussian elimination is not optimal, Numer. Math., 13 (1969), pp. 354-356.
    • (1969) Numer. Math. , vol.13 , pp. 354-356
    • Strassen, V.1
  • 126
    • 34248387649 scopus 로고    scopus 로고
    • Complete inlining of recursive calls: Beyond tail-recursion elimination
    • New York, ACM
    • P. Tang, Complete inlining of recursive calls: Beyond tail-recursion elimination, in Proceedings of the 44th Annual Southeast Regional Conference, ACM-SE 44, New York, ACM, 2006, pp. 579-584.
    • (2006) Proceedings of the 44th Annual Southeast Regional Conference, ACM-SE , vol.44 , pp. 579-584
    • Tang, P.1
  • 128
    • 84867384854 scopus 로고    scopus 로고
    • Linear scaling self-consistent field calculations with millions of atoms in the condensed phase
    • J. VandeVondele, U. Borštnik, and J. Hutter, Linear scaling self-consistent field calculations with millions of atoms in the condensed phase, J. Chem. Theory Comput., (2012).
    • (2012) J. Chem. Theory Comput.
    • Vandevondele, J.1    Borštnik, U.2    Hutter, J.3
  • 129
  • 132
    • 33750378674 scopus 로고    scopus 로고
    • A sharp error estimate for the fast Gauss transform
    • X. Wan and G. E. Karniadakis, A sharp error estimate for the fast Gauss transform, J. Comput. Phys., 219 (2006), pp. 7-12.
    • (2006) J. Comput. Phys. , vol.219 , pp. 7-12
    • Wan, X.1    Karniadakis, G.E.2
  • 133
    • 84967045101 scopus 로고
    • Astrophysical N-body simulations using hierarchical tree data structures
    • Los Alamitos, IEEE
    • M. S. Warren and J. K. Salmon, Astrophysical N-body simulations using hierarchical tree data structures, in Supercomputing '92, Los Alamitos, IEEE, 1992, pp. 570-576.
    • (1992) Supercomputing '92 , pp. 570-576
    • Warren, M.S.1    Salmon, J.K.2
  • 134
    • 84967045101 scopus 로고
    • Astrophysical n-body simulations using hierarchical tree data structures
    • M. S. Warren and J. K. Salmon, Astrophysical n-body simulations using hierarchical tree data structures, in Proceedings of SC Conference, 1992, pp. 570-576.
    • (1992) Proceedings of SC Conference , pp. 570-576
    • Warren, M.S.1    Salmon, J.K.2
  • 137
    • 1842471081 scopus 로고
    • A portable parallel particle program
    • M. S. Warren and J. K. Salmon, A portable parallel particle program, Comput. Phys. Comm., 87 (1995), p. 266.
    • (1995) Comput. Phys. Comm. , vol.87 , pp. 266
    • Warren, M.S.1    Salmon, J.K.2
  • 139
    • 0025460758 scopus 로고
    • Costs of quadtree representation of nondense matrices
    • D. S. Wise and J. Franco, Costs of quadtree representation of nondense matrices, J. Parallel Distributed Computing, 9 (1990), pp. 282-296.
    • (1990) J. Parallel Distributed Computing , vol.9 , pp. 282-296
    • Wise, D.S.1    Franco, J.2
  • 140
    • 0034819362 scopus 로고    scopus 로고
    • Language support for Morton-order matrices
    • D. S. Wise, J. D. Frens, Y. Gu, and G. A. Alexander, Language support for Morton-order matrices, SIGPLAN Not., 36 (2001), pp. 24-33.
    • (2001) SIGPLAN Not. , vol.36 , pp. 24-33
    • Wise, D.S.1    Frens, J.D.2    Gu, Y.3    Alexander, G.A.4
  • 141
    • 0347738087 scopus 로고
    • Representing matrices as quadtrees for parallel processors: Extended abstract
    • D. S. Wise, Representing matrices as quadtrees for parallel processors: Extended abstract, SIGSAM Bull., 18 (1984), pp. 24-25.
    • (1984) SIGSAM Bull. , vol.18 , pp. 24-25
    • Wise, D.S.1
  • 142
    • 84937431996 scopus 로고    scopus 로고
    • Ahnentafel indexing into Morton-ordered arrays, or matrix locality for free
    • A. Bode, T. Ludwig, W. Karl, and R. Wismüller, eds., Lecture Notes in Comput. Sci., Springer, Berlin
    • D. S. Wise, Ahnentafel indexing into Morton-ordered arrays, or matrix locality for free, in Euro-Par 2000 Parallel Processing, A. Bode, T. Ludwig, W. Karl, and R. Wismüller, eds., Lecture Notes in Comput. Sci. 1900, Springer, Berlin, 2000, pp. 774-783.
    • (2000) Euro-Par 2000 Parallel Processing , vol.1900 , pp. 774-783
    • Wise, D.S.1
  • 145
    • 0043144732 scopus 로고
    • A density-matrix divide-and-conquer approach for electronic structure calculations of large molecules
    • W. Yang and T. S. Lee, A density-matrix divide-and-conquer approach for electronic structure calculations of large molecules, J. Chem. Phys., 103 (1995), p. 5674.
    • (1995) J. Chem. Phys. , vol.103 , pp. 5674
    • Yang, W.1    Lee, T.S.2
  • 146
    • 0011621942 scopus 로고
    • Direct calculation of electron density in density-functional theory
    • W. Yang, Direct calculation of electron density in density-functional theory, Phys. Rev. Lett, 66 (1991), pp. 1438-1441.
    • (1991) Phys. Rev. Lett , vol.66 , pp. 1438-1441
    • Yang, W.1
  • 148
    • 34250883179 scopus 로고    scopus 로고
    • Fast sparse matrix multiplication
    • R. Yuster and U. Zwick, Fast sparse matrix multiplication, ACM Trans. Algorithms, 1 (2005), pp. 2-13.
    • (2005) ACM Trans. Algorithms , vol.1 , pp. 2-13
    • Yuster, R.1    Zwick, U.2
  • 149
    • 70350780227 scopus 로고    scopus 로고
    • Survey on real-time collision detection algorithms
    • Y.-S. Zou, G.-F. Ding, M.-H. Xu, and Y. He, Survey on real-time collision detection algorithms, Appl. Res. Comput., 25 (2008), pp. 8-12.
    • (2008) Appl. Res. Comput. , vol.25 , pp. 8-12
    • Zou, Y.-S.1    Ding, G.-F.2    Xu, M.-H.3    He, Y.4
  • 151
    • 84870430661 scopus 로고    scopus 로고
    • Intel Cilk Plus, http://software.intel.com/en-us/articles/intel-cilk- plus/.
    • Intel Cilk Plus
  • 153
    • 84876264103 scopus 로고    scopus 로고
    • Intel SPMD Program Compiler, http://ispc.github.com/.
  • 154
    • 84876211927 scopus 로고    scopus 로고
    • OpenMP, http://openmp.org/.


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.