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Volumn 62, Issue 4, 2011, Pages 1685-1706

Nearly optimal solution of rational linear systems of equations with symbolic lifting and numerical initialization

Author keywords

Hensel's Lifting; Initialization; Iterative refinement; Linear systems of equations; Newton's lifting; Toeplitz matrices

Indexed keywords

HENSEL'S LIFTING; INITIALIZATION; ITERATIVE REFINEMENT; LINEAR SYSTEMS OF EQUATIONS; NEWTON'S LIFTING; TOEPLITZ MATRICES;

EID: 80051801149     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2011.06.006     Document Type: Article
Times cited : (5)

References (53)
  • 2
    • 0000081585 scopus 로고
    • Asymptotically fast solution of Toeplitz and related systems of linear equations
    • R.R. Bitmead, and B.D.O. Anderson Asymptotically fast solution of Toeplitz and related systems of linear equations Linear Algebra and its Applications 34 1980 103 116
    • (1980) Linear Algebra and Its Applications , vol.34 , pp. 103-116
    • Bitmead, R.R.1    Anderson, B.D.O.2
  • 4
    • 84945708089 scopus 로고
    • Approximate algorithms to derive exact solutions to systems of linear equations
    • Lecture Notes in Computer Science Springer Berlin
    • R.T. Moenck, and J.H. Carter Approximate algorithms to derive exact solutions to systems of linear equations Proceedings of EUROSAM Lecture Notes in Computer Science vol. 72 1979 Springer Berlin 63 73
    • (1979) Proceedings of EUROSAM , vol.72 , pp. 63-73
    • Moenck, R.T.1    Carter, J.H.2
  • 5
    • 0001380055 scopus 로고
    • Exact solution of linear equations using p-adic expansions
    • J.D. Dixon Exact solution of linear equations using p-adic expansions Numerische Mathematik 40 1982 137 141
    • (1982) Numerische Mathematik , vol.40 , pp. 137-141
    • Dixon, J.D.1
  • 7
    • 84966259743 scopus 로고
    • Hensel and Newton methods in valuation rings
    • J. von zur Gathen Hensel and Newton methods in valuation rings Mathematics of Computation 42 166 1984 802 824
    • (1984) Mathematics of Computation , vol.42 , Issue.166 , pp. 802-824
    • Von Zur Gathen, J.1
  • 8
    • 0012063917 scopus 로고
    • An improved Newton iteration for the generalized inverse of a matrix, with applications
    • V.Y. Pan, and R. Schreiber An improved Newton iteration for the generalized inverse of a matrix, with applications SIAM Journal on Scientific and Statistical Computing 12 5 1991 1109 1131
    • (1991) SIAM Journal on Scientific and Statistical Computing , vol.12 , Issue.5 , pp. 1109-1131
    • Pan, V.Y.1    Schreiber, R.2
  • 9
    • 49249100712 scopus 로고    scopus 로고
    • Degeneration of integer matrices modulo an integer
    • V.Y. Pan, and X. Wang Degeneration of integer matrices modulo an integer Linear Algebra and its Applications 429 2008 2113 2130
    • (2008) Linear Algebra and Its Applications , vol.429 , pp. 2113-2130
    • Pan, V.Y.1    Wang, X.2
  • 14
    • 80051794201 scopus 로고    scopus 로고
    • On solving linear system with randomized augmentation
    • Graduate Center, the City University of New York
    • V.Y. Pan, G. Qian, On solving linear system with randomized augmentation, Tech. Report TR 2010009, Ph.D. Program in Computer Science, Graduate Center, the City University of New York, 2010. Available at http://www.cs.gc.cuny.edu/ tr/techreport.php?id=352.
    • (2010) Tech. Report TR 2010009, Ph.D. Program in Computer Science
    • Pan, V.Y.1    Qian, G.2
  • 15
    • 79959649941 scopus 로고    scopus 로고
    • Tech. Report TR 2010012, Ph.D. Program in Computer Science, Graduate Center, CUNY
    • V.Y. Pan, G. Qian, A. Zheng, Randomized preconditioning of linear systems of equations, Tech. Report TR 2010012, Ph.D. Program in Computer Science, Graduate Center, CUNY, 2010. Available at http://www.cs.gc.cuny.edu/tr/ techreport.php?id=352.
    • (2010) Randomized Preconditioning of Linear Systems of Equations
    • Pan, V.Y.1    Qian, G.2    Zheng, A.3
  • 18
    • 0038623314 scopus 로고    scopus 로고
    • Acceleration of Euclidean algorithm and rational number reconstruction
    • X. Wang, and V.Y. Pan Acceleration of Euclidean algorithm and rational number reconstruction SIAM Journal on Computing 32 2 2003 548 556
    • (2003) SIAM Journal on Computing , vol.32 , Issue.2 , pp. 548-556
    • Wang, X.1    Pan, V.Y.2
  • 22
    • 38149147254 scopus 로고
    • Complexity of multiplication with vectors for structured matrices
    • I. Gohberg, and V. Olshevsky Complexity of multiplication with vectors for structured matrices Linear Algebra and Its Applications 202 1994 163 192
    • (1994) Linear Algebra and Its Applications , vol.202 , pp. 163-192
    • Gohberg, I.1    Olshevsky, V.2
  • 23
    • 84966229708 scopus 로고
    • On computations with dense structured matrices
    • Proceedings version in Proc. ISSAC'89, 3442, ACM Press, New York, 1989
    • V.Y. Pan On computations with dense structured matrices Mathematics of Computation 55 191 1990 179 190 Proceedings version in Proc. ISSAC'89, 3442, ACM Press, New York, 1989
    • (1990) Mathematics of Computation , vol.55 , Issue.191 , pp. 179-190
    • Pan, V.Y.1
  • 24
    • 84968505008 scopus 로고
    • Fast Gaussian elimination with partial pivoting for matrices with displacement structure
    • I. Gohberg, T. Kailath, and V. Olshevsky Fast Gaussian elimination with partial pivoting for matrices with displacement structure Mathematics of Computation 64 1995 1557 1576
    • (1995) Mathematics of Computation , vol.64 , pp. 1557-1576
    • Gohberg, I.1    Kailath, T.2    Olshevsky, V.3
  • 25
    • 0022122662 scopus 로고
    • Complexity of parallel matrix computations
    • V.Y. Pan Complexity of parallel matrix computations Theoretical Computer Science 54 1987 65 85
    • (1987) Theoretical Computer Science , vol.54 , pp. 65-85
    • Pan, V.Y.1
  • 26
    • 0024031591 scopus 로고
    • Computing the determinant and the characteristic polynomials of a matrix via solving linear systems of equations
    • V.Y. Pan Computing the determinant and the characteristic polynomials of a matrix via solving linear systems of equations Information Processing Letters 28 1988 71 75
    • (1988) Information Processing Letters , vol.28 , pp. 71-75
    • Pan, V.Y.1
  • 27
  • 30
    • 1942477456 scopus 로고    scopus 로고
    • Certified dense linear system solving
    • DOI 10.1016/j.jsc.2003.07.004
    • T. Mulders, and A. Storjohann Certified dense linear system solving Journal of Symbolic Computation 37 4 2004 485 510 (Pubitemid 38516227)
    • (2004) Journal of Symbolic Computation , vol.37 , Issue.4 , pp. 485-510
    • Mulders, T.1    Storjohann, A.2
  • 31
  • 32
    • 79952515563 scopus 로고    scopus 로고
    • Exact solutions to linear systems of equations using output sensitive lifting
    • D.E. Steffy Exact solutions to linear systems of equations using output sensitive lifting ACM Communications in Computer Algebra 44 34 2010 160 182
    • (2010) ACM Communications in Computer Algebra , vol.44 , Issue.34 , pp. 160-182
    • Steffy, D.E.1
  • 34
    • 0004236492 scopus 로고    scopus 로고
    • third ed. Johns Hopkins University Press Baltimore, Maryland
    • G.H. Golub, and C.F. Van Loan Matrix Computations third ed. 1996 Johns Hopkins University Press Baltimore, Maryland
    • (1996) Matrix Computations
    • Golub, G.H.1    Van Loan, C.F.2
  • 35
    • 55749102088 scopus 로고    scopus 로고
    • Schur aggregation for linear systems and determinants
    • D.A. Bini, V.Y. Pan, J. Verschelde, Theoretical Computer Science (special issue)
    • V.Y. Pan, D. Grady, B. Murphy, G. Qian, R.E. Rosholt, and A. Ruslanov Schur aggregation for linear systems and determinants D.A. Bini, V.Y. Pan, J. Verschelde, SymbolicNumerical Algorithms Theoretical Computer Science 409 2 2008 255 268 (special issue)
    • (2008) SymbolicNumerical Algorithms , vol.409 , Issue.2 , pp. 255-268
    • Pan, V.Y.1    Grady, D.2    Murphy, B.3    Qian, G.4    Rosholt, R.E.5    Ruslanov, A.6
  • 36
    • 77950369121 scopus 로고    scopus 로고
    • Randomized preprocessing of homogeneous linear systems of equations
    • V.Y. Pan, and G. Qian Randomized preprocessing of homogeneous linear systems of equations Linear Algebra and its Applications 432 2010 3272 3318
    • (2010) Linear Algebra and Its Applications , vol.432 , pp. 3272-3318
    • Pan, V.Y.1    Qian, G.2
  • 37
    • 0030214595 scopus 로고    scopus 로고
    • Parallel computation of polynomial GCD and some related parallel computations over abstract fields
    • DOI 10.1016/0304-3975(96)00030-8, PII S0304397596000308
    • V.Y. Pan Parallel computation of polynomial GCD and some related parallel computations over abstract fields Theoretical Computer Science 162 2 1996 173 223 (Pubitemid 126413395)
    • (1996) Theoretical Computer Science , vol.162 , Issue.2 , pp. 173-223
    • Pan, V.Y.1
  • 38
    • 84968495217 scopus 로고
    • On Wiedemann's method for solving sparse linear systems
    • Lecture Notes in Computer Science Springer Berlin
    • E. Kaltofen, and B.D. Saunders On Wiedemann's method for solving sparse linear systems Proceedings of AAECC5 Lecture Notes in Computer Science vol. 536 1991 Springer Berlin 29 38
    • (1991) Proceedings of AAECC5 , vol.536 , pp. 29-38
    • Kaltofen, E.1    Saunders, B.D.2
  • 39
    • 84983014125 scopus 로고
    • Fast solution of Toeplitz systems of equations and computation of Padé approximations
    • R.P. Brent, F.G. Gustavson, and D.Y.Y. Yun Fast solution of Toeplitz systems of equations and computation of Padé approximations Journal of Algorithms 1 1980 259 295
    • (1980) Journal of Algorithms , vol.1 , pp. 259-295
    • Brent, R.P.1    Gustavson, F.G.2    Yun, D.Y.Y.3
  • 41
    • 38249013295 scopus 로고
    • On practical algorithms for accelerated matrix multiplication
    • J. Laderman, V.Y. Pan, and H.X. Sha On practical algorithms for accelerated matrix multiplication Linear Algebra and its Applications 162164 1992 557 588
    • (1992) Linear Algebra and Its Applications , vol.162-164 , pp. 557-588
    • Laderman, J.1    Pan, V.Y.2    Sha, H.X.3
  • 42
    • 0038954992 scopus 로고
    • On schemes for the evaluation of products and inverses of matrices
    • (in Russian)
    • V.Y. Pan On schemes for the evaluation of products and inverses of matrices Uspekhi Matematicheskikh Nauk, 27 5 167 1972 249 250 (in Russian)
    • (1972) Uspekhi Matematicheskikh Nauk, 27 , vol.5 , Issue.167 , pp. 249-250
    • Pan, V.Y.1
  • 43
    • 0021452919 scopus 로고
    • How can we speed up matrix multiplication?
    • V.Y. Pan How can we speed up matrix multiplication? SIAM Review 26 3 1984 393 415
    • (1984) SIAM Review , vol.26 , Issue.3 , pp. 393-415
    • Pan, V.Y.1
  • 44
    • 2142700139 scopus 로고    scopus 로고
    • The aggregation and cancellation techniques as a practical tool for faster matrix multiplication
    • I. Kaporin The aggregation and cancellation techniques as a practical tool for faster matrix multiplication Theoretical Computer Science 315 23 2004 469 510
    • (2004) Theoretical Computer Science , vol.315 , Issue.23 , pp. 469-510
    • Kaporin, I.1
  • 45
    • 53249119358 scopus 로고    scopus 로고
    • Solving structured linear systems with large displacement rank
    • A. Bostan, C.P. Jeannerod, and E. Schost Solving structured linear systems with large displacement rank Theoretical Computer Science 407 13 2008 155 181
    • (2008) Theoretical Computer Science , vol.407 , Issue.13 , pp. 155-181
    • Bostan, A.1    Jeannerod, C.P.2    Schost, E.3
  • 46
  • 47
    • 0000835513 scopus 로고
    • Lower bounds for the condition number of Vandermonde matrices
    • W. Gautschi, and G. Inglese Lower bounds for the condition number of Vandermonde matrices Numerische Mathematik 52 1988 241 250
    • (1988) Numerische Mathematik , vol.52 , pp. 241-250
    • Gautschi, W.1    Inglese, G.2
  • 48
    • 21344486577 scopus 로고
    • How bad are Hankel matrices?
    • E.E. Tyrtyshnikov How bad are Hankel matrices? Numerische Mathematik 67 2 1994 261 269
    • (1994) Numerische Mathematik , vol.67 , Issue.2 , pp. 261-269
    • Tyrtyshnikov, E.E.1
  • 52
    • 0032119428 scopus 로고    scopus 로고
    • Modular arithmetic for linear algebra computations in the real field
    • I.Z. Emiris, V.Y. Pan, and Y. Yu Modular arithmetic for linear algebra computations in the real field Journal of Symbolic Computation 21 1998 1 17
    • (1998) Journal of Symbolic Computation , vol.21 , pp. 1-17
    • Emiris, I.Z.1    Pan, V.Y.2    Yu, Y.3
  • 53
    • 0002638254 scopus 로고
    • Inversion of generalized Cauchy matrices and the other classes of structured matrices
    • IMA Volume in Mathematics and Its Applications Springer
    • G. Heinig Inversion of generalized Cauchy matrices and the other classes of structured matrices Linear Algebra for Signal Processing IMA Volume in Mathematics and Its Applications vol. 69 1995 Springer 95 114
    • (1995) Linear Algebra for Signal Processing , vol.69 , pp. 95-114
    • Heinig, G.1


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