메뉴 건너뛰기




Volumn 33, Issue 3, 2012, Pages 837-858

A superfast structured solver for Toeplitz linear systems via randomized sampling

Author keywords

Hierarchically semiseparable matrix; Precomputation; Randomized sampling; Structure preserving rank revealing factorization; Superfast Toeplitz solver

Indexed keywords

PRE-COMPUTATION; RANDOMIZED SAMPLING; RANK-REVEALING; SEMISEPARABLE MATRICES; TOEPLITZ;

EID: 84867319581     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/110831982     Document Type: Article
Times cited : (90)

References (46)
  • 1
    • 0003090115 scopus 로고
    • Superfast solution of real positive definite Toeplitz systems
    • G. S. Ammar and W. B. Gragg, Superfast solution of real positive definite Toeplitz systems, SIAM J. Matrix Anal. Appl., 9 (1988), pp. 61-76.
    • (1988) SIAM J. Matrix Anal. Appl. , vol.9 , pp. 61-76
    • Ammar, G.S.1    Gragg, W.B.2
  • 2
    • 78049415800 scopus 로고    scopus 로고
    • A fast solver for linear systems with displacement structure
    • A. Aricò, and G. Rodriguez, A fast solver for linear systems with displacement structure, Numer. Algorithms, 55 (2010), pp. 529-556.
    • (2010) Numer. Algorithms , vol.55 , pp. 529-556
    • Aricò, A.1    Rodriguez, G.2
  • 4
    • 21844504467 scopus 로고
    • On the stability of the Bareiss and related Toeplitz factorization algorithms
    • A. W. Bojańczyk, R. P. Brent, F. R. de Hoog, and D. R. Sweet, On the stability of the Bareiss and related Toeplitz factorization algorithms, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 40-57.
    • (1995) SIAM J. Matrix Anal. Appl. , vol.16 , pp. 40-57
    • Bojańczyk, A.W.1    Brent, R.P.2    De Hoog, F.R.3    Sweet, D.R.4
  • 5
    • 78651294406 scopus 로고    scopus 로고
    • Numerical experiments on the condition number of the interpolation matrices for radial basis functions
    • J. P. Boyd and K. W. Gildersleeve, Numerical experiments on the condition number of the interpolation matrices for radial basis functions, Appl. Numer. Math., 61 (2011), pp. 443-459.
    • (2011) Appl. Numer. Math. , vol.61 , pp. 443-459
    • Boyd, J.P.1    Gildersleeve, K.W.2
  • 6
    • 0000102913 scopus 로고
    • Stability of methods for solving Toeplitz systems of equations
    • J. R. Bunch, Stability of methods for solving Toeplitz systems of equations, SIAM J. Sci. Statist. Comput., 6 (1985), pp. 349-364.
    • (1985) SIAM J. Sci. Statist. Comput. , vol.6 , pp. 349-364
    • Bunch, J.R.1
  • 7
    • 0009898160 scopus 로고
    • Displacement preconditioners for Toeplitz least squares iterations
    • R. H. Chan, J. G. Nagy, and R. J. Plemmons, Displacement preconditioners for Toeplitz least squares iterations, Electron. Trans. Numer. Anal., 2 (1994), pp. 1-13.
    • (1994) Electron. Trans. Numer. Anal. , vol.2 , pp. 1-13
    • Chan, R.H.1    Nagy, J.G.2    Plemmons, R.J.3
  • 8
    • 0026862959 scopus 로고
    • A look-ahead Levinson algorithm for general Toeplitz systems
    • T. F. Chan and P. C. Hansen, A look-ahead Levinson algorithm for general Toeplitz systems, IEEE Trans. Signal Process., 40 (1992), pp. 1079-1090.
    • (1992) IEEE Trans. Signal Process. , vol.40 , pp. 1079-1090
    • Chan, T.F.1    Hansen, P.C.2
  • 9
    • 0030246195 scopus 로고    scopus 로고
    • Conjugate gradient methods for Toeplitz systems
    • R. H. Chan and M. K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38 (1996) pp. 427-482.
    • (1996) SIAM Rev. , vol.38 , pp. 427-482
    • Chan, R.H.1    Ng, M.K.2
  • 10
    • 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
  • 12
  • 13
    • 0030519188 scopus 로고    scopus 로고
    • Stabilizing the fast generalized Schur algorithm
    • S. Chandrasekaran and A. H. Sayed, Stabilizing the fast generalized Schur algorithm, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 950-983.
    • (1996) SIAM J. Matrix Anal. Appl. , vol.17 , pp. 950-983
    • Chandrasekaran, S.1    Sayed, A.H.2
  • 14
    • 0001112325 scopus 로고
    • The numerical stability of the Levinson-Durbin algorithm for Toeplitz systems of equations
    • G. Cybenko, The numerical stability of the Levinson-Durbin algorithm for Toeplitz systems of equations, SIAM J. Sci. Statist. Comput., 1 (1980), pp. 303-319.
    • (1980) SIAM J. Sci. Statist. Comput. , vol.1 , pp. 303-319
    • Cybenko, G.1
  • 15
    • 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, Math. Comp., 64 (1995), pp. 1557-1576.
    • (1995) Math. Comp. , vol.64 , pp. 1557-1576
    • Gohberg, I.1    Kailath, T.2    Olshevsky, V.3
  • 16
    • 0004236492 scopus 로고    scopus 로고
    • 3rd ed., The Johns Hopkins University Press, Baltimore, MD
    • G. H. Golub and C. V. Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996.
    • (1996) Matrix Computations
    • Golub, G.H.1    Loan, C.V.2
  • 17
    • 0032366597 scopus 로고    scopus 로고
    • Stable and efficient algorithms for structured systems of linear equations
    • M. Gu, Stable and efficient algorithms for structured systems of linear equations, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 279-306.
    • (1998) SIAM J. Matrix Anal. Appl. , vol.19 , pp. 279-306
    • Gu, M.1
  • 18
    • 0003216822 scopus 로고    scopus 로고
    • Efficient algorithms for computing a strong-rank revealing QR factorization
    • M. Gu and S. C. Eisenstat, Efficient algorithms for computing a strong-rank revealing QR factorization, SIAM J. Sci. Comput., 17 (1996), pp. 848-869.
    • (1996) SIAM J. Sci. Comput. , vol.17 , pp. 848-869
    • Gu, M.1    Eisenstat, S.C.2
  • 19
    • 84867289869 scopus 로고    scopus 로고
    • A numerically stable and superfast algorithm for solving Toeplitz systems of linear equations
    • Presentation in the, Monterey, CA
    • M. Gu and J. Xia, A numerically stable and superfast algorithm for solving Toeplitz systems of linear equations, Presentation in the SIAM Conference on Applied Linear Algebra, Monterey, CA, 2009.
    • (2009) SIAM Conference on Applied Linear Algebra
    • Gu, M.1    Xia, J.2
  • 20
    • 79960425522 scopus 로고    scopus 로고
    • Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions
    • N. Halko, P. G. Martinsson, and J. A. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53 (2011), pp. 217-288.
    • (2011) SIAM Rev. , vol.53 , pp. 217-288
    • Halko, N.1    Martinsson, P.G.2    Tropp, J.A.3
  • 21
    • 0002638254 scopus 로고
    • Inversion of generalized Cauchy matrices and other classes of structured matrices
    • Springer, New York
    • G. Heinig, Inversion of generalized Cauchy matrices and other classes of structured matrices, in Linear Algebra for Signal Processing, IMA Vol. Math. Appl. 69, Springer, New York, 1995, pp. 95-114.
    • (1995) Linear Algebra for Signal Processing, IMA Vol. Math. Appl. , vol.69 , pp. 95-114
    • Heinig, G.1
  • 22
    • 0029692982 scopus 로고    scopus 로고
    • Solving Toeplitz systems after extension and transformation
    • G. Heinig, Solving Toeplitz systems after extension and transformation, Calcolo, 33 (1996), pp. 115-129.
    • (1996) Calcolo , vol.33 , pp. 115-129
    • Heinig, G.1
  • 23
    • 21344482533 scopus 로고
    • Displacement structure of pseudoinverses
    • G. Heinig and F. Hellinger, Displacement structure of pseudoinverses, Linear Algebra Appl., 197/198 (1994), pp. 623-649.
    • (1994) Linear Algebra Appl. , vol.197-198 , pp. 623-649
    • Heinig, G.1    Hellinger, F.2
  • 24
    • 0002675246 scopus 로고
    • Algebraic methods for Toeplitz-like matrices and operators
    • Birkh?auser Verlag, Basel
    • G. Heinig and K. Rost, Algebraic methods for Toeplitz-like matrices and operators, Oper. Theory Adv. Appl. 13, Birkh?auser Verlag, Basel, 1984, pp. 109-127.
    • (1984) Oper. Theory Adv. Appl. , vol.13 , pp. 109-127
    • Heinig, G.1    Rost, K.2
  • 25
    • 79960422161 scopus 로고    scopus 로고
    • Rank-deficient nonlinear least squares problems and subset selection
    • I. C. F. Ipsen, C. T. Kelley, and S. R. Pope, Rank-deficient nonlinear least squares problems and subset selection, SIAM J. Numer. Anal., 49 (2011), pp. 1244-1266.
    • (2011) SIAM J. Numer. Anal. , vol.49 , pp. 1244-1266
    • Ipsen, I.C.F.1    Kelley, C.T.2    Pope, S.R.3
  • 26
    • 0017951648 scopus 로고
    • Displacement ranks of matrices and linear equations
    • T. Kailath, S. Y. Kung, and M. Morf, Displacement ranks of matrices and linear equations, J. Math. Anal. Appl., 68 (1979), pp. 395-407.
    • (1979) J. Math. Anal. Appl. , vol.68 , pp. 395-407
    • Kailath, T.1    Kung, S.Y.2    Morf, M.3
  • 30
    • 79952901900 scopus 로고    scopus 로고
    • Fast construction of hierarchical matrix representation from matrix-vector multiplication
    • L. Lin, J. Lu, and L. Ying, Fast construction of hierarchical matrix representation from matrix-vector multiplication, J. Comput. Phys., 230 (2011), pp. 4071-4087.
    • (2011) J. Comput. Phys. , vol.230 , pp. 4071-4087
    • Lin, L.1    Lu, J.2    Ying, L.3
  • 31
    • 84856259101 scopus 로고    scopus 로고
    • A fast randomized algorithm for computing a hierarchically semiseparable representation of a matrix
    • P. G. Martinsson, A fast randomized algorithm for computing a hierarchically semiseparable representation of a matrix, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1251-1274.
    • (2011) SIAM J. Matrix Anal. Appl. , vol.32 , pp. 1251-1274
    • Martinsson, P.G.1
  • 33
    • 26444607857 scopus 로고    scopus 로고
    • A fast algorithm for the inversion of general Toeplitz matrices
    • P. G. Martinsson, V. Rokhlin, and M. Tygert, A fast algorithm for the inversion of general Toeplitz matrices, Comput. Math. Appl., 50 (2005), pp. 741-752.
    • (2005) Comput. Math. Appl. , vol.50 , pp. 741-752
    • Martinsson, P.G.1    Rokhlin, V.2    Tygert, M.3
  • 34
    • 0029693006 scopus 로고    scopus 로고
    • Scientific applications of iterative Toeplitz solvers
    • M. K. Ng and R. H. Chan, Scientific applications of iterative Toeplitz solvers, Calcolo, 33 (1996), pp. 249-267.
    • (1996) Calcolo , vol.33 , pp. 249-267
    • Ng, M.K.1    Chan, R.H.2
  • 36
    • 84882756893 scopus 로고    scopus 로고
    • Fast algorithms for structured matrices: Theory and applications
    • Amer. Math. Soc., Providence, RI
    • V. Olshevski, ed., Fast Algorithms for Structured Matrices: Theory and Applications, Contemp. Math. 323, Amer. Math. Soc., Providence, RI, 2003.
    • (2003) Contemp. Math. , vol.323
    • Olshevski, V.1
  • 37
    • 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 Appl., 432 (2010), pp. 3272-3318.
    • (2010) Linear Algebra Appl. , vol.432 , pp. 3272-3318
    • Pan, V.Y.1    Qian, G.2
  • 38
    • 84966229708 scopus 로고
    • On computations with dense structured matrices
    • V. Y. Pan, On computations with dense structured matrices, Math. Comp., 55 (1990), pp. 179-190.
    • (1990) Math. Comp. , vol.55 , pp. 179-190
    • Pan, V.Y.1
  • 39
    • 0022694911 scopus 로고
    • A proposal for Toeplitz matrix calculations
    • G. Strang, A proposal for Toeplitz matrix calculations, Stud. Appl. Math., 74 (1986), pp. 171-176.
    • (1986) Stud. Appl. Math. , vol.74 , pp. 171-176
    • Strang, G.1
  • 40
    • 3142758633 scopus 로고    scopus 로고
    • A superfast Toeplitz solver with improved numerical stability
    • M. Stewart, A superfast Toeplitz solver with improved numerical stability, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 669-693.
    • (2003) SIAM J. Matrix Anal. Appl. , vol.25 , pp. 669-693
    • Stewart, M.1
  • 41
    • 21144466758 scopus 로고
    • The use of pivoting to improve the numerical performance of algorithms for Toeplitz matrices
    • D. R. Sweet, The use of pivoting to improve the numerical performance of algorithms for Toeplitz matrices, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 468-493.
    • (1993) SIAM J. Matrix Anal. Appl. , vol.14 , pp. 468-493
    • Sweet, D.R.1
  • 42
    • 0000523384 scopus 로고
    • Numerical solution of the eigenvalue problem for Hermitian Toeplitz matrices
    • W. F. Trench, Numerical solution of the eigenvalue problem for Hermitian Toeplitz matrices, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 135-146.
    • (1989) SIAM J. Matrix Anal. Appl. , vol.10 , pp. 135-146
    • Trench, W.F.1
  • 43
    • 0038805356 scopus 로고    scopus 로고
    • Incomplete cross approximation in the mosaic-skeleton method
    • E. E. Tyrtyshnikov, Incomplete cross approximation in the mosaic-skeleton method, Computing, 64 (2000), pp. 367-380.
    • (2000) Computing , vol.64 , pp. 367-380
    • Tyrtyshnikov, E.E.1
  • 44
    • 0036054763 scopus 로고    scopus 로고
    • A stabilized superfast solver for nonsymmetric Toeplitz systems
    • M. Van Barel, G. Heinig, and P. Kravanja, A stabilized superfast solver for nonsymmetric Toeplitz systems, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 494-510.
    • (2001) SIAM J. Matrix Anal. Appl. , vol.23 , pp. 494-510
    • Van Barel, M.1    Heinig, G.2    Kravanja, P.3
  • 45
    • 84865452975 scopus 로고    scopus 로고
    • On the complexity of some hierarchical structured matrix algorithms
    • J. Xia, On the complexity of some hierarchical structured matrix algorithms, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 388-410.
    • (2012) SIAM J. Matrix Anal. Appl. , vol.33 , pp. 388-410
    • Xia, J.1
  • 46


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.