메뉴 건너뛰기




Volumn 39, Issue 3, 1999, Pages 417-438

Bounds for the entries of matrix functions with applications to preconditioning

Author keywords

Band matrices; Exponential decay; Lanczos process; Matrix functions; Preconditioned conjugate gradients; Quadrature rules

Indexed keywords


EID: 0043044851     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022362401426     Document Type: Article
Times cited : (123)

References (40)
  • 2
    • 0001881523 scopus 로고
    • On approximate factorization methods for block matrices suitable for vector and parallel processors
    • O. Axelsson and B. Polman, On approximate factorization methods for block matrices suitable for vector and parallel processors, Linear Algebra Appl., 77 (1986), pp. 3-26.
    • (1986) Linear Algebra Appl. , vol.77 , pp. 3-26
    • Axelsson, O.1    Polman, B.2
  • 3
    • 0030571227 scopus 로고    scopus 로고
    • Some large-scale matrix computation problems
    • Z. Bai, M. Fahey, and G. H. Golub, Some large-scale matrix computation problems, J. Comput. Appl. Math., 74 (1996), pp. 71-89.
    • (1996) J. Comput. Appl. Math. , vol.74 , pp. 71-89
    • Bai, Z.1    Fahey, M.2    Golub, G.H.3
  • 4
    • 0009924116 scopus 로고    scopus 로고
    • Bounds for the trace of the inverse and the determinant of symmetric positive definite matrices
    • Z. Bai and G. H. Golub, Bounds for the trace of the inverse and the determinant of symmetric positive definite matrices, Ann. Numer. Math., 4 (1997), pp. 29-38.
    • (1997) Ann. Numer. Math. , vol.4 , pp. 29-38
    • Bai, Z.1    Golub, G.H.2
  • 5
    • 0041434423 scopus 로고    scopus 로고
    • Bounds for the entries of matrix functions with applications to preconditioning
    • February
    • M. Benzi and G. H. Golub, Bounds for the entries of matrix functions with applications to preconditioning, Stanford University Report SCCM-98-04, February 1998.
    • (1998) Stanford University Report SCCM-98-04
    • Benzi, M.1    Golub, G.H.2
  • 6
    • 0004675411 scopus 로고    scopus 로고
    • A sparse approximate inverse preconditioner for the conjugate gradient method
    • M. Benzi, C. D. Meyer, Jr., and M. Tůma, A sparse approximate inverse preconditioner for the conjugate gradient method, SIAM J. Sci. Comput., 17 (1996), pp. 1135-1149.
    • (1996) SIAM J. Sci. Comput. , vol.17 , pp. 1135-1149
    • Benzi, M.1    Meyer C.D., Jr.2    Tůma, M.3
  • 9
    • 0001646777 scopus 로고
    • Toeplitz equations by conjugate gradients with circulant preconditioner
    • R. H. Chan and G. Strang, Toeplitz equations by conjugate gradients with circulant preconditioner, SIAM J. Sci. Stat. Comput., 10 (1989), pp. 104-119.
    • (1989) SIAM J. Sci. Stat. Comput. , vol.10 , pp. 104-119
    • Chan, R.H.1    Strang, G.2
  • 10
    • 0000853646 scopus 로고    scopus 로고
    • An optimal circulant preconditioner for Toeplitz systems
    • T. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Stat. Comput., 9 (1998), pp. 766-771.
    • (1998) SIAM J. Sci. Stat. Comput. , vol.9 , pp. 766-771
    • Chan, T.1
  • 11
    • 0001630026 scopus 로고    scopus 로고
    • Approximate inverse preconditioned via sparse-sparse iterations
    • E. Chow and Y. Saad, Approximate inverse preconditioned via sparse-sparse iterations, SIAM J. Sci. Comput., 19 (1998), pp. 995-1023.
    • (1998) SIAM J. Sci. Comput. , vol.19 , pp. 995-1023
    • Chow, E.1    Saad, Y.2
  • 12
    • 0001249888 scopus 로고
    • Block preconditioning for the conjugate gradient method
    • P. Concus, G. H. Golub, and G. Meurant, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Stat. Comput., 6 (1985), pp. 220-252.
    • (1985) SIAM J. Sci. Stat. Comput. , vol.6 , pp. 220-252
    • Concus, P.1    Golub, G.H.2    Meurant, G.3
  • 13
    • 0022888917 scopus 로고
    • On computing INV block preconditionings for the conjugate gradient method
    • P. Concus and G. Meurant, On computing INV block preconditionings for the conjugate gradient method, BIT, 26 (1986), pp. 493-504.
    • (1986) BIT , vol.26 , pp. 493-504
    • Concus, P.1    Meurant, G.2
  • 15
    • 0003712558 scopus 로고
    • Bounds for the error of linear systems using the theory of moments
    • G. Dahlquist, S. C. Eisenstat and G. H. Golub, Bounds for the error of linear systems using the theory of moments, J. Math. Anal. Appl., 37 (1972), pp. 151-166.
    • (1972) J. Math. Anal. Appl. , vol.37 , pp. 151-166
    • Dahlquist, G.1    Eisenstat, S.C.2    Golub, G.H.3
  • 17
    • 0001320046 scopus 로고
    • Inverses of band matrices and local convergence of spline projections
    • S. Demko, Inverses of band matrices and local convergence of spline projections, SIAM J. Numer. Anal., 14 (1977), pp. 616-619.
    • (1977) SIAM J. Numer. Anal. , vol.14 , pp. 616-619
    • Demko, S.1
  • 18
    • 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
  • 19
    • 0000446481 scopus 로고
    • Two polynomial methods of calculating functions of symmetric matrices
    • V. L. Druskin and L. A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices, USSR Comput. Maths. Math. Phys., 29 (1989), pp. 112-121.
    • (1989) USSR Comput. Maths. Math. Phys. , vol.29 , pp. 112-121
    • Druskin, V.L.1    Knizhnerman, L.A.2
  • 20
    • 0041434416 scopus 로고
    • Analytic functions of M-matrices and generalizations
    • M. Fiedler and H. Schneider, Analytic functions of M-matrices and generalizations, Linear and Multilinear Algebra, 13 (1983), pp. 185-201.
    • (1983) Linear and Multilinear Algebra , vol.13 , pp. 185-201
    • Fiedler, M.1    Schneider, H.2
  • 22
    • 0002841144 scopus 로고
    • Some uses of the Lanczos algorithm in numerical linear algebra
    • J. H. Miller, ed., Academic Press, New York
    • G. H. Golub, Some uses of the Lanczos algorithm in numerical linear algebra, in Topics in Numerical Analysis, J. H. Miller, ed., Academic Press, New York, 1973, pp. 173-184.
    • (1973) Topics in Numerical Analysis , pp. 173-184
    • Golub, G.H.1
  • 23
    • 0041935774 scopus 로고
    • Bounds for matrix moments
    • G. H. Golub, Bounds for matrix moments, Rocky Mountain J. Math., 4 (1974), pp. 207-211.
    • (1974) Rocky Mountain J. Math. , vol.4 , pp. 207-211
    • Golub, G.H.1
  • 24
    • 0003093333 scopus 로고
    • Matrices, moments and quadratures
    • D. F. Griffiths and G. A. Watson, eds., Pitman Research Notes in Mathematics, Essex, England
    • G. H. Golub and G. Meurant, Matrices, moments and quadratures, in Numerical Analysis 1993, D. F. Griffiths and G. A. Watson, eds., Pitman Research Notes in Mathematics, Vol. 303, Essex, England, 1994, pp. 105-156.
    • (1994) Numerical Analysis 1993 , vol.303 , pp. 105-156
    • Golub, G.H.1    Meurant, G.2
  • 25
    • 0000391978 scopus 로고    scopus 로고
    • Matrices, moments and quadratures II; How to compute the norm of the error in iterative methods
    • G. H. Golub and G. Meurant, Matrices, moments and quadratures II; how to compute the norm of the error in iterative methods, BIT, 37 (1997), pp. 687-705.
    • (1997) BIT , vol.37 , pp. 687-705
    • Golub, G.H.1    Meurant, G.2
  • 26
    • 0000918880 scopus 로고
    • Estimates in quadratic formulas
    • G. H. Golub and Z. Strakoš, Estimates in quadratic formulas, Numer. Algorithms, 8 (1994), pp. 241-268.
    • (1994) Numer. Algorithms , vol.8 , pp. 241-268
    • Golub, G.H.1    Strakoš, Z.2
  • 28
    • 0001678475 scopus 로고
    • Quadratically constrained least squares and quadratic problems
    • G. H. Golub and U. von Matt, Quadratically constrained least squares and quadratic problems, Numer. Math., 59 (1991), pp. 561-580.
    • (1991) Numer. Math. , vol.59 , pp. 561-580
    • Golub, G.H.1    Von Matt, U.2
  • 29
    • 0000223746 scopus 로고
    • Calculation of Gauss quadrature rules
    • G. H. Golub and J. H. Welsh, Calculation of Gauss quadrature rules. Math. Comp., 23 (1969), pp. 221-230.
    • (1969) Math. Comp. , vol.23 , pp. 221-230
    • Golub, G.H.1    Welsh, J.H.2
  • 30
    • 0031129197 scopus 로고    scopus 로고
    • Parallel preconditioning with sparse approximate inverses
    • M. Grote and T. Huckle, Parallel preconditioning with sparse approximate inverses, SIAM J. Sci. Comput., 18 (1997), pp. 838-853.
    • (1997) SIAM J. Sci. Comput. , vol.18 , pp. 838-853
    • Grote, M.1    Huckle, T.2
  • 31
    • 0008025436 scopus 로고    scopus 로고
    • The simple Lanczos procedure: Estimates of the error of the Gauss quadrature formula and their applications
    • L. A. Knizhnerman, The simple Lanczos procedure: estimates of the error of the Gauss quadrature formula and their applications, Comput. Math. Math. Phys., 36 (1996), pp. 1481-1492.
    • (1996) Comput. Math. Math. Phys. , vol.36 , pp. 1481-1492
    • Knizhnerman, L.A.1
  • 32
    • 21144474663 scopus 로고
    • Factorized sparse approximate inverse pre-conditionings I. Theory
    • L. Yu. Kolotilina and A. Yu. Yeremin, Factorized sparse approximate inverse pre-conditionings I. Theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45-58.
    • (1993) SIAM J. Matrix Anal. Appl. , vol.14 , pp. 45-58
    • Kolotilina, L.Yu.1    Yeremin, A.Yu.2
  • 33
    • 0000747319 scopus 로고
    • On a family of two-level preconditionings of the incomplete block factorization type
    • L. Yu. Kolotilina and A. Yu. Yeremin, On a family of two-level preconditionings of the incomplete block factorization type, Sov. J. Numer. Anal. Math. Modelling, 1 (1986), pp. 293-320.
    • (1986) Sov. J. Numer. Anal. Math. Modelling , vol.1 , pp. 293-320
    • Kolotilina, L.Yu.1    Yeremin, A.Yu.2
  • 35
    • 0000809242 scopus 로고
    • A review on the inverse of symmetric tridiagonal and block tridiagonal matrices
    • G. Meurant, A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 707-728.
    • (1992) SIAM J. Matrix Anal. Appl. , vol.13 , pp. 707-728
    • Meurant, G.1
  • 36
    • 21144480809 scopus 로고
    • Variational bounds on the entries of the inverse of a matrix
    • P. D. Robinson and A. Wathen, Variational bounds on the entries of the inverse of a matrix, IMA J. Numer. Anal., 12 (1992), pp. 463-486.
    • (1992) IMA J. Numer. Anal. , vol.12 , pp. 463-486
    • Robinson, P.D.1    Wathen, A.2
  • 38
    • 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
  • 39
    • 0041434421 scopus 로고
    • Nonnegatively posed problems and completely monotonic functions
    • R. S. Varga, Nonnegatively posed problems and completely monotonic functions, Linear Algebra Appl., 1 (1968), pp. 329-347.
    • (1968) Linear Algebra Appl. , vol.1 , pp. 329-347
    • Varga, R.S.1
  • 40
    • 0001646445 scopus 로고
    • An iterative solution method for solving f(A)x = b, using Krylov subspace information obtained for the symmetric positive definite matrix A
    • H. A. van der Vorst, An iterative solution method for solving f(A)x = b, using Krylov subspace information obtained for the symmetric positive definite matrix A, J. Comput. Appl. Math., 18 (1987), pp. 249-263.
    • (1987) J. Comput. Appl. Math. , vol.18 , pp. 249-263
    • Van Der Vorst, H.A.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.