메뉴 건너뛰기




Volumn 13, Issue , 2002, Pages 119-147

The interplay between classical analysis and (numerical) linear algebra - A tribute to Gene H. Golub

Author keywords

Bounds for matrix functionals; Eigenvalue vector characterizations; Gauss type quadratures; Iterative methods; Lanczos algorithm; Least squares problem; Modification algorithms; Orthogonal polynomials; Polynomials orthogonal on several intervals

Indexed keywords


EID: 2942620338     PISSN: 10689613     EISSN: 10689613     Source Type: Journal    
DOI: None     Document Type: Article
Times cited : (56)

References (48)
  • 1
    • 0009924116 scopus 로고    scopus 로고
    • Bounds for the trace of the inverse and the determinant of symmetric positive definite matrices
    • The Heritage of P. L. Chebyshev: A Festschrift in Honor of the 70th Birthday of T. J. Rivlin, C. A. Micchelli, ed.
    • Z. BAI AND G. H. GOLUB, Bounds for the trace of the inverse and the determinant of symmetric positive definite matrices, in The Heritage of P. L. Chebyshev: A Festschrift in Honor of the 70th Birthday of T. J. Rivlin, C. A. Micchelli, ed., Annals Numer. Math., 4 (1997), pp. 29-38.
    • (1997) Annals Numer. Math. , vol.4 , pp. 29-38
    • Bai, Z.1    Golub, G.H.2
  • 2
    • 0030571227 scopus 로고    scopus 로고
    • Some large-scale matrix computation problems
    • Z. BAI, M. FAHEY, AND G. 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.3
  • 3
    • 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
  • 4
    • 0040901903 scopus 로고
    • Estimating the largest singular values of large sparse matrices via modified moments
    • M. BERRY AND G. GOLUB, Estimating the largest singular values of large sparse matrices via modified moments, Numer. Algorithms, 1 (1991), pp. 353-374.
    • (1991) Numer. Algorithms , vol.1 , pp. 353-374
    • Berry, M.1    Golub, G.2
  • 5
    • 21344496401 scopus 로고
    • An adaptive Chebyshev iterative method for nonsymmetric linear systems based on modified moments
    • D. CALVETTI, G. H. GOLUB, AND L. REICHEL, An adaptive Chebyshev iterative method for nonsymmetric linear systems based on modified moments, Numer. Math., 67 (1994), pp. 21-40.
    • (1994) Numer. Math. , vol.67 , pp. 21-40
    • Calvetti, D.1    Golub, G.H.2    Reichel, L.3
  • 6
    • 0033098565 scopus 로고    scopus 로고
    • A computable error bound for matrix functionals
    • D. CALVETTI, G. H. GOLUB, AND L. REICHEL, A computable error bound for matrix functionals, J. Comput. Appl. Math., 103 (1999), pp. 301-306.
    • (1999) J. Comput. Appl. Math. , vol.103 , pp. 301-306
    • Calvetti, D.1    Golub, G.H.2    Reichel, L.3
  • 7
    • 0000924088 scopus 로고    scopus 로고
    • Estimation of the L-curve via Lanczos bidiagonalization
    • D. CALVETTI, G. H. GOLUB, AND L. REICHEL, Estimation of the L-curve via Lanczos bidiagonalization, BIT, 39 (1999), pp. 603-619.
    • (1999) BIT , vol.39 , pp. 603-619
    • Calvetti, D.1    Golub, G.H.2    Reichel, L.3
  • 8
    • 0034423846 scopus 로고    scopus 로고
    • Computation of Gauss-Kronrod quadrature rules
    • D. CALVETTI, G. H. GOLUB, W. B. GRAGG, AND L. REICHEL, Computation of Gauss-Kronrod quadrature rules, Math. Comp., 69 (2000), 1035-1052.
    • (2000) Math. Comp. , vol.69 , pp. 1035-1052
    • Calvetti, D.1    Golub, G.H.2    Gragg, W.B.3    Reichel, L.4
  • 9
    • 0035151353 scopus 로고    scopus 로고
    • An iterative method with error estimators
    • Numerical Analysis 2000, Quadrature and Orthogonal Polynomials, W. Gautschi, F. Marcellán, and L. Reichel, eds.
    • D. CALVETTI, S. MORIGI, L. REICHEL, AND F. SGALLARI, An iterative method with error estimators, in Numerical Analysis 2000, Vol. 5, Quadrature and Orthogonal Polynomials, W. Gautschi, F. Marcellán, and L. Reichel, eds., J. Comput. Appl. Math., 127 (2001), pp. 93-119.
    • (2001) J. Comput. Appl. Math. , vol.5-127 , pp. 93-119
    • Calvetti, D.1    Morigi, S.2    Reichel, L.3    Sgallari, F.4
  • 10
    • 0003712558 scopus 로고
    • Bounds for the error of linear systems of equations using the theory of moments
    • G. DAHLQUIST, S. C. EISENSTAT AND G. H. GOLUB, Bounds for the error of linear systems of equations 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
  • 11
    • 0041590254 scopus 로고
    • Bounds for the error in linear systems
    • Semi-Infinite Programming (Proc. Workshop, Bad Honnef, 1978), R. Hettich, ed., Springer-Verlag, Berlin
    • G. DAHLQUIST, G. H. GOLUB, AND S. G. NASH, Bounds for the error in linear systems, in Semi-Infinite Programming (Proc. Workshop, Bad Honnef, 1978), R. Hettich, ed., Lecture Notes in Control and Information Sci., Vol. 15, Springer-Verlag, Berlin, 1979, pp. 154-172.
    • (1979) Lecture Notes in Control and Information Sci. , vol.15 , pp. 154-172
    • Dahlquist, G.1    Golub, G.H.2    Nash, S.G.3
  • 12
    • 0000149521 scopus 로고
    • Updating and downdating of orthogonal polynomials with data fitting applications
    • S. ELHAY, G. H. GOLUB, AND J. KAUTSKY, Updating and downdating of orthogonal polynomials with data fitting applications, SIAM J. Matrix Anal. Appl., 12 (1991), pp. 327-353.
    • (1991) SIAM J. Matrix Anal. Appl. , vol.12 , pp. 327-353
    • Elhay, S.1    Golub, G.H.2    Kautsky, J.3
  • 13
    • 0038860753 scopus 로고
    • Jacobi matrices for sums of weight functions
    • S. ELHAY, G. H. GOLUB, AND J. KAUTSKY, Jacobi matrices for sums of weight functions, BIT, 32 (1992), pp. 143-166.
    • (1992) BIT , vol.32 , pp. 143-166
    • Elhay, S.1    Golub, G.H.2    Kautsky, J.3
  • 14
    • 84966259408 scopus 로고
    • On generating polynomials which are orthogonal over several intervals
    • B. FISCHER AND G. H. GOLUB, On generating polynomials which are orthogonal over several intervals, Math. Comp., 56 (1991), pp. 711-730.
    • (1991) Math. Comp. , vol.56 , pp. 711-730
    • Fischer, B.1    Golub, G.H.2
  • 15
    • 0006327956 scopus 로고
    • How to generate unknown orthogonal polynomials out of known orthogonal polynomials
    • B. FISCHER AND G. H. GOLUB, How to generate unknown orthogonal polynomials out of known orthogonal polynomials, J. Comput. Appl. Math., 43 (1992), pp. 99-115.
    • (1992) J. Comput. Appl. Math. , vol.43 , pp. 99-115
    • Fischer, B.1    Golub, G.H.2
  • 16
    • 84968490637 scopus 로고
    • An implementation of Christoffel's theorem in the theory of orthogonal polynomials
    • D. GALANT, An implementation of Christoffel's theorem in the theory of orthogonal polynomials, Math. Comp., 25 (1971), pp. 111-113.
    • (1971) Math. Comp. , vol.25 , pp. 111-113
    • Galant, D.1
  • 17
    • 0037823843 scopus 로고    scopus 로고
    • Adaptive quadrature -revisited
    • W. GANDER AND W. GAUTSCHI, Adaptive quadrature -revisited, BIT, 40 (2000), pp. 84-101.
    • (2000) BIT , vol.40 , pp. 84-101
    • Gander, W.1    Gautschi, W.2
  • 18
    • 0345779164 scopus 로고    scopus 로고
    • AMS Chelsea Publishing, Providence, RI, [Translated from the Russian by K. A. Hirsch. Reprint of the 1959 translation.]
    • F. R. GANTMACHER, The theory of matrices, Vol. 1, AMS Chelsea Publishing, Providence, RI, 1998. [Translated from the Russian by K. A. Hirsch. Reprint of the 1959 translation.]
    • (1998) The Theory of Matrices , vol.1
    • Gantmacher, F.R.1
  • 19
    • 0002308241 scopus 로고
    • E. B. Christoffel (Aachen/Monschau, 1979), P L. Butzer and F. Fehér, eds., Birkhäuser, Basel
    • W. GAUTSCHI, A survey of Gauss-Christoffel quadrature formulae, in E. B. Christoffel (Aachen/Monschau, 1979), P L. Butzer and F. Fehér, eds., Birkhäuser, Basel, 1981, pp. 72-147.
    • (1981) A Survey of Gauss-Christoffel Quadrature Formulae , pp. 72-147
    • Gautschi, W.1
  • 20
    • 0001849258 scopus 로고
    • An algorithmic implementation of the generalized Christoffel theorem
    • Numerical Integration, G. Hämmerlin, ed., Birkhäuser, Basel
    • W. GAUTSCHI, An algorithmic implementation of the generalized Christoffel theorem, in Numerical Integration, G. Hämmerlin, ed., Internat. Ser. Numer. Math., Vol. 57, 1982, Birkhäuser, Basel, pp. 89-106.
    • (1982) Internat. Ser. Numer. Math. , vol.57 , pp. 89-106
    • Gautschi, W.1
  • 21
    • 0001505159 scopus 로고
    • Ongenerating orthogonal polynomials
    • W. GAUTSCHI, Ongenerating orthogonal polynomials, SIAM J. Sci. Statist. Comput., 3 (1982), pp. 289-317.
    • (1982) SIAM J. Sci. Statist. Comput. , vol.3 , pp. 289-317
    • Gautschi, W.1
  • 22
    • 0031270469 scopus 로고    scopus 로고
    • s-orthogonality and construction of Gauss-Túran-type quadrature formulae
    • W. GAUTSCHI AND G. V. MILOVANOVIĆ, s-orthogonality and construction of Gauss-Túran-type quadrature formulae, J. Comput. Appl. Math., 86 (1997), pp. 205-218.
    • (1997) J. Comput. Appl. Math. , vol.86 , pp. 205-218
    • Gautschi, W.1    Milovanović, G.V.2
  • 23
    • 0001070999 scopus 로고
    • Some modified matrix eigenvalue problems
    • G. H. GOLUB, Some modified matrix eigenvalue problems, SIAM Rev., 15 (1973), pp. 318-334.
    • (1973) SIAM Rev. , vol.15 , pp. 318-334
    • Golub, G.H.1
  • 24
    • 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
  • 25
    • 2942621702 scopus 로고
    • Matrix computation and the theory of moments
    • (Zürich, 1994), S. D. Chatterji, ed., Birkhäuser, Basel
    • G. H. GOLUB, Matrix computation and the theory of moments, in Proceedings of the International Congress of Mathematicians (Zürich, 1994), S. D. Chatterji, ed., Vol. 2, Birkhäuser, Basel, 1995, pp. 1440-1448.
    • (1995) Proceedings of the International Congress of Mathematicians , vol.2 , pp. 1440-1448
    • Golub, G.H.1
  • 27
    • 0042675294 scopus 로고
    • Modified moments for indefinite weight functions
    • G. H. GOLUB AND M. H. GUTKNECHT, Modified moments for indefinite weight functions, Numer. Math., 57 (1990), pp. 607-624.
    • (1990) Numer. Math. , vol.57 , pp. 607-624
    • Golub, G.H.1    Gutknecht, M.H.2
  • 28
    • 0001494898 scopus 로고
    • Calculation of Gauss quadratures with multiple free and fixed knots
    • G. H. GOLUB AND J. KAUTSKY, Calculation of Gauss quadratures with multiple free and fixed knots, Numer. Math., 41 (1983), pp. 147-163.
    • (1983) Numer. Math. , vol.41 , pp. 147-163
    • Golub, G.H.1    Kautsky, J.2
  • 29
    • 84968500647 scopus 로고
    • Estimates of eigenvalues for iterative methods
    • G. H. GOLUB AND M. D. KENT, Estimates of eigenvalues for iterative methods, Math. Comp., 53 (1989), pp. 619-626.
    • (1989) Math. Comp. , vol.53 , pp. 619-626
    • Golub, G.H.1    Kent, M.D.2
  • 30
    • 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
  • 31
    • 0003093333 scopus 로고
    • Matrices, moments and quadrature
    • Numerical Analysis 1993 (Dundee, 1993), Longman Sci. Tech., Harlow
    • G. H. GOLUB AND G. MEURANT, Matrices, moments and quadrature, in Numerical Analysis 1993 (Dundee, 1993), Pitman Res. Notes Math. Ser., Vol. 303, Longman Sci. Tech., Harlow, 1994.
    • (1994) Pitman Res. Notes Math. Ser. , vol.303
    • Golub, G.H.1    Meurant, G.2
  • 32
    • 0000391978 scopus 로고
    • Matrices, moments and quadrature. II. How to compute the norm of the error in iterative methods
    • G. H. GOLUB AND G. MEURANT, Matrices, moments and quadrature. II. How to compute the norm of the error in iterative methods, BIT, 37 (1994), pp. 687-705.
    • (1994) BIT , vol.37 , pp. 687-705
    • Golub, G.H.1    Meurant, G.2
  • 33
    • 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
  • 34
    • 0000223746 scopus 로고
    • Calculation of Gauss quadrature rules
    • G. H. GOLUB AND J. H. WELSCH, Calculation of Gauss quadrature rules, Math. Comp., 23 (1969), pp. 221-230.
    • (1969) Math. Comp. , vol.23 , pp. 221-230
    • Golub, G.H.1    Welsch, J.H.2
  • 35
    • 53949095744 scopus 로고    scopus 로고
    • Addendum
    • [Addendum: loose microfiche suppl. A1-A10.]
    • Loose Microfiche , Issue.SUPPL.
  • 36
    • 0000072192 scopus 로고
    • Error bounds in equilibrium statistical mechanics
    • R. G. GORDON, Error bounds in equilibrium statistical mechanics, J. Mathematical Phys., 9 (1968), pp. 655-663.
    • (1968) J. Mathematical Phys. , vol.9 , pp. 655-663
    • Gordon, R.G.1
  • 37
    • 0001190030 scopus 로고
    • The numerically stable reconstruction of Jacobi matrices from spectral data
    • W. B. GRAGG AND W. J. HARROD, The numerically stable reconstruction of Jacobi matrices from spectral data, Numer. Math., 44 (1994), pp. 317-335.
    • (1994) Numer. Math. , vol.44 , pp. 317-335
    • Gragg, W.B.1    Harrod, W.J.2
  • 38
    • 0000135303 scopus 로고
    • Methods of conjugate gradients for solving linear systems
    • M. R. HESTENES AND E. STIEFEL, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards, 49 (1952), pp. 409-436.
    • (1952) J. Research Nat. Bur. Standards , vol.49 , pp. 409-436
    • Hestenes, M.R.1    Stiefel, E.2
  • 39
    • 0001300342 scopus 로고
    • Calculation of the weights of interpolatory quadratures
    • J. KAUTSKY AND S. ELHAY, Calculation of the weights of interpolatory quadratures, Numer. Math., 40 (1982), pp. 407-422.
    • (1982) Numer. Math. , vol.40 , pp. 407-422
    • Kautsky, J.1    Elhay, S.2
  • 40
    • 0039546363 scopus 로고
    • On the calculation of Jacobi matrices
    • J. KAUTSKY AND G. H. GOLUB, On the calculation of Jacobi matrices, Linear Algebra Appl., 52/53 (1983), pp. 439-455.
    • (1983) Linear Algebra Appl. , vol.52-53 , pp. 439-455
    • Kautsky, J.1    Golub, G.H.2
  • 42
    • 0031520388 scopus 로고    scopus 로고
    • Calculation of Gauss-Kronrod quadrature rules
    • D. P. LAURIE, Calculation of Gauss-Kronrod quadrature rules, Math. Comp., 66 (1997), pp. 1133-1145.
    • (1997) Math. Comp. , vol.66 , pp. 1133-1145
    • Laurie, D.P.1
  • 43
    • 0003674259 scopus 로고
    • Classical Orthogonal Polynomials of a Discrete Variable
    • Springer-Verlag, Berlin, [Translated from the Russian.]
    • A. F. NIKIFOROV, S. K. SUSLOV, AND V. B. UVAROV, Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991. [Translated from the Russian.]
    • (1991) Springer Series in Computational Physics
    • Nikiforov, A.F.1    Suslov, S.K.2    Uvarov, V.B.3
  • 44
    • 0003358142 scopus 로고    scopus 로고
    • The Symmetric Eigenvalue Problem
    • SIAM, Philadelphia, PA, [Corrected reprint of the 1980 original.]
    • B. N. PARLETT, The Symmetric Eigenvalue Problem, Classics in Applied Mathematics, Vol. 20, SIAM, Philadelphia, PA, 1998. [Corrected reprint of the 1980 original.]
    • (1998) Classics in Applied Mathematics , vol.20
    • Parlett, B.N.1
  • 46
    • 0002115888 scopus 로고
    • Kernel polynomials in linear algebra and their numerical applications
    • Further Contributions to the Solution of Simultaneous Linear Equations and the Determination of Eigenvalues, U.S. Government Printing Office, Washington, DC
    • E. L. STIEFEL, Kernel polynomials in linear algebra and their numerical applications, in Further Contributions to the Solution of Simultaneous Linear Equations and the Determination of Eigenvalues, Nat. Bur. Standards Appl. Math. Ser., Vol. 49, U.S. Government Printing Office, Washington, DC, 1958, pp. 1-22.
    • (1958) Nat. Bur. Standards Appl. Math. Ser. , vol.49 , pp. 1-22
    • Stiefel, E.L.1
  • 47
    • 0003960161 scopus 로고
    • John Wiley, New York, [Reprinted in 1978 by Dover Publications, New York.]
    • H. S. WILF, Mathematics for the Physical Sciences, John Wiley, New York, 1962. [Reprinted in 1978 by Dover Publications, New York.]
    • (1962) Mathematics for the Physical Sciences
    • Wilf, H.S.1
  • 48
    • 53949122556 scopus 로고
    • Personal communication
    • H. S. WILF, Personal communication, 1980.
    • (1980)
    • Wilf, H.S.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.