메뉴 건너뛰기




Volumn 123, Issue 1-2, 2000, Pages 35-65

Eigenvalue computation in the 20th century

Author keywords

65F15; 65N25; Arnoldi's method; Jacobi's method; Jacobi Davidson; Lanczos' method; Perturbation theory; Power iteration; QR; QZ; RKS; SVD

Indexed keywords

COMPUTATIONAL METHODS; CONVERGENCE OF NUMERICAL METHODS; ITERATIVE METHODS; LINEAR ALGEBRA; LINEAR SYSTEMS; NUMERICAL ANALYSIS; PERTURBATION TECHNIQUES; PROBLEM SOLVING; VECTORS;

EID: 0034321492     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(00)00413-1     Document Type: Article
Times cited : (342)

References (164)
  • 1
    • 0002807741 scopus 로고
    • The principle of minimized iteration in the solution of the matrix eigenproblem
    • Arnoldi W.E. The principle of minimized iteration in the solution of the matrix eigenproblem. Quart. Appl. Math. 9:1951;17-29.
    • (1951) Quart. Appl. Math. , vol.9 , pp. 17-29
    • Arnoldi, W.E.1
  • 5
    • 84985384413 scopus 로고
    • Progress in the numerical solution of the nonsymmetric eigenvalue problem
    • Bai Z. Progress in the numerical solution of the nonsymmetric eigenvalue problem. Numer. Linear Algebra Appl. 2:1995;219-234.
    • (1995) Numer. Linear Algebra Appl. , vol.2 , pp. 219-234
    • Bai, Z.1
  • 7
    • 0000482693 scopus 로고
    • Das Verfahren der Treppeniteration und verwandte Verfahren zur Lösung algebraischers Eigenwertprobleme
    • Bauer F.L. Das Verfahren der Treppeniteration und verwandte Verfahren zur Lösung algebraischers Eigenwertprobleme. Z. Angew. Math. Phys. 8:1957;214-235.
    • (1957) Z. Angew. Math. Phys. , vol.8 , pp. 214-235
    • Bauer, F.L.1
  • 8
    • 4644369885 scopus 로고
    • Norms and exclusion theorems
    • Bauer F.L., Fike C.T. Norms and exclusion theorems. Numer. Math. 2:1960;137-141.
    • (1960) Numer. Math. , vol.2 , pp. 137-141
    • Bauer, F.L.1    Fike, C.T.2
  • 9
    • 0039287162 scopus 로고
    • The two-sided block Jacobi method on hypercube architectures
    • M.T. Heath. Philadelphia, PA: SIAM
    • Bischof C.H. The two-sided block Jacobi method on hypercube architectures. Heath M.T. Hypercube Multiprocessors. 1987;SIAM, Philadelphia, PA.
    • (1987) Hypercube Multiprocessors
    • Bischof, C.H.1
  • 13
    • 0001595001 scopus 로고    scopus 로고
    • Computing the field of values and pseudospectra using the Lanczos method with continuation
    • Braconnier T., Higham N.J. Computing the field of values and pseudospectra using the Lanczos method with continuation. BIT. 36:1996;422-440.
    • (1996) BIT , vol.36 , pp. 422-440
    • Braconnier, T.1    Higham, N.J.2
  • 14
    • 0002115256 scopus 로고
    • The solution of singular value and symmetric eigenvalue problems on multiprocessor arrays
    • Brent R.P., Luk F.T. The solution of singular value and symmetric eigenvalue problems on multiprocessor arrays. SIAM J. Sci. Statist. Comput. 6:1985;69-84.
    • (1985) SIAM J. Sci. Statist. Comput. , vol.6 , pp. 69-84
    • Brent, R.P.1    Luk, F.T.2
  • 15
    • 84945709007 scopus 로고
    • Algorithm 358: Singular value decomposition of a complex matrix
    • Businger P.A., Golub G.H. Algorithm 358: Singular value decomposition of a complex matrix. Comm. Assoc. Comp. Mach. 12:1969;564-565.
    • (1969) Comm. Assoc. Comp. Mach. , vol.12 , pp. 564-565
    • Businger, P.A.1    Golub, G.H.2
  • 16
    • 0007949423 scopus 로고
    • Spectral portrait for non-Hermitian large sparse matrices
    • Carpraux J.F., Erhel J., Sadkane M. Spectral portrait for non-Hermitian large sparse matrices. Computing. 53:1994;301-310.
    • (1994) Computing , vol.53 , pp. 301-310
    • Carpraux, J.F.1    Erhel, J.2    Sadkane, M.3
  • 24
    • 0040145952 scopus 로고
    • A practical procedure for computing eigenvalues of large sparse nonsymmetric matrices
    • J. Cullum, & R.A. Willoughby. North-Holland: Amsterdam
    • Cullum J., Willoughby R.A. A practical procedure for computing eigenvalues of large sparse nonsymmetric matrices. Cullum J., Willoughby R.A. Large Scale Eigenvalue Problems. 1986;193-240 Amsterdam, North-Holland.
    • (1986) Large Scale Eigenvalue Problems , pp. 193-240
    • Cullum, J.1    Willoughby, R.A.2
  • 25
    • 0000659575 scopus 로고
    • A divide and conquer method for the symmetric eigenproblem
    • Cuppen J.J.M. A divide and conquer method for the symmetric eigenproblem. Numer. Math. 36:1981;177-195.
    • (1981) Numer. Math. , vol.36 , pp. 177-195
    • Cuppen, J.J.M.1
  • 26
    • 5344230920 scopus 로고
    • The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real symmetric matrices
    • Davidson E.R. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real symmetric matrices. J. Comput. Phys. 17:1975;87-94.
    • (1975) J. Comput. Phys. , vol.17 , pp. 87-94
    • Davidson, E.R.1
  • 32
    • 0000417182 scopus 로고
    • A fully parallel algorithm for the symmetric eigenvalue problem
    • Dongarra J.J., Sorensen D.C. A fully parallel algorithm for the symmetric eigenvalue problem. SIAM J. Sci. Statist. Comput. 8:1987;139-154.
    • (1987) SIAM J. Sci. Statist. Comput. , vol.8 , pp. 139-154
    • Dongarra, J.J.1    Sorensen, D.C.2
  • 33
    • 0029324485 scopus 로고
    • Software libraries for linear algebra computations on high performance computers
    • Dongarra J.J., Walker D.W. Software libraries for linear algebra computations on high performance computers. SIAM Rev. 37:1995;151-180.
    • (1995) SIAM Rev. , vol.37 , pp. 151-180
    • Dongarra, J.J.1    Walker, D.W.2
  • 34
    • 0000942136 scopus 로고
    • A minimax theory for overdamped networks
    • Duffin R.J. A minimax theory for overdamped networks. J. Rational Mech. Anal. 4:1955;221-233.
    • (1955) J. Rational Mech. Anal. , vol.4 , pp. 221-233
    • Duffin, R.J.1
  • 35
    • 0000129084 scopus 로고
    • Iterative methods in eigenvalue problems
    • D'yakonov E.G. Iterative methods in eigenvalue problems. Math. Notes. 34:1983;945-953.
    • (1983) Math. Notes , vol.34 , pp. 945-953
    • D'yakonov, E.G.1
  • 36
    • 0040471978 scopus 로고
    • Solution to the complex eigenproblem by a norm-reducing Jacobi-type method
    • Eberlein P.J. Solution to the complex eigenproblem by a norm-reducing Jacobi-type method. Numer. Math. 14:1970;232-245.
    • (1970) Numer. Math. , vol.14 , pp. 232-245
    • Eberlein, P.J.1
  • 38
    • 84972535723 scopus 로고
    • Block diagonally dominant matrices in generalizations of the Gerschgorin circle theorem
    • Feingold D.G., Varga R.S. Block diagonally dominant matrices in generalizations of the Gerschgorin circle theorem. Pacific J. Math. 12:1962;1241-1250.
    • (1962) Pacific J. Math. , vol.12 , pp. 1241-1250
    • Feingold, D.G.1    Varga, R.S.2
  • 39
    • 0032131701 scopus 로고    scopus 로고
    • Jacobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils
    • Fokkema D.R., Sleijpen G.L.G., van der Vorst H.A. Jacobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM J. Sci. Comput. 20:1996;94-125.
    • (1996) SIAM J. Sci. Comput. , vol.20 , pp. 94-125
    • Fokkema, D.R.1    Sleijpen, G.L.G.2    Van Der Vorst, H.A.3
  • 40
    • 0000394782 scopus 로고
    • The QR transformation: A unitary analogue to the LR transformation
    • 332-345
    • J.G.F. Francis, The QR transformation: a unitary analogue to the LR transformation, Parts I and II Comput. J. 4 1961 265-272, 332-345.
    • (1961) Parts I and II Comput. J. , vol.4 , pp. 265-272
    • Francis, J.G.F.1
  • 42
    • 38249009170 scopus 로고
    • Quasi-kernel polynomials and their use in non-Hermitian matrix iterations
    • Freund R.W. Quasi-kernel polynomials and their use in non-Hermitian matrix iterations. J. Comput. Appl. Math. 43:1992;135-158.
    • (1992) J. Comput. Appl. Math. , vol.43 , pp. 135-158
    • Freund, R.W.1
  • 44
  • 47
    • 0002841144 scopus 로고
    • Some uses of the Lanczos algorithm in numerical linear algebra
    • J.J.H. Miller. New York: Academic Press
    • Golub G.H. Some uses of the Lanczos algorithm in numerical linear algebra. Miller J.J.H. Topics in Numerical Analysis. 1973;173-184 Academic Press, New York.
    • (1973) Topics in Numerical Analysis , pp. 173-184
    • Golub, G.H.1
  • 49
    • 0000288016 scopus 로고
    • Calculating the singular values and pseudo-inverse of a matrix
    • Golub G.H., Kahan W. Calculating the singular values and pseudo-inverse of a matrix. SIAM J. Numer. Anal. 2:1965;205-224.
    • (1965) SIAM J. Numer. Anal. , vol.2 , pp. 205-224
    • Golub, G.H.1    Kahan, W.2
  • 50
    • 0002302093 scopus 로고    scopus 로고
    • 150 Years old and still alive: Eigenproblems
    • in: I.S. Duff, G.A. Watson (Eds.) Clarendon Press, Oxford
    • G.H. Golub, H.A. van der Vorst, 150 Years old and still alive: Eigenproblems, in: I.S. Duff, G.A. Watson (Eds.), The State of the Art in Numerical Analysis, Clarendon Press, Oxford, 1997, pp. 93-119.
    • (1997) The State of the Art in Numerical Analysis , pp. 93-119
    • Golub, G.H.1    Van Der Vorst, H.A.2
  • 52
    • 0017011163 scopus 로고
    • Ill-conditioned eigensystems and the computation of the Jordan canonical form
    • Golub G.H., Wilkinson J.H. Ill-conditioned eigensystems and the computation of the Jordan canonical form. SIAM Rev. 18:1976;578-619.
    • (1976) SIAM Rev. , vol.18 , pp. 578-619
    • Golub, G.H.1    Wilkinson, J.H.2
  • 53
    • 0040471953 scopus 로고
    • A method for finding roots of arbitrary matrices
    • Greenstadt J. A method for finding roots of arbitrary matrices. Math. Tables Aids Comput. 9:1955;47-52.
    • (1955) Math. Tables Aids Comput. , vol.9 , pp. 47-52
    • Greenstadt, J.1
  • 54
    • 84968505027 scopus 로고
    • Computing eigenvalues and eigenvectors of a symmetric matrix on the ILLIAC
    • Gregory R.T. Computing eigenvalues and eigenvectors of a symmetric matrix on the ILLIAC. Math. Tables Aids Comput. 7:1953;215-220.
    • (1953) Math. Tables Aids Comput. , vol.7 , pp. 215-220
    • Gregory, R.T.1
  • 55
    • 21844526695 scopus 로고
    • A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem
    • Gu M., Eisenstat S.C. A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. SIAM J. Matrix Anal. Appl. 16:1995;172-191.
    • (1995) SIAM J. Matrix Anal. Appl. , vol.16 , pp. 172-191
    • Gu, M.1    Eisenstat, S.C.2
  • 56
    • 21844500966 scopus 로고
    • Numerical solutions for large sparse eigenvalue problems
    • Guo J.-S., Lin W.-W., Wang C.-S. Numerical solutions for large sparse eigenvalue problems. Linear Algebra Appl. 225:1995;57-89.
    • (1995) Linear Algebra Appl. , vol.225 , pp. 57-89
    • Guo, J.-S.1    Lin, W.-W.2    Wang, C.-S.3
  • 57
    • 0000497979 scopus 로고
    • A completed theory of the unsymmetric Lanczos process and related algorithms, Part I
    • Gutknecht M.H. A completed theory of the unsymmetric Lanczos process and related algorithms, Part I. SIAM J. Matrix Anal. Appl. 13:1992;594-639.
    • (1992) SIAM J. Matrix Anal. Appl. , vol.13 , pp. 594-639
    • Gutknecht, M.H.1
  • 59
    • 0040742107 scopus 로고
    • On the speed of convergence of cyclic and quasicyclic Jacobi methods for computing the eigenvalues of Hermitian matrices
    • Henrici P. On the speed of convergence of cyclic and quasicyclic Jacobi methods for computing the eigenvalues of Hermitian matrices. SIAM J. Appl. Math. 6:1958;144-162.
    • (1958) SIAM J. Appl. Math. , vol.6 , pp. 144-162
    • Henrici, P.1
  • 61
    • 0000678479 scopus 로고
    • A new proof of global convergence for the tridiagonal QL algorithm
    • Hoffmann W., Parlett B.N. A new proof of global convergence for the tridiagonal QL algorithm. SIAM J. Numer. Anal. 15:1978;929-937.
    • (1978) SIAM J. Numer. Anal. , vol.15 , pp. 929-937
    • Hoffmann, W.1    Parlett, B.N.2
  • 62
    • 0039287076 scopus 로고
    • A method of factor analysis by means of which all coordinates of the factor matrix are given simultaneously
    • Horst P. A method of factor analysis by means of which all coordinates of the factor matrix are given simultaneously. Psychometrika. 2:1937;225-236.
    • (1937) Psychometrika , vol.2 , pp. 225-236
    • Horst, P.1
  • 63
    • 0000652188 scopus 로고
    • Unitary triangularization of a nonsymmetric matrix
    • Householder A.S. Unitary triangularization of a nonsymmetric matrix. J. ACM. 5:1958;339-342.
    • (1958) J. ACM , vol.5 , pp. 339-342
    • Householder, A.S.1
  • 65
    • 84868361941 scopus 로고
    • Über ein leichtes Verfahren, die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen
    • C.G.J. Jacobi, Über ein leichtes Verfahren, die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen, J. Reine Angew. Math. 30 (1846) 51-94.
    • (1846) J. Reine Angew. Math. , vol.30 , pp. 51-94
    • Jacobi, C.G.J.1
  • 66
    • 0021564159 scopus 로고
    • The generalized singular value decomposition and the general A-λ B problem
    • Kågström B. The generalized singular value decomposition and the general. A-λ B problem BIT. 24:1985;568-583.
    • (1985) BIT , vol.24 , pp. 568-583
    • Kågström, B.1
  • 67
    • 0000169752 scopus 로고
    • RGSVD: An algorithm for computing the Kronecker structure and reducing subspaces of singular A-λ B pencils
    • Kågström B. RGSVD: an algorithm for computing the Kronecker structure and reducing subspaces of singular. A-λ B pencils SIAM J. Sci. Statist. Comput. 7:1986;185-211.
    • (1986) SIAM J. Sci. Statist. Comput. , vol.7 , pp. 185-211
    • Kågström, B.1
  • 68
    • 0008441955 scopus 로고
    • How far should you go with the Lanczos process
    • J.R. Bunch, & D.J. Rose. New York: Academic Press
    • Kahan W., Parlett B.N. How far should you go with the Lanczos process. Bunch J.R., Rose D.J. Sparse Matrix Computations. 1976;Academic Press, New York.
    • (1976) Sparse Matrix Computations
    • Kahan, W.1    Parlett, B.N.2
  • 69
    • 84968508687 scopus 로고
    • Estimates for some computational techniques in linear algebra
    • Kaniel S. Estimates for some computational techniques in linear algebra. Math. Comput. 20:1966;369-378.
    • (1966) Math. Comput. , vol.20 , pp. 369-378
    • Kaniel, S.1
  • 70
    • 1642278713 scopus 로고
    • On the upper and lower bounds of eigenvalues
    • Kato T. On the upper and lower bounds of eigenvalues. J. Phys. Soc. Japan. 4:1949;334-339.
    • (1949) J. Phys. Soc. Japan , vol.4 , pp. 334-339
    • Kato, T.1
  • 71
    • 0000522446 scopus 로고    scopus 로고
    • Preconditioned eigensolvers - An oxymoron?
    • Knyazev A.V. Preconditioned eigensolvers - an oxymoron? Electron. Trans. Numer. Anal. 7:1998;104-123.
    • (1998) Electron. Trans. Numer. Anal. , vol.7 , pp. 104-123
    • Knyazev, A.V.1
  • 72
    • 33645149298 scopus 로고    scopus 로고
    • Preconditioned eigensolvers
    • in: Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, H. van der Vorst (Eds.), SIAM, Philadelphia, PA, in press
    • A. Knyazev, Preconditioned eigensolvers, in: Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, H. van der Vorst (Eds.), Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide, SIAM, Philadelphia, PA, 2000, in press.
    • (2000) Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide
    • Knyazev, A.1
  • 73
    • 0347108088 scopus 로고
    • O cislennom resenii uravnenija, kotorym v techniceskih voprasah opredeljajutsja castoty malyh kolebanii material'nyh sistem
    • A.N. Krylov, O cislennom resenii uravnenija, kotorym v techniceskih voprasah opredeljajutsja castoty malyh kolebanii material'nyh sistem, Izv. Akad. Nauk SSSR. Ser. Fiz.-Mat. 4 (1931) 491-539.
    • (1931) Izv. Akad. Nauk SSSR. Ser. Fiz.-Mat. , vol.4 , pp. 491-539
    • Krylov, A.N.1
  • 74
    • 0039966628 scopus 로고
    • On some algorithms for the solution of the complete eigenvalue problem
    • Kublanovskaja V.N. On some algorithms for the solution of the complete eigenvalue problem. Zh. Vych. Mat. 1:1961;555-570.
    • (1961) Zh. Vych. Mat. , vol.1 , pp. 555-570
    • Kublanovskaja, V.N.1
  • 75
    • 0039287073 scopus 로고
    • On an application of Newton's method to the determination of eigenvalues of λ -matrices
    • Kublanovskaja V.N. On an application of Newton's method to the determination of eigenvalues of. λ -matrices Soviet Math. Dokl. 10:1969;1240-1241.
    • (1969) Soviet Math. Dokl. , vol.10 , pp. 1240-1241
    • Kublanovskaja, V.N.1
  • 76
    • 0004523003 scopus 로고
    • On an approach to the solution of the generalized latent value problem for λ -matrices
    • Kublanovskaja V.N. On an approach to the solution of the generalized latent value problem for. λ -matrices SIAM J. Numer. Anal. 7:1970;532-537.
    • (1970) SIAM J. Numer. Anal. , vol.7 , pp. 532-537
    • Kublanovskaja, V.N.1
  • 78
    • 0000094594 scopus 로고
    • An iteration method for the solution of the eigenvalue problem of linear differential and integral operators
    • Lanczos C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Natl. Bur. Stand. 45:1950;225-280.
    • (1950) J. Res. Natl. Bur. Stand , vol.45 , pp. 225-280
    • Lanczos, C.1
  • 79
    • 51249188481 scopus 로고
    • On Szegö's eigenvalue distribution theory and non-Hermitian kernels
    • Landau H.J. On Szegö's eigenvalue distribution theory and non-Hermitian kernels. J. Anal. Math. 28:1975;335-357.
    • (1975) J. Anal. Math. , vol.28 , pp. 335-357
    • Landau, H.J.1
  • 80
    • 21844500067 scopus 로고
    • A practical upper bound for departure from normality
    • Lee S.L. A practical upper bound for departure from normality. SIAM J. Matrix Anal. Appl. 16:1995;462-468.
    • (1995) SIAM J. Matrix Anal. Appl. , vol.16 , pp. 462-468
    • Lee, S.L.1
  • 82
    • 84968468997 scopus 로고
    • Homotopy-determinant algorithm for solving nonsymmetric eigenvalue problems
    • Li T.-Y., Zeng Z. Homotopy-determinant algorithm for solving nonsymmetric eigenvalue problems. Math. Comput. 59:1992;483-502.
    • (1992) Math. Comput. , vol.59 , pp. 483-502
    • Li, T.-Y.1    Zeng, Z.2
  • 83
    • 0041066069 scopus 로고    scopus 로고
    • Nonlinear eigenvalue problems with orthogonality constraints
    • in: Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, H. van der Vorst (Eds.), SIAM, Philadelphia, PA, in press
    • R. Lippert, A. Edelman, Nonlinear eigenvalue problems with orthogonality constraints, in: Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, H. van der Vorst (Eds.), Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide, SIAM, Philadelphia, PA, 2000, in press.
    • (2000) Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide
    • Lippert, R.1    Edelman, A.2
  • 84
    • 0031082730 scopus 로고    scopus 로고
    • Computation of pseudospectra by continuation
    • Lui S.H. Computation of pseudospectra by continuation. SIAM J. Sci. Comput. 18:1997;565-573.
    • (1997) SIAM J. Sci. Comput. , vol.18 , pp. 565-573
    • Lui, S.H.1
  • 85
    • 0041066067 scopus 로고
    • Homotopy method for the numerical solution of the eigenvalue problem of self-adjoint partial differential operators
    • Lui S.H., Golub G.H. Homotopy method for the numerical solution of the eigenvalue problem of self-adjoint partial differential operators. Numer. Algorithms. 10:1995;363-378.
    • (1995) Numer. Algorithms , vol.10 , pp. 363-378
    • Lui, S.H.1    Golub, G.H.2
  • 86
    • 0031496956 scopus 로고    scopus 로고
    • Homotopy method for the large sparse real nonsymmetric eigenvalue problem
    • S.H. Lui, H.B. Keller, W.C. Kwok, Homotopy method for the large sparse real nonsymmetric eigenvalue problem, SIAM J. Matrix Anal. Appl. 18 (1997) 312-333.
    • (1997) SIAM J. Matrix Anal. Appl. , vol.18 , pp. 312-333
    • Lui, S.H.1    Keller, H.B.2    Kwok, W.C.3
  • 89
    • 0031539675 scopus 로고    scopus 로고
    • The restarted Arnoldi method applied to iterative linear system solvers for the computation of rightmost eigenvalues
    • Meerbergen K., Roose D. The restarted Arnoldi method applied to iterative linear system solvers for the computation of rightmost eigenvalues. SIAM J. Matrix Anal. Appl. 18:1997;1-20.
    • (1997) SIAM J. Matrix Anal. Appl. , vol.18 , pp. 1-20
    • Meerbergen, K.1    Roose, D.2
  • 90
    • 0039287071 scopus 로고
    • Singular value analysis of cryptograms
    • Moler C.B., Morrison D. Singular value analysis of cryptograms. Amer. Math. Monthly. 90:1983;78-87.
    • (1983) Amer. Math. Monthly , vol.90 , pp. 78-87
    • Moler, C.B.1    Morrison, D.2
  • 91
    • 0000356673 scopus 로고
    • An algorithm for generalized matrix eigenvalue problems
    • Moler C.B., Stewart G.W. An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10:1973;241-256.
    • (1973) SIAM J. Numer. Anal. , vol.10 , pp. 241-256
    • Moler, C.B.1    Stewart, G.W.2
  • 92
    • 0000965239 scopus 로고
    • Computing interior eigenvalues of large matrices
    • Morgan R.B. Computing interior eigenvalues of large matrices. Linear Algebra Appl. 154-156:1991;289-309.
    • (1991) Linear Algebra Appl. , vol.154-156 , pp. 289-309
    • Morgan, R.B.1
  • 93
    • 0001398614 scopus 로고
    • Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices
    • Morgan R.B., Scott D.S. Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices. SIAM J. Sci. Statist. Comput. 7(3):1986;817-825.
    • (1986) SIAM J. Sci. Statist. Comput. , vol.7 , Issue.3 , pp. 817-825
    • Morgan, R.B.1    Scott, D.S.2
  • 94
    • 0001742428 scopus 로고
    • Passing the one-billion limit in full configuration-interaction (FCI) calculations
    • Olsen J., J ø rgensen P., Simons J. Passing the one-billion limit in full configuration-interaction (FCI) calculations. Chem. Phys. Lett. 169:1990;463-472.
    • (1990) Chem. Phys. Lett. , vol.169 , pp. 463-472
    • Olsen, J.1    Rgensen, P.2    Simons, J.3
  • 95
    • 34250951234 scopus 로고    scopus 로고
    • On the convergence of the Rayleigh quotient iteration for the computation of characteristic roots and vectors I-VI
    • A.M. Ostrowski, On the convergence of the Rayleigh quotient iteration for the computation of characteristic roots and vectors I-VI, Arch. Rational. Mesh. Anal. I 1:233-241, II 2:423-428, III 3:325-340, IV 3:341-347, V 3:472-481, VI 4:153-165, 1958-1959.
    • Arch. Rational. Mesh. Anal. I , vol.1 , pp. 233-241
    • Ostrowski, A.M.1
  • 96
    • 34250951234 scopus 로고    scopus 로고
    • A.M. Ostrowski, On the convergence of the Rayleigh quotient iteration for the computation of characteristic roots and vectors I-VI, Arch. Rational. Mesh. Anal. I 1:233-241, II 2:423-428, III 3:325-340, IV 3:341-347, V 3:472-481, VI 4:153-165, 1958-1959.
    • Arch. Rational. Mesh. Anal. II , vol.2 , pp. 423-428
  • 97
    • 34250951234 scopus 로고    scopus 로고
    • A.M. Ostrowski, On the convergence of the Rayleigh quotient iteration for the computation of characteristic roots and vectors I-VI, Arch. Rational. Mesh. Anal. I 1:233-241, II 2:423-428, III 3:325-340, IV 3:341-347, V 3:472-481, VI 4:153-165, 1958-1959.
    • Arch. Rational. Mesh. Anal. III , vol.3 , pp. 325-340
  • 98
    • 34250951234 scopus 로고    scopus 로고
    • A.M. Ostrowski, On the convergence of the Rayleigh quotient iteration for the computation of characteristic roots and vectors I-VI, Arch. Rational. Mesh. Anal. I 1:233-241, II 2:423-428, III 3:325-340, IV 3:341-347, V 3:472-481, VI 4:153-165, 1958-1959.
    • Arch. Rational. Mesh. Anal. IV , vol.3 , pp. 341-347
  • 99
    • 34250951234 scopus 로고    scopus 로고
    • A.M. Ostrowski, On the convergence of the Rayleigh quotient iteration for the computation of characteristic roots and vectors I-VI, Arch. Rational. Mesh. Anal. I 1:233-241, II 2:423-428, III 3:325-340, IV 3:341-347, V 3:472-481, VI 4:153-165, 1958-1959.
    • Arch. Rational. Mesh. Anal. V , vol.3 , pp. 472-481
  • 100
    • 34250951234 scopus 로고    scopus 로고
    • 1958-1959.
    • A.M. Ostrowski, On the convergence of the Rayleigh quotient iteration for the computation of characteristic roots and vectors I-VI, Arch. Rational. Mesh. Anal. I 1:233-241, II 2:423-428, III 3:325-340, IV 3:341-347, V 3:472-481, VI 4:153-165, 1958-1959.
    • Arch. Rational. Mesh. Anal. VI , vol.4 , pp. 153-165
  • 101
    • 0005637341 scopus 로고
    • An eigenvalue algorithm for skew symmetric matrices
    • Paardekooper M.H.C. An eigenvalue algorithm for skew symmetric matrices. Numer. Math. 17:1971;189-202.
    • (1971) Numer. Math. , vol.17 , pp. 189-202
    • Paardekooper, M.H.C.1
  • 104
    • 0016555955 scopus 로고
    • Solution of sparse indefinite systems of linear equations
    • Paige C.C., Saunders M.A. Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12:1975;617-629.
    • (1975) SIAM J. Numer. Anal. , vol.12 , pp. 617-629
    • Paige, C.C.1    Saunders, M.A.2
  • 106
    • 0039287072 scopus 로고
    • Misconvergence in the Lanczos algorithm
    • in: M.G. Cox, S. Hammarling (Eds.), Clarendon Press, Oxford, UK, (Chapter 1)
    • B.N. Parlett, Misconvergence in the Lanczos algorithm, in: M.G. Cox, S. Hammarling (Eds.), Reliable Numerical Computation, Clarendon Press, Oxford, UK, 1990, pp. 7-24 (Chapter 1).
    • (1990) Reliable Numerical Computation , pp. 7-24
    • Parlett, B.N.1
  • 107
    • 0012756718 scopus 로고
    • Tracking the progress of the Lanczos algorithm for large symmetric eigenproblems
    • Parlett B.N., Reid J.K. Tracking the progress of the Lanczos algorithm for large symmetric eigenproblems. IMA J. Numer. Anal. 1:1981;135-155.
    • (1981) IMA J. Numer. Anal. , vol.1 , pp. 135-155
    • Parlett, B.N.1    Reid, J.K.2
  • 108
    • 84968487118 scopus 로고
    • The Lanczos algorithm with selective orthogonalization
    • Parlett B.N., Scott D.S. The Lanczos algorithm with selective orthogonalization. Math. Comput. 33:1979;217-238.
    • (1979) Math. Comput. , vol.33 , pp. 217-238
    • Parlett, B.N.1    Scott, D.S.2
  • 109
    • 84966231631 scopus 로고
    • A look-ahead Lanczos algorithm for unsymmetric matrices
    • Parlett B.N., Taylor D.R., Liu Z.A. A look-ahead Lanczos algorithm for unsymmetric matrices. Math. Comput. 44:1985;105-124.
    • (1985) Math. Comput. , vol.44 , pp. 105-124
    • Parlett, B.N.1    Taylor, D.R.2    Liu, Z.A.3
  • 110
    • 0039878919 scopus 로고
    • Maximizing functions of rotations: Experiments concerning speed of diagonalization of symmetric matrices using Jacobi's method
    • Pope D.A., Tompkins C. Maximizing functions of rotations: experiments concerning speed of diagonalization of symmetric matrices using Jacobi's method. J. ACM. 4:1957;459-466.
    • (1957) J. ACM , vol.4 , pp. 459-466
    • Pope, D.A.1    Tompkins, C.2
  • 111
    • 34250946417 scopus 로고
    • A minimax theory for overdamped systems
    • Rogers E.H. A minimax theory for overdamped systems. Arch. Rational Mech. Anal. 19:1964;89-96.
    • (1964) Arch. Rational Mech. Anal. , vol.19 , pp. 89-96
    • Rogers, E.H.1
  • 112
    • 0000219783 scopus 로고
    • Algorithms for the nonlinear eigenvalue problem
    • Ruhe A. Algorithms for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 10:1973;674-689.
    • (1973) SIAM J. Numer. Anal. , vol.10 , pp. 674-689
    • Ruhe, A.1
  • 113
    • 84968508607 scopus 로고
    • Implementation aspects of band Lanczos algorithms for computation of eigenvalues of large sparse symmetric matrices
    • Ruhe A. Implementation aspects of band Lanczos algorithms for computation of eigenvalues of large sparse symmetric matrices. Math. Comput. 33:1979;680-687.
    • (1979) Math. Comput. , vol.33 , pp. 680-687
    • Ruhe, A.1
  • 114
    • 0001409002 scopus 로고
    • The rational Krylov algorithm for nonsymmetric eigenvalue problems. iii. complex shifts for real matrices
    • Ruhe A. The rational Krylov algorithm for nonsymmetric eigenvalue problems. iii. complex shifts for real matrices. BIT. 34:1994;165-176.
    • (1994) BIT , vol.34 , pp. 165-176
    • Ruhe, A.1
  • 115
    • 43949153380 scopus 로고
    • Rational Krylov algorithms for nonsymmetric eigenvalue problems. ii. Matrix pairs
    • Ruhe A. Rational Krylov algorithms for nonsymmetric eigenvalue problems. ii. Matrix pairs. Linear Algebra Appl. 197 / 198:1994;283-295.
    • (1994) Linear Algebra Appl. , vol.197-198 , pp. 283-295
    • Ruhe, A.1
  • 116
    • 0002392285 scopus 로고
    • Solution of eigenvalue problems with the LR-transformation
    • Rutishauser H. Solution of eigenvalue problems with the LR-transformation. Nat. Bur. Standards Appl. Math. Ser. 49:1958;47-81.
    • (1958) Nat. Bur. Standards Appl. Math. Ser. , vol.49 , pp. 47-81
    • Rutishauser, H.1
  • 117
    • 0002198379 scopus 로고
    • Computational aspects of F.L. Bauer's simultaneous iteration method
    • Rutishauser H. Computational aspects of F.L. Bauer's simultaneous iteration method. Numer. Math. 13:1969;4-13.
    • (1969) Numer. Math. , vol.13 , pp. 4-13
    • Rutishauser, H.1
  • 118
    • 0040471905 scopus 로고
    • Simultaneous iteration method for symmetric matrices
    • J.H. Wilkinson, & C. Reinsch. Heidelberg: Springer Verlag
    • Rutishauser H. Simultaneous iteration method for symmetric matrices. Wilkinson J.H., Reinsch C. Handbook for Automatic Computation, Vol. 2, Linear Algebra. 1971;284-302 Springer Verlag, Heidelberg.
    • (1971) Handbook for Automatic Computation, Vol. 2, Linear Algebra , pp. 284-302
    • Rutishauser, H.1
  • 119
    • 0000767453 scopus 로고
    • On the rates of convergence of the Lanczos and the block Lanczos methods
    • Saad Y. On the rates of convergence of the Lanczos and the block Lanczos methods. SIAM J. Numer. Anal. 17:1980;687-706.
    • (1980) SIAM J. Numer. Anal. , vol.17 , pp. 687-706
    • Saad, Y.1
  • 120
    • 49149143501 scopus 로고
    • Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices
    • Saad Y. Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices. Linear Algebra Appl. 34:1980;269-295.
    • (1980) Linear Algebra Appl. , vol.34 , pp. 269-295
    • Saad, Y.1
  • 122
    • 0000048673 scopus 로고
    • GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems
    • Saad Y., Schultz M.H. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7:1986;856-869.
    • (1986) SIAM J. Sci. Statist. Comput. , vol.7 , pp. 856-869
    • Saad, Y.1    Schultz, M.H.2
  • 123
    • 84968513725 scopus 로고
    • On Jacobi and Jacobi-like algorithms for a parallel computer
    • Sameh A. On Jacobi and Jacobi-like algorithms for a parallel computer. Math. Comput. 25:1971;579-590.
    • (1971) Math. Comput. , vol.25 , pp. 579-590
    • Sameh, A.1
  • 124
    • 4243810288 scopus 로고
    • Zur Konvergenz des Jacobi-Verfahrens
    • Schönhage A. Zur Konvergenz des Jacobi-Verfahrens. Numer. Math. 3:1961;374-380.
    • (1961) Numer. Math. , vol.3 , pp. 374-380
    • Schönhage, A.1
  • 125
    • 1542498668 scopus 로고    scopus 로고
    • Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems
    • Sleijpen G.L.G., Booten J.G.L., Fokkema D.R., van der Vorst H.A. Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT. 36:1996;595-633.
    • (1996) BIT , vol.36 , pp. 595-633
    • Sleijpen, G.L.G.1    Booten, J.G.L.2    Fokkema, D.R.3    Van Der Vorst, H.A.4
  • 126
    • 0030560293 scopus 로고    scopus 로고
    • A Jacobi-Davidson iteration method for linear eigenvalue problems
    • Sleijpen G.L.G., van der Vorst H.A. A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17:1996;401-425.
    • (1996) SIAM J. Matrix Anal. Appl. , vol.17 , pp. 401-425
    • Sleijpen, G.L.G.1    Van Der Vorst, H.A.2
  • 127
    • 0001917562 scopus 로고    scopus 로고
    • Efficient expansion of subspaces in the Jacobi-Davidson method for standard and generalized eigenproblems
    • Sleijpen G.L.G., van der Vorst H.A., Meijerink E. Efficient expansion of subspaces in the Jacobi-Davidson method for standard and generalized eigenproblems. Electron. Trans. Numer. Anal. 7:1998;75-89.
    • (1998) Electron. Trans. Numer. Anal. , vol.7 , pp. 75-89
    • Sleijpen, G.L.G.1    Van Der Vorst, H.A.2    Meijerink, E.3
  • 130
    • 0000322706 scopus 로고
    • Implicit application of polynomial filters in a k-step Arnoldi method
    • Sorensen D.C. Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13(1):1992;357-385.
    • (1992) SIAM J. Matrix Anal. Appl. , vol.13 , Issue.1 , pp. 357-385
    • Sorensen, D.C.1
  • 131
    • 0000946417 scopus 로고
    • Simultaneous iteration for computing invariant subspaces of non-Hermitian matrices
    • Stewart G.W. Simultaneous iteration for computing invariant subspaces of non-Hermitian matrices. Numer. Math. 25:1976;123-136.
    • (1976) Numer. Math. , vol.25 , pp. 123-136
    • Stewart, G.W.1
  • 133
    • 0027879547 scopus 로고
    • On the early history of the singular value decomposition
    • Stewart G.W. On the early history of the singular value decomposition. SIAM Rev. 35:1993;551-566.
    • (1993) SIAM Rev. , vol.35 , pp. 551-566
    • Stewart, G.W.1
  • 136
    • 0000004244 scopus 로고
    • A recurring theorem on determinants
    • Taussky O. A recurring theorem on determinants. Amer. Math. Monthly. 56:1949;672-676.
    • (1949) Amer. Math. Monthly , vol.56 , pp. 672-676
    • Taussky, O.1
  • 137
    • 0008477857 scopus 로고
    • The accuracy of Rayleigh's method of calculating the natural frequencies of vibrating systems
    • Temple G. The accuracy of Rayleigh's method of calculating the natural frequencies of vibrating systems. Proc. Roy. Soc. A. 211:1952;204-224.
    • (1952) Proc. Roy. Soc. a , vol.211 , pp. 204-224
    • Temple, G.1
  • 139
    • 0033293693 scopus 로고    scopus 로고
    • A parallel divide and conquer algorithm for the symmetric eigenvalue problem on distributed memory architectures
    • Tisseur F., Dongarra J. A parallel divide and conquer algorithm for the symmetric eigenvalue problem on distributed memory architectures. SIAM J. Sci. Comput. 20:1999;2223-2236.
    • (1999) SIAM J. Sci. Comput. , vol.20 , pp. 2223-2236
    • Tisseur, F.1    Dongarra, J.2
  • 140
    • 0000992781 scopus 로고
    • Pseudospectra of matrices
    • D.F. Griffiths, & G.A. Watson. Longman
    • Trefethen L.N. Pseudospectra of matrices. Griffiths D.F., Watson G.A. Numerical Analysis 1991. 1992;234-266 Longman.
    • (1992) Numerical Analysis 1991 , pp. 234-266
    • Trefethen, L.N.1
  • 141
    • 85011519633 scopus 로고    scopus 로고
    • Computation of pseudospectra
    • Trefethen L.N. Computation of pseudospectra. Acta Numer. 8:1999;247-295.
    • (1999) Acta Numer. , vol.8 , pp. 247-295
    • Trefethen, L.N.1
  • 144
    • 0023433933 scopus 로고
    • An instability phenomenon in spectral methods
    • Trefethen L.N., Trummer M.R. An instability phenomenon in spectral methods. SIAM J. Numer. Anal. 24:1987;1008-1023.
    • (1987) SIAM J. Numer. Anal. , vol.24 , pp. 1008-1023
    • Trefethen, L.N.1    Trummer, M.R.2
  • 147
    • 38249033965 scopus 로고
    • The convergence behavior of Ritz values in the presence of close eigenvalues
    • van der Sluis A., van der Vorst H.A. The convergence behavior of Ritz values in the presence of close eigenvalues. Linear Algebra Appl. 88 / 89:1987;651-694.
    • (1987) Linear Algebra Appl. , vol.8889 , pp. 651-694
    • Van Der Sluis, A.1    Van Der Vorst, H.A.2
  • 148
    • 0000769280 scopus 로고
    • The computation of Kronecker's canonical form of a singular pencil
    • Van Dooren P. The computation of Kronecker's canonical form of a singular pencil. Linear Algebra Appl. 27:1979;103-140.
    • (1979) Linear Algebra Appl. , vol.27 , pp. 103-140
    • Van Dooren, P.1
  • 149
    • 0019529778 scopus 로고
    • The generalized eigenstructure problem in linear system theory
    • Van Dooren P. The generalized eigenstructure problem in linear system theory. IEEE Trans. Automat. Control AC. 26:1981;111-128.
    • (1981) IEEE Trans. Automat. Control AC , vol.26 , pp. 111-128
    • Van Dooren, P.1
  • 150
    • 0000669144 scopus 로고
    • A generalized eigenvalue approach of solving Ricatti equations
    • Van Dooren P. A generalized eigenvalue approach of solving Ricatti equations. SIAM J. Sci. Statist. Comput. 2:1981;121-135.
    • (1981) SIAM J. Sci. Statist. Comput. , vol.2 , pp. 121-135
    • Van Dooren, P.1
  • 151
    • 84976789456 scopus 로고
    • Algorithm 590, DUSBSP AND EXCHQZZ: FORTRAN subroutines for computing deflating subspaces with specified spectrum
    • Van Dooren P. Algorithm 590, DUSBSP AND EXCHQZZ: FORTRAN subroutines for computing deflating subspaces with specified spectrum. ACM Trans. Math. Software. 8:1982;376-382.
    • (1982) ACM Trans. Math. Software , vol.8 , pp. 376-382
    • Van Dooren, P.1
  • 152
    • 0041037295 scopus 로고
    • On quadratic convergence of the special cyclic Jacobi method
    • van Kempen H.P.M. On quadratic convergence of the special cyclic Jacobi method. Numer. Math. 9:1966;19-22.
    • (1966) Numer. Math. , vol.9 , pp. 19-22
    • Van Kempen, H.P.M.1
  • 156
    • 0027657519 scopus 로고
    • Some perspectives on the eigenvalue problem
    • Watkins D.S. Some perspectives on the eigenvalue problem. SIAM Rev. 35:1993;430-471.
    • (1993) SIAM Rev. , vol.35 , pp. 430-471
    • Watkins, D.S.1
  • 157
    • 0000630233 scopus 로고
    • Chasing algorithms for the eigenvalue problem
    • Watkins D.S., Elsner L. Chasing algorithms for the eigenvalue problem. SIAM J. Matrix Anal. Appl. 12:1991;374-384.
    • (1991) SIAM J. Matrix Anal. Appl. , vol.12 , pp. 374-384
    • Watkins, D.S.1    Elsner, L.2
  • 160
    • 0000318599 scopus 로고
    • Linear differential equations and Kronecker's canonical form
    • C. de Boor, & G.H. Golub. New York: Academic Press
    • Wilkinson J.H. Linear differential equations and Kronecker's canonical form. de Boor C., Golub G.H. Recent advances in Numerical Analysis. 1978;231-265 Academic Press, New York.
    • (1978) Recent Advances in Numerical Analysis , pp. 231-265
    • Wilkinson, J.H.1
  • 161
    • 0040876964 scopus 로고
    • Kronecker's canonical form and the QZ algorithm
    • Wilkinson J.H. Kronecker's canonical form and the QZ algorithm. Linear Algebra Appl. 28:1979;285-303.
    • (1979) Linear Algebra Appl. , vol.28 , pp. 285-303
    • Wilkinson, J.H.1
  • 162
    • 0039877633 scopus 로고
    • Eigenvalue problems
    • A. Iserles, & M.J.D. Powell. Oxford: Clarendon Press
    • Wilkinson J.H. Eigenvalue problems. Iserles A., Powell M.J.D. The State of the Art in Numerical Analysis. 1987;1-39 Clarendon Press, Oxford.
    • (1987) The State of the Art in Numerical Analysis , pp. 1-39
    • Wilkinson, J.H.1
  • 164
    • 0039287052 scopus 로고    scopus 로고
    • On the homotopy method for perturbed symmetric generalized eigenvalue problems
    • Zhang T., Law K.H., Golub G.H. On the homotopy method for perturbed symmetric generalized eigenvalue problems. SIAM J. Numer. Anal. 35(1):1998;300-319.
    • (1998) SIAM J. Numer. Anal. , vol.35 , Issue.1 , pp. 300-319
    • Zhang, T.1    Law, K.H.2    Golub, G.H.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.