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Volumn 48, Issue 1, 2006, Pages 3-40

Convergence analysis of Krylov subspace iterations with methods from potential theory

Author keywords

Constrained equilibrium; Eigenvalue distribution; Equilibrium measure; Krylov subspace iterations; Potential theory; Ritz values

Indexed keywords

APPROXIMATION THEORY; CALCULATIONS; CONVERGENCE OF NUMERICAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; LINEAR SYSTEMS; MATRIX ALGEBRA; POLYNOMIALS;

EID: 33644583290     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036144504445376     Document Type: Article
Times cited : (74)

References (81)
  • 1
    • 0035824491 scopus 로고    scopus 로고
    • Asymptotics of discrete orthogonal polynomials and the continuum limit of the Toda lattice
    • A.I. Aptekarev and W. Van Assche, Asymptotics of discrete orthogonal polynomials and the continuum limit of the Toda lattice, J. Phys. A, 34 (2001), pp. 10627-10637.
    • (2001) J. Phys. A , vol.34 , pp. 10627-10637
    • Aptekarev, A.I.1    Van Assche, W.2
  • 7
    • 8444226333 scopus 로고    scopus 로고
    • Convergence of restarted Krylov subspaces to invariant subspaces
    • C. Beattie, M. Embree, and J. Rossi, Convergence of restarted Krylov subspaces to invariant subspaces, SIAM J. Matrix Anal. Appl., 25 (2004), pp. 1074-1109.
    • (2004) SIAM J. Matrix Anal. Appl. , vol.25 , pp. 1074-1109
    • Beattie, C.1    Embree, M.2    Rossi, J.3
  • 8
    • 0034343596 scopus 로고    scopus 로고
    • On a conjecture of E.A. Rakhmanov
    • B. Beckermann, On a conjecture of E.A. Rakhmanov, Constr. Approx., 16 (2000), pp. 427-448.
    • (2000) Constr. Approx. , vol.16 , pp. 427-448
    • Beckermann, B.1
  • 10
    • 0035733975 scopus 로고    scopus 로고
    • Superlinear convergence of conjugate gradients
    • B. Beckermann and A.B.J. Kuijlaars, Superlinear convergence of conjugate gradients, SIAM J. Numer. Anal., 39 (2001), pp. 300-329.
    • (2001) SIAM J. Numer. Anal. , vol.39 , pp. 300-329
    • Beckermann, B.1    Kuijlaars, A.B.J.2
  • 11
    • 3042580768 scopus 로고    scopus 로고
    • On the sharpness of an asymptotic error estimate for conjugate gradients
    • B. Beckermann and A.B.J. Kuijlaars, On the sharpness of an asymptotic error estimate for conjugate gradients, BIT, 41 (2001), pp. 856-867.
    • (2001) BIT , vol.41 , pp. 856-867
    • Beckermann, B.1    Kuijlaars, A.B.J.2
  • 14
    • 84963026891 scopus 로고
    • Jentzsch-Szegö type theorems for the zeros of best approximants
    • H.P. Blatt, E.B. Saff, and M. Simkani, Jentzsch-Szego type theorems for the zeros of best approximants, J. London Math. Soc., 38 (1988), pp. 307-316.
    • (1988) J. London Math. Soc. , vol.38 , pp. 307-316
    • Blatt, H.P.1    Saff, E.B.2    Simkani, M.3
  • 15
    • 0033241008 scopus 로고    scopus 로고
    • Families of equilibrium measures with external field on the real axis
    • V.S. Buyarov and E.A. Rakhmanov, Families of equilibrium measures with external field on the real axis, Sb. Math., 190 (1999), pp. 791-802.
    • (1999) Sb. Math. , vol.190 , pp. 791-802
    • Buyarov, V.S.1    Rakhmanov, E.A.2
  • 16
    • 0002654780 scopus 로고
    • An implicitly restarted Lanczos method for large symmetric eigenvalue problems
    • D. Calvetti, L. Reichel, and D.C. Sorensen, An implicitly restarted Lanczos method for large symmetric eigenvalue problems, Electron. Trans. Numer. Anal., 2 (1994), pp. 1-21.
    • (1994) Electron. Trans. Numer. Anal. , vol.2 , pp. 1-21
    • Calvetti, D.1    Reichel, L.2    Sorensen, D.C.3
  • 17
    • 22844454647 scopus 로고    scopus 로고
    • The support of the equilibrium measure in the presence of a monomial external field on [-1, 1]
    • S.B. Damelin and A.B.J. Kuijlaars, The support of the equilibrium measure in the presence of a monomial external field on [-1, 1], Trans. Amer. Math. Soc., 351 (1999), pp. 4561-4584.
    • (1999) Trans. Amer. Math. Soc. , vol.351 , pp. 4561-4584
    • Damelin, S.B.1    Kuijlaars, A.B.J.2
  • 19
    • 84967476031 scopus 로고
    • Four lectures on random matrix theory
    • A.M. Vershik, ed., Lecture Notes in Math. Springer-Verlag, Berlin, 2003
    • P. Deift, Four lectures on random matrix theory, in Asymptotic Combinatorics with Applications to Mathematical Physics, A.M. Vershik, ed., Lecture Notes in Math. 1815, Springer-Verlag, Berlin, 2003, pp. 21-52.
    • (1815) Asymptotic Combinatorics With Applications to Mathematical Physics , pp. 21-52
    • Deift, P.1
  • 22
    • 0000614428 scopus 로고    scopus 로고
    • Constrained energy problems with applications to orthogonal polynomials of a discrete variable
    • P.D. Dragnev and E.B. Saff, Constrained energy problems with applications to orthogonal polynomials of a discrete variable, J. Anal. Math., 72 (1997), pp. 223-259.
    • (1997) J. Anal. Math. , vol.72 , pp. 223-259
    • Dragnev, P.D.1    Saff, E.B.2
  • 23
    • 0001282264 scopus 로고    scopus 로고
    • A problem in potential theory and zero asymptotics of Krawtchouk polynomials
    • P.D. Dragnev and E.B. Saff, A problem in potential theory and zero asymptotics of Krawtchouk polynomials, J. Approx. Theory, 102 (2000), pp. 120-140.
    • (2000) J. Approx. Theory , vol.102 , pp. 120-140
    • Dragnev, P.D.1    Saff, E.B.2
  • 24
    • 0032163793 scopus 로고    scopus 로고
    • From potential theory to matrix iterations in six steps
    • T.A. Driscoll, K.-C. Toh, and L.N. Trefethen, From potential theory to matrix iterations in six steps, SIAM Rev., 40 (1998), pp. 547-578.
    • (1998) SIAM Rev. , vol.40 , pp. 547-578
    • Driscoll, T.A.1    Toh, K.-C.2    Trefethen, L.N.3
  • 25
    • 0030507854 scopus 로고    scopus 로고
    • Minimal residual method stronger than polynomial preconditioning
    • V. Faber, W. Joubert, E. Knill, and T. Manteuffel, Minimal residual method stronger than polynomial preconditioning, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 707-729.
    • (1996) SIAM J. Matrix Anal. Appl. , vol.17 , pp. 707-729
    • Faber, V.1    Joubert, W.2    Knill, E.3    Manteuffel, T.4
  • 28
    • 0010922542 scopus 로고
    • Potential d'équilibre et capacité des ensembles
    • Ph.D. thesis, Lund University, Sweden
    • O. Frostman, Potential d'équilibre et capacité des ensembles, Ph.D. thesis, Lund University, Sweden, 1935.
    • (1935)
    • Frostman, O.1
  • 29
    • 0002451703 scopus 로고    scopus 로고
    • Eigenvalue curves of asymmetric tridiagonal random matrices
    • I.Ya. Goldsheid and B.A. Khoruzhenko, Eigenvalue curves of asymmetric tridiagonal random matrices, Electron. J. Probab., 5 (16) (2000), pp. 1-28.
    • (2000) Electron. J. Probab. , vol.5 , Issue.16 , pp. 1-28
    • Goldsheid, I.Ya.1    Khoruzhenko, B.A.2
  • 30
    • 0041742512 scopus 로고    scopus 로고
    • Regular spacings of complex eigenvalues in the one-dimensional non-Hermitian Anderson model
    • I.Ya. Goldsheid and B.A. Khoruzhenko, Regular spacings of complex eigenvalues in the one-dimensional non-Hermitian Anderson model, Comm. Math. Phys., 238 (2003), pp. 505-524.
    • (2003) Comm. Math. Phys. , vol.238 , pp. 505-524
    • Goldsheid, I.Ya.1    Khoruzhenko, B.A.2
  • 31
    • 0004236492 scopus 로고    scopus 로고
    • 3rd ed., Johns Hopkins University Press, Baltimore, MD
    • G.H. Golub and C.F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
    • (1996) Matrix Computations
    • Golub, G.H.1    Van Loan, C.F.2
  • 32
    • 0034321492 scopus 로고    scopus 로고
    • Eigenvalue computation in the 20th century
    • G.H. Golub and H. van der Vorst, Eigenvalue computation in the 20th century, J. Comput. Appl. Math., 123 (2000), pp. 35-65.
    • (2000) J. Comput. Appl. Math. , vol.123 , pp. 35-65
    • Golub, G.H.1    van der Vorst, H.2
  • 33
    • 0002863868 scopus 로고
    • Equilibrium measure and the distribution of zeros of extremal polynomials
    • A.A. Gonchar and E.A. Rakhmanov, Equilibrium measure and the distribution of zeros of extremal polynomials, Mat. Sb., 125 (1984), pp. 117-127.
    • (1984) Mat. Sb. , vol.125 , pp. 117-127
    • Gonchar, A.A.1    Rakhmanov, E.A.2
  • 35
    • 0001645763 scopus 로고
    • Max-min properties of matrix factor norms
    • A. Greenbaum and L. Gurvits, Max-min properties of matrix factor norms, SIAM J. Sci. Comput., 15 (1994), pp. 348-358.
    • (1994) SIAM J. Sci. Comput. , vol.15 , pp. 348-358
    • Greenbaum, A.1    Gurvits, L.2
  • 36
    • 0000580113 scopus 로고
    • GMRES/CR and Arnoldi/Lanczos as matrix approximation problems
    • A. Greenbaum and L.N. Trefethen, GMRES/CR and Arnoldi/Lanczos as matrix approximation problems, SIAM J. Sci. Comput., 15 (1994), pp. 359-368.
    • (1994) SIAM J. Sci. Comput. , vol.15 , pp. 359-368
    • Greenbaum, A.1    Trefethen, L.N.2
  • 37
    • 0039165794 scopus 로고    scopus 로고
    • Random-matrix theories in quantum physics: Common concepts
    • T. Guhr, A. Müller-Groeling, and H.A. Weidenmüller, Random-matrix theories in quantum physics: Common concepts, Phys. Rep., 299 (1998), pp. 189-425.
    • (1998) Phys. Rep. , vol.299 , pp. 189-425
    • Guhr, T.1    Müller-Groeling, A.2    Weidenmüller, H.A.3
  • 39
    • 31544439100 scopus 로고    scopus 로고
    • A numerical solution of the constrained energy problem
    • to appear
    • S. Helsen and M. Van Barel, A numerical solution of the constrained energy problem, J. Comput. Appl. Math., to appear.
    • J. Comput. Appl. Math.
    • Helsen, S.1    Van Barel, M.2
  • 41
    • 0035626398 scopus 로고    scopus 로고
    • Discrete orthogonal polynomial ensembles and the Plancherel measure
    • K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. (2), 153 (2001), pp. 259-296.
    • (2001) Ann. of Math. , vol.153 , Issue.2 , pp. 259-296
    • Johansson, K.1
  • 42
    • 0036628218 scopus 로고    scopus 로고
    • Non-intersecting paths, random tilings and random matrices
    • K. Johansson, Non-intersecting paths, random tilings and random matrices, Probab. Theory Related Fields, 123 (2002), pp. 225-280.
    • (2002) Probab. Theory Related Fields , vol.123 , pp. 225-280
    • Johansson, K.1
  • 43
    • 84968508687 scopus 로고
    • Estimates for some computational techniques in linear algebra
    • S. Kaniel, Estimates for some computational techniques in linear algebra, Math. Comp., 20 (1966), pp. 369-378.
    • (1966) Math. Comp. , vol.20 , pp. 369-378
    • Kaniel, S.1
  • 44
    • 0034353662 scopus 로고    scopus 로고
    • Which eigenvalues are found by the Lanczos method?
    • A.B.J. Kuijlaars, Which eigenvalues are found by the Lanczos method?, SIAM J. Matrix Anal. Appl., 22 (2000), pp. 306-321.
    • (2000) SIAM J. Matrix Anal. Appl. , vol.22 , pp. 306-321
    • Kuijlaars, A.B.J.1
  • 45
    • 0034664307 scopus 로고    scopus 로고
    • On the finite-gap ansatz in the continuum limit of the Toda lattice
    • A.B.J. Kuijlaars, On the finite-gap ansatz in the continuum limit of the Toda lattice, Duke Math. J., 104 (2000), pp. 433-462.
    • (2000) Duke Math. J. , vol.104 , pp. 433-462
    • Kuijlaars, A.B.J.1
  • 46
    • 0032207940 scopus 로고    scopus 로고
    • Zero distributions for discrete orthogonal polynomials
    • A.B.J. Kuijlaars and E.A. Rakhmanov, Zero distributions for discrete orthogonal polynomials, J. Comp. Appl. Math., 99 (1998), pp. 255-274.
    • (1998) J. Comp. Appl. Math. , vol.99 , pp. 255-274
    • Kuijlaars, A.B.J.1    Rakhmanov, E.A.2
  • 49
    • 4043177137 scopus 로고    scopus 로고
    • The worst-case GMRES for normal matrices
    • J. Liesen and P. Tichý, The worst-case GMRES for normal matrices, BIT, 44 (2004), pp. 79-98.
    • (2004) BIT , vol.44 , pp. 79-98
    • Liesen, J.1    Tichý, P.2
  • 50
    • 0004163930 scopus 로고
    • 2nd ed., Academic Press, Boston
    • M.L. Mehta, Random Matrices, 2nd ed., Academic Press, Boston, 1991.
    • (1991) Random Matrices
    • Mehta, M.L.1
  • 51
    • 84967754997 scopus 로고
    • Extremal problems for polynomials with exponential weights
    • H.N. Mhaskar and E.B. Saff, Extremal problems for polynomials with exponential weights, Trans. Amer. Math. Soc., 285 (1984), pp. 204-234.
    • (1984) Trans. Amer. Math. Soc. , vol.285 , pp. 204-234
    • Mhaskar, H.N.1    Saff, E.B.2
  • 52
    • 0002534477 scopus 로고
    • Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials)
    • H.N. Mhaskar and E.B. Saff, Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials), Constr. Approx., 1 (1985), pp. 71-91.
    • (1985) Constr. Approx. , vol.1 , pp. 71-91
    • Mhaskar, H.N.1    Saff, E.B.2
  • 53
    • 0001918897 scopus 로고
    • Weighted analogues of capacity, transfinite diameter, and Chebyshev constant
    • H.N. Mhaskar and E.B. Saff, Weighted analogues of capacity, transfinite diameter, and Chebyshev constant, Constr. Approx., 8 (1992), pp. 105-124.
    • (1992) Constr. Approx. , vol.8 , pp. 105-124
    • Mhaskar, H.N.1    Saff, E.B.2
  • 55
    • 0003358142 scopus 로고    scopus 로고
    • The Symmetric Eigenvalue Problem
    • SIAM, Philadelphia
    • B.N. Parlett, The Symmetric Eigenvalue Problem, Classics Appl. Math. 20, SIAM, Philadelphia, 1998.
    • (1998) Classics Appl. Math. , vol.20
    • Parlett, B.N.1
  • 56
    • 84941498586 scopus 로고
    • Über den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen
    • G. Pölya and G. Szegö, Über den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen, J. Reine Angew. Math., 165, (1931), pp. 4-49.
    • (1931) J. Reine Angew. Math. , vol.165 , pp. 4-49
    • Pölya, G.1    Szegö, G.2
  • 57
    • 0002048208 scopus 로고
    • On asymptotic properties of polynomials orthogonal on the real axis
    • English transl. Math. USSR-Sb., 47 (1984), pp. 155-193
    • E.A. Rakhmanov, On asymptotic properties of polynomials orthogonal on the real axis, Mat. Sb., 119 (1982), pp. 163-203. English transl. Math. USSR-Sb., 47 (1984), pp. 155-193.
    • (1982) Mat. Sb. , vol.119 , pp. 163-203
    • Rakhmanov, E.A.1
  • 58
    • 0030305556 scopus 로고    scopus 로고
    • Equilibrium measure and the distribution of zeros of the extremal polynomials of a discrete variable
    • E.A. Rakhmanov, Equilibrium measure and the distribution of zeros of the extremal polynomials of a discrete variable, Sb. Math., 187 (1996), pp. 1213-1228.
    • (1996) Sb. Math. , vol.187 , pp. 1213-1228
    • Rakhmanov, E.A.1
  • 60
    • 0000767453 scopus 로고
    • On the rates of convergence of the Lanczos and the block-Lanczos methods
    • Y. Saad, On the rates of convergence of the Lanczos and the block-Lanczos methods, SIAM J. Numer. Anal., 17 (1980), pp. 687-706.
    • (1980) SIAM J. Numer. Anal. , vol.17 , pp. 687-706
    • Saad, Y.1
  • 64
    • 0034144763 scopus 로고    scopus 로고
    • Locally X matrices, spectral distributions, preconditioning, and applications
    • S. Serra Capizzano, Locally X matrices, spectral distributions, preconditioning, and applications, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1354-1388.
    • (2000) SIAM J. Matrix Anal. Appl. , vol.21 , pp. 1354-1388
    • Serra Capizzano, S.1
  • 65
    • 0035501546 scopus 로고    scopus 로고
    • Spectral behavior of matrix sequences and discretized boundary value problems
    • S. Serra Capizzano, Spectral behavior of matrix sequences and discretized boundary value problems, Linear Algebra Appl., 337 (2001), pp. 37-78.
    • (2001) Linear Algebra Appl. , vol.337 , pp. 37-78
    • Serra Capizzano, S.1
  • 66
    • 0037409955 scopus 로고    scopus 로고
    • Generalized locally Toeplitz sequences: Spectral analysis and applications to discretized partial differential equations
    • S. Serra Capizzano, Generalized locally Toeplitz sequences: Spectral analysis and applications to discretized partial differential equations, Linear Algebra Appl., 366 (2003), pp. 371-402.
    • (2003) Linear Algebra Appl. , vol.366 , pp. 371-402
    • Serra Capizzano, S.1
  • 67
    • 21444444506 scopus 로고    scopus 로고
    • Further results on the convergence behavior of conjugate gradients and Ritz values
    • G.L.G. Sleijpen and A. van der Sluis, Further results on the convergence behavior of conjugate gradients and Ritz values, Linear Algebra Appl., 246 (1996), pp. 233-278.
    • (1996) Linear Algebra Appl. , vol.246 , pp. 233-278
    • Sleijpen, G.L.G.1    van der Sluis, A.2
  • 68
    • 0001403122 scopus 로고
    • The rate of convergence of conjugate gradients
    • A. van der Sluis and H.A. van der Vorst, The rate of convergence of conjugate gradients, Numer. Math., 48 (1986), pp. 543-560.
    • (1986) Numer. Math. , vol.48 , pp. 543-560
    • van der Sluis, A.1    van der Vorst, H.A.2
  • 69
    • 38249033965 scopus 로고
    • The convergence behavior of Ritz values in the presence of close eigenvalues
    • A. van der Sluis and H.A. van der Vorst, The convergence behavior of Ritz values in the presence of close eigenvalues, Linear Algebra Appl., 88/89 (1987), pp. 651-694.
    • (1987) Linear Algebra Appl. , vol.88-89 , pp. 651-694
    • van der Sluis, A.1    van der Vorst, H.A.2
  • 71
  • 72
    • 0004082558 scopus 로고
    • Weighted Approximation with Varying Weight
    • Lecture Notes in Math. 1569, Springer-Verlag, Berlin
    • V. Totik, Weighted Approximation with Varying Weight, Lecture Notes in Math. 1569, Springer-Verlag, Berlin, 1994.
    • (1994)
    • Totik, V.1
  • 73
    • 0000649044 scopus 로고
    • Introduction to random matrices
    • G.F. Helminck, ed., Springer-Verlag, Berlin
    • C. Tracy and H. Widom, Introduction to random matrices, in Geometric and Quantum Aspects of Integrable Systems, G.F. Helminck, ed., Springer-Verlag, Berlin, 1993, pp. 103-130.
    • (1993) Geometric and Quantum Aspects of Integrable Systems , pp. 103-130
    • Tracy, C.1    Widom, H.2
  • 74
    • 0001683904 scopus 로고    scopus 로고
    • Universality of the distribution functions of random Matrix theory
    • J. Harnad, G. Sabidussi, and P. Winternitz, eds., CRM Proc. Lecture Notes 26, AMS, Providence, RI
    • C. Tracy and H. Widom, Universality of the distribution functions of random Matrix theory, in Integrable Systems: From Classical to Quantum, J. Harnad, G. Sabidussi, and P. Winternitz, eds., CRM Proc. Lecture Notes 26, AMS, Providence, RI, 2000, pp. 251-264.
    • (2000) Integrable Systems: From Classical to Quantum , pp. 251-264
    • Tracy, C.1    Widom, H.2
  • 76
    • 0035582035 scopus 로고    scopus 로고
    • Spectra, pseudospectra, and localization for random bidiagonal matrices
    • L.N. Trefethen, M. Contedini, and M. Embree, Spectra, pseudospectra, and localization for random bidiagonal matrices, Comm. Pure Appl. Math., 54 (2001), pp. 595-623.
    • (2001) Comm. Pure Appl. Math. , vol.54 , pp. 595-623
    • Trefethen, L.N.1    Contedini, M.2    Embree, M.3
  • 80
    • 0027657519 scopus 로고
    • Some perspectives on the eigenvalue problem
    • D.S. Watkins, Some perspectives on the eigenvalue problem, SIAM Rev., 35 (1993), pp. 430-471.
    • (1993) SIAM Rev. , vol.35 , pp. 430-471
    • Watkins, D.S.1


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