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Volumn 32, Issue 3, 2010, Pages 393-428

Extremal Rational Functions on Symmetric Discrete Sets and Superlinear Convergence of the ADI Method

Author keywords

ADI; Logarithmic potential theory; Minimal energy problems with constraint

Indexed keywords


EID: 77957300827     PISSN: 01764276     EISSN: 14320940     Source Type: Journal    
DOI: 10.1007/s00365-010-9087-6     Document Type: Article
Times cited : (20)

References (39)
  • 3
    • 0034343596 scopus 로고    scopus 로고
    • On a conjecture of E.A. Rakhmanov
    • Beckermann, B.: On a conjecture of E. A. Rakhmanov. Constr. Approx. 16, 427-448 (2000).
    • (2000) Constr. Approx. , vol.16 , pp. 427-448
    • Beckermann, B.1
  • 4
    • 77957314417 scopus 로고    scopus 로고
    • Beckermann, B.: Singular values of small displacement rank matrices. Talk at conference Structured Numerical Linear Algebra Problems: Algorithms and Applications, Cortona, 2004.
  • 5
    • 33750162548 scopus 로고    scopus 로고
    • Discrete orthogonal polynomials and superlinear convergence of Krylov subspace methods in numerical linear algebra
    • Lecture Notes in Mathematics, F. Marcellan and W. AsscheVan (Eds.), Berlin: Springer
    • Beckermann, B.: Discrete orthogonal polynomials and superlinear convergence of Krylov subspace methods in numerical linear algebra. In: Marcellan, F., van Assche, W. (eds.) Orthogonal Polynomial and Special Functions. Lecture Notes in Mathematics, vol. 1883, pp. 119-185. Springer, Berlin (2006).
    • (2006) Orthogonal Polynomial and Special Functions , vol.1883 , pp. 119-185
    • Beckermann, B.1
  • 6
    • 0035733975 scopus 로고    scopus 로고
    • Superlinear convergence of conjugate gradients
    • Beckermann, B., Kuijlaars, A. B. J.: Superlinear convergence of conjugate gradients. SIAM J. Numer. Anal. 39, 300-329 (2001).
    • (2001) SIAM J. Numer. Anal. , vol.39 , pp. 300-329
    • Beckermann, B.1    Kuijlaars, A.B.J.2
  • 7
    • 3042580768 scopus 로고    scopus 로고
    • On the sharpness of an asymptotic error estimate for conjugate gradients
    • Beckermann, B., Kuijlaars, A. B. J.: On the sharpness of an asymptotic error estimate for conjugate gradients. BIT 41, 856-867 (2001).
    • (2001) Bit , vol.41 , pp. 856-867
    • Beckermann, B.1    Kuijlaars, A.B.J.2
  • 9
    • 84931151127 scopus 로고
    • Implicit alternating direction methods
    • Birkhoff, G., Varga, R. S.: Implicit alternating direction methods. Trans. Am. Math. Soc. 92, 13-24 (1959).
    • (1959) Trans. Am. Math. Soc. , vol.92 , pp. 13-24
    • Birkhoff, G.1    Varga, R.S.2
  • 10
    • 0003696141 scopus 로고
    • Springer Series in Computational Mathematics, Berlin: Springer
    • Braess, D.: Nonlinear Approximation Theory. Springer Series in Computational Mathematics, vol. 7. Springer, Berlin (1980).
    • (1980) Nonlinear Approximation Theory , vol.7
    • Braess, D.1
  • 11
    • 0346264755 scopus 로고
    • Rational approximation of Stieltjes functions by the Carathéodory-Fejer method
    • Braess, D.: Rational approximation of Stieltjes functions by the Carathéodory-Fejer method. Constr. Approx. 3, 43-50 (1987).
    • (1987) Constr. Approx. , vol.3 , pp. 43-50
    • Braess, D.1
  • 12
    • 0033241008 scopus 로고    scopus 로고
    • Families of equilibrium measures with external field on the real axis
    • Buyarov, V., Rakhmanov, E. A.: Families of equilibrium measures with external field on the real axis. Mat. Sb. 190, 791-802 (1999).
    • (1999) Mat. Sb. , vol.190 , pp. 791-802
    • Buyarov, V.1    Rakhmanov, E.A.2
  • 13
    • 0000614428 scopus 로고    scopus 로고
    • Constrained energy problems with applications to orthogonal polynomials of a discrete variable
    • Dragnev, P. D., Saff, E. B.: Constrained energy problems with applications to orthogonal polynomials of a discrete variable. J. Anal. Math. 72, 223-259 (1997).
    • (1997) J. Anal. Math. , vol.72 , pp. 223-259
    • Dragnev, P.D.1    Saff, E.B.2
  • 14
    • 0032163793 scopus 로고    scopus 로고
    • From potential theory to matrix iteration in six steps
    • Driscoll, T. A., Toh, K.-C., Trefethen, L. N.: From potential theory to matrix iteration in six steps. SIAM Rev. 40, 547-578 (1998).
    • (1998) SIAM Rev. , vol.40 , pp. 547-578
    • Driscoll, T.A.1    Toh, K.-C.2    Trefethen, L.N.3
  • 15
    • 0033130160 scopus 로고    scopus 로고
    • The asymptotic distribution of minimal Blaschke products
    • Fisher, S. D., Saff, E. B.: The asymptotic distribution of minimal Blaschke products. J. Approx. Theory 98, 104-116 (1999).
    • (1999) J. Approx. Theory , vol.98 , pp. 104-116
    • Fisher, S.D.1    Saff, E.B.2
  • 16
    • 0004236492 scopus 로고    scopus 로고
    • 3rd edn., Johns Hopkins Studies in the Mathematical Sciences, Baltimore: Johns Hopkins University Press
    • Golub, G. H., van Loan, F.: Matrix Computations, 3rd edn. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (1996).
    • (1996) Matrix Computations
    • Golub, G.H.1    van Loan, F.2
  • 17
    • 22744435259 scopus 로고
    • On the speed of rational approximation of some analytic functions
    • Gonchar, A. A.: On the speed of rational approximation of some analytic functions. Mat. Sb. 34, 131-145 (1978).
    • (1978) Mat. Sb. , vol.34 , pp. 131-145
    • Gonchar, A.A.1
  • 19
    • 0034353662 scopus 로고    scopus 로고
    • Which eigenvalues are found by the Lanczos method
    • Kuijlaars, A. B. J.: Which eigenvalues are found by the Lanczos method? SIAM J. Matrix Anal. Appl. 22, 306-321 (2000).
    • (2000) SIAM J. Matrix Anal. Appl. , vol.22 , pp. 306-321
    • Kuijlaars, A.B.J.1
  • 20
    • 33644583290 scopus 로고    scopus 로고
    • Convergence analysis of Krylov subspace iterations with methods from potential theory
    • Kuijlaars, A. B. J.: Convergence analysis of Krylov subspace iterations with methods from potential theory. SIAM Rev. 48, 3-40 (2006).
    • (2006) SIAM Rev. , vol.48 , pp. 3-40
    • Kuijlaars, A.B.J.1
  • 21
    • 22644450485 scopus 로고    scopus 로고
    • Equilibrium problems associated with fast decreasing polynomials
    • Kuijlaars, A. B. J., Dragnev, P. D.: Equilibrium problems associated with fast decreasing polynomials. Proc. Am. Math. Soc. 127, 1065-1074 (1999).
    • (1999) Proc. Am. Math. Soc. , vol.127 , pp. 1065-1074
    • Kuijlaars, A.B.J.1    Dragnev, P.D.2
  • 22
    • 33751038742 scopus 로고    scopus 로고
    • Support of the extremal measure in a vector equilibrium problem
    • Lapik, M. A.: Support of the extremal measure in a vector equilibrium problem. Mat. Sb. 197, 1205-1219 (2006).
    • (2006) Mat. Sb. , vol.197 , pp. 1205-1219
    • Lapik, M.A.1
  • 23
    • 21344481063 scopus 로고
    • A generalized ADI iterative method
    • Levenberg, N., Reichel, L.: A generalized ADI iterative method. Numer. Math. 66, 215-233 (1993).
    • (1993) Numer. Math. , vol.66 , pp. 215-233
    • Levenberg, N.1    Reichel, L.2
  • 24
    • 0035701842 scopus 로고    scopus 로고
    • Green equilibrium measures and representations of an external field
    • Levin, A. L., Lubinsky, D. S.: Green equilibrium measures and representations of an external field. J. Approx. Theory 113, 298-323 (2001).
    • (2001) J. Approx. Theory , vol.113 , pp. 298-323
    • Levin, A.L.1    Lubinsky, D.S.2
  • 25
    • 0042907519 scopus 로고    scopus 로고
    • The distribution of zeros and poles of asymptotically extremal rational functions for Zolotarev's problem
    • Levin, A. L., Saff, E. B.: The distribution of zeros and poles of asymptotically extremal rational functions for Zolotarev's problem. J. Approx. Theory 110, 88-108 (2001).
    • (2001) J. Approx. Theory , vol.110 , pp. 88-108
    • Levin, A.L.1    Saff, E.B.2
  • 26
    • 33748771540 scopus 로고    scopus 로고
    • Potential theoretic tools in polynomial and rational approximation in harmonic analysis and rational approximation
    • Fournier, Grimm, Leblond, and Partington (Eds.), Berlin: Springer
    • Levin, A. L., Saff, E. B.: Potential theoretic tools in polynomial and rational approximation in harmonic analysis and rational approximation. In: Fournier, Grimm, Leblond, Partington (eds.) Lecture Notes in Control and Information Sciences, vol. 327, pp. 71-94. Springer, Berlin (2006).
    • (2006) Lecture Notes in Control and Information Sciences , vol.327 , pp. 71-94
    • Levin, A.L.1    Saff, E.B.2
  • 27
    • 0002534477 scopus 로고
    • Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials)
    • Mhaskar, H. N., Saff, E. B.: Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials). Constr. Approx. 1, 71-91 (1985).
    • (1985) Constr. Approx. , vol.1 , pp. 71-91
    • Mhaskar, H.N.1    Saff, E.B.2
  • 29
    • 0002058827 scopus 로고
    • The numerical solution of elliptic and parabolic differential equations
    • Peaceman, D., Rachford, H.: The numerical solution of elliptic and parabolic differential equations. J. Soc. Ind. Appl. Math. 3, 28-41 (1955).
    • (1955) J. Soc. Ind. Appl. Math. , vol.3 , pp. 28-41
    • Peaceman, D.1    Rachford, H.2
  • 30
    • 0001082478 scopus 로고    scopus 로고
    • Eigenvalue decay bound for solutions of Lyapounov equations: The symmetric case
    • Penzl, T.: Eigenvalue decay bound for solutions of Lyapounov equations: the symmetric case. Syst. Control Lett. 40, 139-144 (2000).
    • (2000) Syst. Control Lett. , vol.40 , pp. 139-144
    • Penzl, T.1
  • 31
    • 0030305556 scopus 로고    scopus 로고
    • Equilibrium measure and the distribution of zeros of the extremals polynomials of a discrete variable
    • Rakhmanov, E. A.: Equilibrium measure and the distribution of zeros of the extremals polynomials of a discrete variable. Mat. Sb. 187, 1213-1228 (1996).
    • (1996) Mat. Sb. , vol.187 , pp. 1213-1228
    • Rakhmanov, E.A.1
  • 32
    • 0003525517 scopus 로고
    • London Math. Soc. Student Text, Cambridge: Cambridge University Press
    • Ransford, T.: Potential Theory in the Complex Plane. London Math. Soc. Student Text, vol. 28. Cambridge University Press, Cambridge (1995).
    • (1995) Potential Theory in the Complex Plane , vol.28
    • Ransford, T.1
  • 33
  • 34
    • 0001831337 scopus 로고
    • Approximation theory and numerical linear algebra
    • J. C. Mason and M. G. Cox (Eds.), London: Chapman & Hall
    • Trefethen, L. N.: Approximation theory and numerical linear algebra. In: Mason, J. C., Cox, M. G. (eds.) Algorithms for Approximation II. Chapman & Hall, London (1990).
    • (1990) Algorithms for Approximation II
    • Trefethen, L.N.1
  • 37
    • 0005653922 scopus 로고
    • Extended application of alternating direction implicit iteration model problem theory
    • Wachspress, E. L.: Extended application of alternating direction implicit iteration model problem theory. J. Soc. Ind. Appl. Math. 11, 994-1016 (1963).
    • (1963) J. Soc. Ind. Appl. Math. , vol.11 , pp. 994-1016
    • Wachspress, E.L.1
  • 38
    • 84856029226 scopus 로고
    • Solution of the generalized ADI minmax problem
    • Mathematics Software, Amsterdam: North-Holland
    • Wachspress, E. L.: Solution of the generalized ADI minmax problem. In: Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968). Mathematics Software, vol. 1, pp. 99-105. North-Holland, Amsterdam (1969).
    • (1969) Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) , vol.1 , pp. 99-105
    • Wachspress, E.L.1
  • 39
    • 77957307682 scopus 로고
    • (in Russian), Moscow: USSR Acad. Sci
    • Zolotarev, E. I.: Collected Work, vol. 2. USSR Acad. Sci., Moscow (1932) (in Russian).
    • (1932) Collected Work , vol.2
    • Zolotarev, E.I.1


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