Abstract
There is a very fruitful interaction between numerical linear algebra and logarithmic potential theory. For instance, we may describe weak asympotics for a polynomial extremal problem occurring in the convergence analysis of conjugate gradients (CG) or of Ritz values; here the link with a constrained minimal energy problem allows to quantify the effect of superlinear convergence.
In the present paper, we introduce an extremal problem for rational functions on two discrete sets E N , F N , the so-called third Zolotarev problem. Roughly speaking, we look for rational functions of a prescribed degree which are as small as possible on E N and as large as possible on F N . Such a problem occurs naturally in the convergence analysis for the so-called alternating direction implicit (ADI) method for solving Lyapunov equations, but also in the approximation of particular matrix functions, and in the decay rate of singular values of matrices with small displacement rank.
Whereas asymptotics for this extremal problem on continuous sets is well studied, we require new tools in order to handle the case of discrete sets. The main contribution of this paper is to show that a minimal energy problem for signed and constrained measures allows us to describe the asymptotics. This new minimal energy problem is a natural extension of that of Kuijlaars and Beckermann used to describe weak asymptotics for the polynomial extremal problem related to CG.
We discuss the sharpness of our asymptotic findings for general discrete sets in the complex plane and suggest new formulas for the extremal constant in the particular case where the two discrete sets are real and symmetric with respect to the origin. Finally, by discussing a model example, we show the impact of our findings for analyzing the rate of superlinear convergence of the ADI method applied to a Lyapunov equation.
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References
Achieser, N.I.: Theory of Approximation. Ungar, New York (1956)
Akhieser, N.I.: Elements of the Theory of Elliptic Functions. Transl. of Math. Monographs, vol. 79. AMS, Providence (1990)
Beckermann, B.: On a conjecture of E.A. Rakhmanov. Constr. Approx. 16, 427–448 (2000)
Beckermann, B.: Singular values of small displacement rank matrices. Talk at conference Structured Numerical Linear Algebra Problems: Algorithms and Applications, Cortona, 2004
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)
Beckermann, B., Kuijlaars, A.B.J.: Superlinear convergence of conjugate gradients. SIAM J. Numer. Anal. 39, 300–329 (2001)
Beckermann, B., Kuijlaars, A.B.J.: On the sharpness of an asymptotic error estimate for conjugate gradients. BIT 41, 856–867 (2001)
Beckermann, B., Kuijlaars, A.B.J.: Superlinear CG convergence for special right-hand sides. Electron. Trans. Numer. Anal. 14, 1–19 (2002)
Birkhoff, G., Varga, R.S.: Implicit alternating direction methods. Trans. Am. Math. Soc. 92, 13–24 (1959)
Braess, D.: Nonlinear Approximation Theory. Springer Series in Computational Mathematics, vol. 7. Springer, Berlin (1980)
Braess, D.: Rational approximation of Stieltjes functions by the Carathéodory–Fejer method. Constr. Approx. 3, 43–50 (1987)
Buyarov, V., Rakhmanov, E.A.: Families of equilibrium measures with external field on the real axis. Mat. Sb. 190, 791–802 (1999)
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)
Driscoll, T.A., Toh, K.-C., Trefethen, L.N.: From potential theory to matrix iteration in six steps. SIAM Rev. 40, 547–578 (1998)
Fisher, S.D., Saff, E.B.: The asymptotic distribution of minimal Blaschke products. J. Approx. Theory 98, 104–116 (1999)
Golub, G.H., Van Loan, F.: Matrix Computations, 3rd edn. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (1996)
Gonchar, A.A.: On the speed of rational approximation of some analytic functions. Mat. Sb. 34, 131–145 (1978)
Higham, H.J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)
Kuijlaars, A.B.J.: Which eigenvalues are found by the Lanczos method? SIAM J. Matrix Anal. Appl. 22, 306–321 (2000)
Kuijlaars, A.B.J.: Convergence analysis of Krylov subspace iterations with methods from potential theory. SIAM Rev. 48, 3–40 (2006)
Kuijlaars, A.B.J., Dragnev, P.D.: Equilibrium problems associated with fast decreasing polynomials. Proc. Am. Math. Soc. 127, 1065–1074 (1999)
Lapik, M.A.: Support of the extremal measure in a vector equilibrium problem. Mat. Sb. 197, 1205–1219 (2006)
Levenberg, N., Reichel, L.: A generalized ADI iterative method. Numer. Math. 66, 215–233 (1993)
Levin, A.L., Lubinsky, D.S.: Green equilibrium measures and representations of an external field. J. Approx. Theory 113, 298–323 (2001)
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)
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)
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)
Nikishin, E.M., Sorokin, V.N.: Rational Approximations and Orthogonality. AMS Translations of Mathematical Monographs, vol. 92. AMS, Providence (1991)
Peaceman, D., Rachford, H.: The numerical solution of elliptic and parabolic differential equations. J. Soc. Ind. Appl. Math. 3, 28–41 (1955)
Penzl, T.: Eigenvalue decay bound for solutions of Lyapounov equations: the symmetric case. Syst. Control Lett. 40, 139–144 (2000)
Rakhmanov, E.A.: Equilibrium measure and the distribution of zeros of the extremals polynomials of a discrete variable. Mat. Sb. 187, 1213–1228 (1996)
Ransford, T.: Potential Theory in the Complex Plane. London Math. Soc. Student Text, vol. 28. Cambridge University Press, Cambridge (1995)
Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. A Series of Comprehensive Studies in Mathematics, vol. 316. Springer, Berlin (1997)
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)
Trefethen, L.N., Bau, D. III: Numerical Linear Algebra. SIAM, Philadelphia (1997)
van den Eshof, J., Frommer, A., Lippert, Th., Schilling, K., van der Vorst, H.A.: Numerical methods for the QCD overlap operator: I. Sign-function and error bounds. Comput. Phys. Commun. 146, 203–224 (2002)
Wachspress, E.L.: Extended application of alternating direction implicit iteration model problem theory. J. Soc. Ind. Appl. Math. 11, 994–1016 (1963)
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)
Zolotarev, E.I.: Collected Work, vol. 2. USSR Acad. Sci., Moscow (1932) (in Russian)
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Communicated by Edward B. Saff.
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Beckermann, B., Gryson, A. Extremal Rational Functions on Symmetric Discrete Sets and Superlinear Convergence of the ADI Method. Constr Approx 32, 393–428 (2010). https://doi.org/10.1007/s00365-010-9087-6
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DOI: https://doi.org/10.1007/s00365-010-9087-6