Numerische Mathematik

, Volume 66, Issue 1, pp 215–233 | Cite as

A generalized ADI iterative method

  • N. Levenberg
  • L. Reichel


The ADI iterative method for the solution of Sylvester's equationAX−XB=C proceeds by strictly alternating between the solution of the two equations
$$\begin{gathered} \left( {A - \delta _{k + 1} I} \right)X_{2k + 1} = X_{2k} \left( {B - \delta _{k + 1} I} \right) + C, \hfill \\ X_{2k + 2} \left( {B - \tau _{k + 1} I} \right) = \left( {A - \tau _{k + 1} I} \right)X_{2k + 1} - C, \hfill \\ \end{gathered} $$
fork=0, 1, 2,... HereXo is a given initial approximate solution, and the δ k andτ k are real or complex parameters chosen so that the computed approximate solutionsX k converge rapidly to the solutionX of the Sylvester equation ask increases. This paper discusses the possibility of solving one of the equations in the ADI iterative method more often than the other one, i.e., relaxing the strict alternation requirement, in order to achieve a higher rate of convergence. Our analysis based on potential theory shows that this generalization of the ADI iterative method can give faster convergence than when strict alternation is required.

Mathematics Subject Classification (1991)



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  1. 1.
    Arnoldi, W.E. (1951) The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quart. Appl. Math.9, 17–29Google Scholar
  2. 2.
    Bagby, T. (1967): The modulus of a plane condenser. J. Math. Mech.17, 315–329Google Scholar
  3. 3.
    Bagby, T. (1969): On interpolation by rational functions. Duke J. Math.36, 95–104Google Scholar
  4. 4.
    Bagby, T. (1973): Rational interpolation with restricted poles. J. Approx. Theory7, 1–7Google Scholar
  5. 5.
    Bartels, R., Stewart, G.W. (1972): Algorithm 432: solution of the matrix equationAX+XB=C. Comm. ACM.15, 820–826Google Scholar
  6. 6.
    Bickley, W.G., McNamee, J. (1960): Matrix and other direct methods for the solution of systems of linear difference equations. Proc. Royal Soc.252, 70–131Google Scholar
  7. 7.
    Birkhoff, G., Varga, R.S. (1959): Implicit alternating direction methods. Trans. Amer. Math. Soc.92, 13–24Google Scholar
  8. 8.
    Birkhoff, G., Varga, R.S., Young, D. (1962): Alternating direction implicit methods. In: Advances in Computing, vol. 3, 189–273. Academic Press, New YorkGoogle Scholar
  9. 9.
    de Boor, C., Rice, R.C. (1963): Chebyshev approximation by\(a\prod {\frac{{x - r_r }}{{x + s_1 }}} \) with application to ADI iteration. J. Soc. Indust. Appl. Math.11, 159–169Google Scholar
  10. 10.
    Calvetti, D., Golub, G.H., Reichel, L. Adaptive Chebyshev iterative methods for nonsymmetric linear systems based on modified moments, Numer. Math., to appearGoogle Scholar
  11. 11.
    Chin, R.C.Y., Manteuffel, T.A., de Pillis, J. (1984): ADI as a preconditioning for solving the convection-diffusion equation. SIAM J. Sci. Statist. Comput.5, 281–299Google Scholar
  12. 12.
    Datta, B.N., Datta, K. (1986): Theoretical and computational aspects of some linear algebra problems in control theory. in: C.I. Byrnes, A. Lindquist, ed., Computational and combinatorial methods in systems theory. 201–212. Elsevier, AmsterdamGoogle Scholar
  13. 13.
    Ellner, N.S., Wachspress, E.L. (1991): Alternating direction implicit iteration for systems with complex spectra. SIAM J. Numer. Anal.28, 859–870Google Scholar
  14. 14.
    Elman, H.C., Saad, Y., Saylor, P.E. (1986): A hybrid Chebyshev Krylov subspace algorithm for solving nonsymmetric systems of linear equations. SIAM J. Sci. Statist. Comput.7, 840–855Google Scholar
  15. 15.
    Gaier, D., Todd, J. (1967): On the rate of convergence of optimal ADI processes. Numer. Math.9, 452–459Google Scholar
  16. 16.
    Ganelius, T. (1981): Rational functions, capacity and approximation. in: D.A. Brannan, J.G. Clunie, eds., Aspects of contemporary complex analysis. 409–414. Academic Press, Orlando, FLGoogle Scholar
  17. 17.
    Gantmacher, F.R. (1986): Matrizentheorie, Springer, Berlin Heidelburg New YorkGoogle Scholar
  18. 18.
    Golub, G.H., Nash, S., Van Loan, C. (1979): A Hessenberg-Schur method for the problemAX+XB=C, IEEE Trans. Automat. Control. AC-24 909–913Google Scholar
  19. 19.
    Gonchar, A.A. (1969): Zolotarev problems connected with rational functions. Math. USSR Sbornik,7, 623–635Google Scholar
  20. 20.
    Gonchar, A.A., Rakhmanov, E.A. (1989): Equilibrium distributions and degree of rational approximation of analytic functions. Math. USSR Sbornik62, 305–348Google Scholar
  21. 21.
    Hu, D.Y., Reichel, L. (1992): Krylov subspace methods for the Sylvester equation. Linear Algebra Appl.172, 283–313Google Scholar
  22. 22.
    Jiang, H., Wong, Y.S. (1991): A parallel alternating direction implicit preconditioning method. J. Comput. Appl. Math.36, 209–226Google Scholar
  23. 23.
    Kloke, H. (1985): On the capacity of a plane condenser and conformal mapping. J. Reine Angew. Math.358, 179–201Google Scholar
  24. 24.
    Landkof, N. (1972): Foundations of modern potential theory, Springer, Berlin Heidelberg New YorkGoogle Scholar
  25. 25.
    Manteuffel, T.A. (1978): Adaptive procedure for estimating parameters for the nonsymmetric Tchebyshev iteration. Numer. Math.31, 183–208Google Scholar
  26. 26.
    Nachtigal, N.M., Reichel, L., Trefethen, N.L. (1992): A hybrid GMRES algorithm for nonsymmetric linear systems. SIAM J. Matrix Anal. Appl.13, 796–825Google Scholar
  27. 27.
    Peaceman, D.W., Rachford, H.H. (1955): The numerical solution of parabolic and elliptic differential equations. J. Soc. Indust. Appl. Math.3, 28–41Google Scholar
  28. 28.
    Phillips, T.N. (1987): The smoothing properties of the alternating direction implicit method in multigrid iteration. Applied Numer. Math.3, 513–522Google Scholar
  29. 29.
    Reichel, L., Trefethen, L.N. (1992): Eigenvalues and pseudo-eigenvalues of Toeplitz matrices, Linear Algebra Appl.162–164, 153–185Google Scholar
  30. 30.
    Saad, Y. (1980): Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices. Linear Algebra Appl.34, 269–285Google Scholar
  31. 31.
    Saff, E.B., Stahl, H. (1991): Sequences in the Walsh table for ⋎xα, Report ICM #91-015. Institute for Constructive Mathematics. University of South Florida, Tampa, FLGoogle Scholar
  32. 32.
    Saylor, P.E., Smolarski, D.C. (1991): Implementation of an adaptive algorithm for Richardson's method. Linear Algebra Appl.154–156, 615–646Google Scholar
  33. 33.
    Starke, G. (1989): Rationale minimierungsprobleme in der komplexen Fbene im Zusammenhang mit der Bestimmung optimaler ADI-Parameter, Ph.D. Thesis, Institut für Praktische Mathematik, Universität Karlsruhe, Karlsruhe, GermanyGoogle Scholar
  34. 34.
    Starke, G. (1991): Optimal alternating direction implicit parameters for nonsymmetric systems of linear equations. SIAM J. Numer. Anal.28, 1431–1445Google Scholar
  35. 35.
    Starke, G., Niethammer, W. (1991): SOR forAX−XB=C, Linear Algebra Appl.,154–156, 355–375Google Scholar
  36. 36.
    Starke, G., Varga, R.S. (1992) A hybrid Arnoldi-Faber iterative method for nonsymmetric systems of linear equations. Numer. Math. (to appear)Google Scholar
  37. 37.
    Todd, J. (1984): Application of transformation theory: a legacy from Zolotarev (1847–1878). in: S. P. Singh, ed., Approximation theory and spline functions. 207–245 Reidel, BostonGoogle Scholar
  38. 38.
    Trefethen, L.N. (1990): Approximation theory and numerical linear algebra. in: J.C. Mason, M.G. Cox, eds., Algorithms for approximation II. 336–360 Chapman and Hall, LondonGoogle Scholar
  39. 39.
    Tsuji, M. (1959): Potential theory in modern function theory. Maruzen, TokyoGoogle Scholar
  40. 40.
    Varga, R.S. (1962): Matrix iterative analysis. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  41. 41.
    Wachspress, E.L. (1962): Optimum alternating-direction-implicit iteration parameters for a model problem. J. Soc. Indust. Appl. Math.10, 339–350Google Scholar
  42. 42.
    Wachspress, E.L. (1963): Extended application of alternating direction implicit iteration model problem theory. J. Soc. Indust. Appl. Math.11, 994–1016Google Scholar
  43. 43.
    Wachspress, E.L. (1990): The ADI minimax problem for complex spectra, in: D.R. Kincaid, L.J. Hayes, eds., Iterative methods for large linear systems. 251–271 Academic Press, San Diego, CAGoogle Scholar
  44. 44.
    Wachspress, E.L. (1992): Optimum parameters for two-variable ADI iteration. Ann. Nucl. Energy,19, 765–778Google Scholar
  45. 45.
    Walsh, J.L. (1965): Interpolation and approximation by rational functions in the complex domain, 4th Amer. Math. Soc., Providence, RIGoogle Scholar
  46. 46.
    Widlund, O.B. (1971): On the effect of scaling of the Peaceman-Rachford method. Math. Comput.25, 33–41Google Scholar
  47. 47.
    Widom, H. (1972): Rational approximation andn-dimensional diameter. J. Approx. Theory5, 343–361Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • N. Levenberg
    • 1
  • L. Reichel
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of AucklandAucklandNew Zealand
  2. 2.Department of Mathematics and Computer ScienceKent State UniverswityKentUSA

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