The Peaceman–Rachford Model Problem

  • Eugene Wachspress


Early analysis of ADI iteration introduced by Peaceman and Rachford led to application of Chebyshev minimax theory to determine optimal parameters. Elliptic functions play a crucial role. It was discovered belatedly that this problem had been solved in 1877 by Zolotarev.


Elliptic Function Initial Error Error Reduction Elliptic Integral Preconditioned Conjugate Gradient 
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  1. Abramowitz M, Stegun IA (1964) Handbook of mathematical functions, NBS AMS - 55; NIST Handbook of mathematical functions. Cambridge University Press, 2010Google Scholar
  2. Achieser NI (1967) Theory of approximation. Ungar, New YorkGoogle Scholar
  3. Birkhoff G, Varga RS, Young DM (1962) Alternating direction implicit methods. Advances in computers, vol 3. Academic, New York, pp 189–273Google Scholar
  4. Cauer W (1958) Synthesis of linear communication networks. McGraw-Hill, New YorkMATHGoogle Scholar
  5. Cesari L (1937) Sulla risoluzione dei sistemi di equazioni lineari par aprossimazioni successive. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. VI S–25:422–428Google Scholar
  6. Ellner NS, Wachspress EL (1986) Alternating direction implicit iteration for systems with complex spectra. SIAM J. Numer Anal 28(3):859–870 (1991); also New ADI model problem applications. In: Proceedings of fall joint computer conference, IEEE Computer Society Press, 1986 and Master’s thesis of N. Saltzman (Ellner), on ADI parameters for some complex spectra, University of Tennessee, Knoxville, TN, 1987Google Scholar
  7. Guilinger WH (1965) Peaceman-Rachford method for small mesh increments. J Math Anal Appl 11:261–277MathSciNetMATHCrossRefGoogle Scholar
  8. Istace MP, Thiran JP (1993) On the third and fourth Zolotarev problems in the complex plane. Math ComputGoogle Scholar
  9. Lu A, Wachspress EL (1991) Solution of Lyapunov equations by ADI iteration. Comput Math Appl 21(9):43–58; Wachspress EL (1987) Iterative solution of the Lyapunov matrix equation. Appl Math Ltrs (1):87–90Google Scholar
  10. Lynch RE, Rice JR (1968) Convergence rates of ADI methods with smooth initial error. Math Comp 22(102):331–335MathSciNetGoogle Scholar
  11. (2010) NIST Handbook of mathematical functions, chapter 19 by B.C. Carlson and chapter 22 by W.P. Reinhardt and P.L. Walke. Cambridge University Press, CambridgeGoogle Scholar
  12. Peaceman DW, Rachford HH Jr (1955) The numerical solution of parabolic and elliptic differential equation. J SIAM 3:28–41MathSciNetMATHGoogle Scholar
  13. Saltzman N (1987) ADI parameters for some complex spectra. University of Tennessee Master’s ThesisGoogle Scholar
  14. Todd J (1984) Applications of transformation theory: A legacy from Zolotarev (1847–1878). Approximation theory and spline functions. D.Reidel, Dordrecht, pp 207–245Google Scholar
  15. Wachspress EL (1957) CURE: A generalized two-space-dimension multigroup coding for the IBM-704. Knolls Atomic Power Report KAPL-1724, AEC R&D Physics and Mathematics TID-4500, 13th edn. Knolls Atomic Power Lab, SchenectadyGoogle Scholar
  16. Wachspress EL (1963) Extended application of alternating-direction-implicit iteration model problem theory. J Soc Indust Appl Math 11:994–1016MathSciNetCrossRefGoogle Scholar
  17. Wachspress EL (1966) Iterative solution of elliptic systems. Prentice Hall, Englewood CliffsMATHGoogle Scholar
  18. Wachspress EL (1984) Generalized ADI preconditioning. Comput Math Appl 10(6):457–461MathSciNetMATHCrossRefGoogle Scholar
  19. Wachspress EL (1990) The ADI minimax problem for complex spectra. In: Kincaid D, Hayes L, (eds) Iterative methods for large linear system. Academic, New York, pp 251–271Google Scholar
  20. Wachspress EL, Habetler GJ (1960) An alternating-direction-implicit iteration technique. J Soc Indust Appl Math 8(2):403–424MathSciNetMATHCrossRefGoogle Scholar
  21. Pemerburiskoi, Z, Oeuvres de G. Zolotarev (1877) Akademuu Nauk XXX, (5), pp 1–59Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Eugene Wachspress
    • 1
  1. 1.East WindsorUSA

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