On the effects of scaling of the Peaceman-Rachford method

Author:
Olof B. Widlund

Journal:
Math. Comp. **25** (1971), 33-41

MSC:
Primary 65N10

DOI:
https://doi.org/10.1090/S0025-5718-1971-0303754-8

MathSciNet review:
0303754

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Abstract: The alternating direction method of Peaceman and Rachford is considered for elliptic difference schemes of second order and with two independent variables. An earlier result of the author's on the rapid convergence of multi-parameter noncommutative problems is described and a connection is established between that result and theorems on optimal scaling of band matrices. Simple algorithms to decrease the condition number and increase the rate of convergence are discussed.

**[1]**G. Birkhoff & R. S. Varga, "Implicit alternating direction methods,"*Trans. Amer. Math. Soc.*, v. 92, 1959, pp. 13-24. MR**21**#4549. MR**0105814 (21:4549)****[2]**B. L. Buzbee, G. H. Golub & C. W. Nielson,*The Method of Odd/Even Reduction and Factorization with Application to Poisson's Equation*, Stanford Computer Science Department Report, 1969.**[3]**R. Courant & D. Hilbert,*Methods of Mathematical Physics*. Vol. I, Interscience, New York, 1953. MR**13**, 800. MR**0065391 (16:426a)****[4]**J. Douglas, Jr., "Alternating direction methods for three space variables,"*Numer. Math.*, v. 4, 1962, pp. 41-63. MR**24**#B2122. MR**0136083 (24:B2122)****[5]**G. E. Forsythe & E. G. Straus, "On best conditioned matrices,"*Proc. Amer. Math. Soc.*, v. 6, 1955, pp. 340-345. MR**16**, 1054. MR**0069585 (16:1054j)****[6]**P. R. Garabedian,*Partial Differential Equations*, Wiley, New York, 1964. MR**28**#5247. MR**0162045 (28:5247)****[7]**G. H. Golub, "Comparison of the variance of minimum variance and weighted least squares regression coefficients,"*Ann. Math. Statist.*, v. 34, 1963, pp. 984-991. MR**27**#5336. MR**0155402 (27:5336)****[8]**W. H. Guilinger, Jr., "The Peaceman-Rachford method for small mesh increments,"*J. Math. Anal. Appl.*, v. 11, 1965, pp. 261-277. MR**32**#607. MR**0183125 (32:607)****[9]**J. E. Gunn, "On the two-stage iterative method of Douglas for mildly nonlinear elliptic difference equations,"*Numer. Math.*, v. 6, 1964, pp. 243-249. MR**29**#6637. MR**0169387 (29:6637)****[10]**R. W. Hockney, "A fast direct solution of Poisson's equation using Fourier analysis,"*J. Assoc. Comput. Mach.*, v. 12, 1965, pp. 95-113. MR**35**#3913. MR**0213048 (35:3913)****[11]**W. Kahan & J. Varah,*Two Working Algorithms for the Eigenvalues of a Symmetric Tridiagonal Matrix*, Stanford Computer Science Department Report, 1966.**[12]**D. W. Peaceman & H. H. Rachford, Jr., "The numerical solution of parabolic and elliptic differential equations,"*J. Soc. Indust. Appl. Math.*, v. 3, 1955, pp. 28-41. MR**17**, 196. MR**0071874 (17:196d)****[13]**C. M. Pearcy, "On the convergence of alternating direction procedures,"*Numer. Math.*, v. 4, 1962, pp. 172-176. MR**26**#3206. MR**0145677 (26:3206)****[14]**A. van der Sluis, "Condition numbers and equilibration of matrices." (To appear.) MR**0253546 (40:6760)****[15]**R. S. Varga,*Matrix Iterative Analysis*, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR**28**#1725. MR**0158502 (28:1725)****[16]**E. L. Wachspress,*Iterative Solution of Elliptic Systems, and Applications to the Neutron Diffusion Equations of Reactor Physics*, Prentice-Hall, Englewood Cliffs, N. J., 1966. MR**38**#2965. MR**0234649 (38:2965)****[17]**E. L. Wachpress & G. J. Habetler, "An alternating-direction-implicit iteration technique,"*J. Soc. Indust. Appl. Math.*, v. 8, 1960, pp. 403-424. MR**22**#5132. MR**0114308 (22:5132)****[18]**O. B. Widlund, "On the rate of convergence of an alternating direction implicit method in a noncommutative case,"*Math. Comp.*, v. 20, 1966, pp. 500-515. MR**37**#7104. MR**0231551 (37:7104)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1971-0303754-8

Keywords:
Alternating direction implicit methods,
Peaceman-Rachford method,
elliptic of second order,
multi-parameter,
noncommutative,
separation-of-variables,
condition number,
rate of convergence,
optimal scaling

Article copyright:
© Copyright 1971
American Mathematical Society