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Block AOR iteration for nonsymmetric matrices

Author: Theodore S. Papatheodorou
Journal: Math. Comp. 41 (1983), 511-525
MSC: Primary 65F50; Secondary 65F10
MathSciNet review: 717699
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Abstract: We consider a class of matrices that are of interest to numerical applications and are large, sparse, but not symmetric or diagonally dominant. We give a criterion for the existence of (and we actually construct) the inverse matrix in terms of powers of a "small" matrix. We use this criterion to find that the spectral radius of the Jacobi iteration matrix, corresponding to a block tridiagonal partition, is in general $ \geqslant 1$. We also derive conditions (that are satisfied in cases of interest to applications) for the Jacobi matrix to have spectral radius = 1. We introduce convergent "block AOR" iterative schemes such as extrapolated Jacobi and extrapolated Gauss-Seidel schemes with optimum (under) relaxation parameter $ \omega = .5$. A numerical example pertaining to the solution of Poisson's equation is given, as a demonstration of some of our hypotheses and results. A comparison with SOR, applied to the 5-point finite difference method, is also included.

References [Enhancements On Off] (What's this?)

  • [1] R. Balart, E. N. Houstis & T. S. Papatheodorou, On the Iterative Solution of Collocation Method Equations, Proc. IMACS World Congress, Montreal, Aug. 1982, Vol. 1, pp. 98-100.
  • [2] A. Hadjidimos, "Accelerated overrelaxation method," Math. Comp., v. 32 1978, pp. 149-157. MR 0483340 (58:3353)
  • [3] A. Hadjidimos, "A new method for the solution of linear systems arising from the discretization of P.D.E.'s," Proc. Advances in Computer Methods for Partial Differential Equations IV (Vichnevetsky and Stepleman, eds.,), IMACS, 1981, pp. 74-79.
  • [4] E. N. Houstis, R. E. Lynch, T. S. Papatheodorou & J. R. Rice, "Evaluation of numerical methods for elliptic partial differential equations," J. Comput. Phys., v. 27, 1978, pp. 323-350. MR 496854 (80g:65119)
  • [5] T. S. Papatheodorou, "Inverses for a class of banded matrices and applications to piecewise cubic approximation," J. Comput. Appl. Math., v. 8, 1982, pp. 285-288. MR 694660 (84c:65055)
  • [6] J. R. Rice, Matrix Computations and Mathematical Software, McGraw-Hill Computer Science Series, New York, 1981. MR 634029 (83g:65003)
  • [7] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1982. MR 0158502 (28:1725)
  • [8] D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1981. MR 0305568 (46:4698)

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Keywords: AOR iteration
Article copyright: © Copyright 1983 American Mathematical Society

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