Summary
Using a recently derived classical type general functional equation, relating the eigenvalues of a weakly cyclic Jacobi iteration matrix to the eigenvalues of its associated Unsymmetric Successive Overrelaxation (USSOR) iteration matrix, we obtain bounds for the convergence of the USSOR method, when applied to systems with ap-cyclic coefficient matrix.
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References
Alefeld, G., Varga, R.S.: Zur Konvergenz des symmetrischen Relaxationsverfahren. Numer. Math.25, 291–295 (1976)
Axelsson, O.: On preconditioning and convergence acceleration in sparse matrix problems. CERN 74-10, Geneva 1974
Axelsson, O.: Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations. Linear Algebra Appl.29, 1–16 (1980)
Axelsson, O.: A survey of preconditioned iterative methods for linear systems of algebraic equations. BIT25, 166–187 (1985)
D'Sylva, E., Miles, G.A.: The SSOR iteration scheme for equations withσ 1-ordering. Comput. J.6, 366–367 (1964)
Eirman, M., Niethammer, W., Ruttan, A.: Optimal successive overrelaxation iterative methods forp-cyclic matrices. Numer. Math. (to appear)
Forsythe, G.E., Wasow, W.R.: Finite difference methods for partial differential equations, 1st Ed. Wiley, New York 1960
Fox, L.: Numerical Solution of Ordinary and Partial differential Equations, 1st Ed. Pergamon, Oxford 1962
Gong, L., Cai, D.Y.: Relationship between eigenvalues of Jacobi and SSOR iterative matrix withp-weak cyclic matrix. J. Comput. Math. Colleges Univ.1, 79–84 (1985) (in Chinese)
Hageman, L.A., Young, D.M.: Applied iterative methods, 1st Ed. Academic Press, New York 1981
Krishna, L.B.: Some new results on unsymmetric successive overrelaxation method. Numer. Math.42, 155–160 (1983)
Krishna, L.B.: On the convergence of the symmetric successive overrelaxation method. Linear Algebra Appl.56, 185–194 (1984).
Li, X., Varga, R.S.: A note on the SSOR and USSOR iterative methods applied top-cyclic matrices. Numer. Math. (to appear)
Neumaier, A., Varga, R.S.: Exact convergence and divergence domains for the symmetric successive overrelaxation iterative (SSOR) method applied toH-matrices. Linear Algebra Appl.58, 261–272 (1984)
Newmann, M.: On bounds for the convergence of the SSOR method forH-matrices. Linear Multilinear Algebra15, 13–21 (1984).
Pierce, D.J., Hadjidimos, A., Plemmons, R.J.: Optimality relationships forp-cyclic SOR. Numer. Math. (to appear)
Saridakis, Y.G.: On the analysis of the unsymmetric successive overrelaxation method when applied top-cyclic matrices. Numer. Math.49, 461–472 (1986)
Saridakis, Y.G., Varga, R.S.: Regions of exact convergence of the USSOR method applied on generalized consistently ordered matrices. Linear Algebra Appl. (to appear)
Sheldon, J.: On the numerical solution of elliptic difference equations. M. T. A. C.9, 101–112 (1955)
Varga, R.S.: Matrix iterative analysis, 1st Ed. Prentice-Hall, New Jersey 1962
Varga, R.S., Niethammer, W., Cai, D.Y.:p-cyclic matrices and the symmetric successive overrelaxation method. Linear Algebra Appl.58, 425–439 (1984)
Verner, J.H., Bernal, M.J.M.: On generalizations of the theory of consistent ordering for successive over-relaxation methods. Numer. Math.12, 215–222 (1968)
Young, D.M.: Iterative solution of large linear systems, 1st Ed. Academic, New York 1971
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Saridakis, Y.G. Domains of divergence of the USSOR method applied onp-cyclic matrices. Numer. Math. 57, 405–412 (1990). https://doi.org/10.1007/BF01386419
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DOI: https://doi.org/10.1007/BF01386419