Abstract
In this article, we introduce two new asynchronous multisplitting methods for solving the system of weakly nonlinear equations Ax = G(x) in which A is an n × n real matrix and G(x) = (g 1(x), g 2(x), . . . , g n (x))T is a P-bounded mapping. First, by generalized accelerated overrelaxation (GAOR) technique, we introduce the asynchronous parallel multisplitting GAOR method (including the synchronous parallel multisplitting AOR method as a special case) for solving the system of weakly nonlinear equations. Second, asynchronous parallel multisplitting method based on symmetric successive overrelaxation (SSOR) multisplitting is introduced, which is called asynchronous parallel multisplitting SSOR method. Then under suitable conditions, we establish the convergence of the two introduced methods. The given results contain synchronous multisplitting iterations as a special case.
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References
Arnal J., Migallón V., Penadés J.: Non-stationary parallel multisplitting algorithms for almost linear systems. Numer. Linear Algebra Appl. 6, 79–92 (1999)
Bai Z.Z.: A class of two-stage iterative methods for systems of weakly nonlinear equations. Numer. Algor. 14, 295–319 (1997)
Bai Z.Z.: Parallel multisplitting two-stage iterative methods for large sparse systems of weakly nonlinear equations. Numer. Algor. 15, 347–372 (1997)
Bai Z.Z., Huang Y.G.: Asynchronous multisplitting two-stage iterations for systems of weakly nonlinear equations. J. Comput. Appl. Math. 93, 13–33 (1998)
Bai Z.Z.: Asynchronous parallel nonlinear multisplitting relaxation methods for large sparse nonlinear complementarity problems. Appl. Math. Comput. 92, 85–100 (1998)
Bai Z.Z.: The convergence of the two-stage iterative method for hermitian positive definite linear systems. Appl. Math. Lett. 11, 1–5 (1998)
Bai Z.Z.: Asynchronous multisplitting AOR methods for a class of systems of weakly nonlinear equations. Appl. Math. Comput. 98, 49–59 (1999)
Bai Z.Z., Wang C.L.: Convergence theorems for parallel multisplitting twostage iterative methods for mildly nonlinear systems. Linear Algebra Appl. 362, 237–250 (2003)
Berman A., Plemmons R.J.: Non-negative matrices in the mathematical sciences, third ed. SIAM, Philadelphia (1994)
Bru R., Migallón V., Penad́s J., Szyld D.B.: Parallel synchronous and asychronous two-stage multisplitting methods. Electronic Transactions on Numerical Analysis 3, 24–38 (1995)
Bru R., Elsner L., Neumann M.: Models of parallel chaotic iteration methods. Linear Algebra Appl. 103, 175–192 (1998)
Cao Z.H.: Nonstationary two-stage multisplitting methods with overlapping blocks. Linear Algebra Appl. 285, 153–163 (1998)
Cao Z.H., Liu Z.Y.: Symmetric multisplitting of a symmetric positive definite matrix. Linear Algebra Appl. 285, 309–319 (1998)
Chang D.W.: The parallel multisplitting TOR (MTOR) method for linear systems. Comput. Math. Appl. 41, 215–227 (2001)
Datta B.N.: Numerical linear algebra and applications. Brooks / Cole Publishing Co., Pacific Grove, California (1995)
Elsner L., Koltracht I., Neumann M.: On the convergence of asynchronous paracontractions with application to tomographic reconstruction from incomplete data. Linear Algebra Appl. 130, 65–82 (1990)
Frommer A., Mayer G.: Convergence of relaxed parallel multisplitting methods. Linear Algebra Appl. 119, 141–152 (1989)
R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge, Cambridge UP, 1985.
R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.
James K.R.: Convergence of matrix iterations subject to diagonal dominance. SIAM J. Numer. Anal. 12, 478–484 (1973)
Jones M.T., Szyld D.B.: Two-stage multisplitting methods with overlapping blocks. Numer. Linear Algebra Appl. 3, 113–124 (1996)
Li C.L., Zeng J.P.: Multisplitting Iteration Schemes for Solving a Class of Nonlinear Complementarity Problems. Acta Math. Sinica, Ser. A 23, 79–90 (2007)
Liu Z.Y., Lin L., Shi C.C.: Nonstationary two-stage multisplitting methods for symmetric positive definite matrices. Appl. Math. Lett. 13, 49–54 (2000)
Memon R.A.: Multisplitting accelerated overrelaxation methods for systems of weakly nonlinear equations. J. Shanghai Univ. Sci. Tech. 17, 213–224 (1994)
Neumann M., Plemmons R.J.: Convergence of parallel multisplitting iterative methods for M-matrices. Linear Algebra Appl. 88/89, 559–573 (1987)
O’Leary D.P., White R.E.: Multi-splittings of matrices and parallel solution of linear systems. SIAM J. Algebra Discrete Methods 6, 630–640 (1985)
Ortega J.M., Rheinboldt W.C.: Iterative solution of nonlinear equations in several variables. Academic Press, New York (1970)
Song Y.: On the convergence of the generalized AOR method. Linear Algebra Appl. 256, 199–218 (1997)
Todd M.D., Vohraan S.T.: Alternative approach to poincarè sectioning in weakly nonlinear systems. International Journal of Bifurcation and Chaos 9, 953–962 (1999)
Varga R.S.: Matrix iterative analysis. Prentice-Hall, Englewood Cliffs (1962)
Wang D., Bai Z.Z., Evans D.J.: On the monotone convergence of multisplitting method for a class of systems of weakly nonlinear equations. Int. J. Comput. Math. 60, 229–242 (1996)
Wang Y.M.: Parallel multisplitting explicit AOR methods for numerical solutions of semilinear elliptic boundary value problems. Comput. Math. Appl. 38, 55–66 (1999)
Wang Y.M.: Parallel multisplitting methods for a class of systems of weakly nonlinear equations without isotone mapping. Appl. Math. Comput. 109, 135–150 (2000)
Wu M., Wang L., Song Y.: Preconditioned AOR iterative method for linear systems. Appl. Num. Math. 57, 672–685 (2007)
Xinmin W.: Comparison theorems for a class of parallel multisplitting AOR type iterative methods. Linear Algebra Appl. 269, 1–16 (1998)
Yangfeng S.: Generalized multisplitting asynchronous iteration. Linear Algebra Appl. 235, 77–92 (1996)
Yun J.H.: Convergence of SSOR multisplitting method for an H-matrix. J. Comput. Appl. Math. 217, 252–258 (2008)
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Dehghan, M., Hajarian, M. Asynchronous Multisplitting GAOR Method and Asynchronous Multisplitting SSOR Method for Systems of Weakly Nonlinear Equations. Mediterr. J. Math. 7, 209–223 (2010). https://doi.org/10.1007/s00009-010-0047-y
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DOI: https://doi.org/10.1007/s00009-010-0047-y