The matrix analogue of the scalar AOR iterative method

The Accelerated Overrelaxation (AOR) and the Generalized AOR (GAOR) iterative methods for the solution of linear systems of algebraic equations ( A x = b , A ? C n i? n , det ( A ) ? 0 , b ? C n ) have been around for about four decades and a plethora of variations of them have been proposed. In this work a novel algorithm is introduced, the Matrix Analogue of the AOR (MAAOR) iterative method, which is analysed and studied. The MAAOR method generalizes both the AOR and the GAOR. Sufficient convergence conditions for the GAOR method are determined when the coefficient matrix A of the linear system to be solved is a Hermitian matrix with positive diagonal elements. Similarly, sufficient convergence conditions for the MAAOR method are determined when A is a nonsingular H -matrix. The new convergence conditions are the most general ones so far. Numerical examples are presented in support of the theory developed.

[1]  K. R. James.  Convergence of Matrix Iterations Subject to Diagonal Dominance , 1973 .

[2]  Zhong-xi Gao,et al.  A Note on the Convergence of the Generalized AOR Iterative Method for Linear Systems , 2012 .

[3]  Richard S. Varga,et al.  Matrix Iterative Analysis , 2000, The Mathematical Gazette.

[4]  Apostolos Hadjidimos,et al.  On some extensions of the accelerated overrelaxation (AOR) theory , 1982 .

[5]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .

[6]  Apostolos Hadjidimos,et al.  Best Cyclic Repartitioning for Optimal Successive Overrelaxation Convergence , 1992, SIAM J. Matrix Anal. Appl..

[7]  The Young-Eidson algorithm: applications and extensions , 1990 .

[8]  Ting Wang,et al.  Convergence of GAOR Iterative Method with Strictly α Diagonally Dominant Matrices , 2011, J. Appl. Math..

[9]  Shi-Liang Wu,et al.  Modified Preconditioned GAOR Methods for Systems of Linear Equations , 2013, J. Appl. Math..

[10]  W. Niethammer.  On Different Splittings and the Associated Iteration Methods , 1979 .

[11]  On the equivalence of extrapolation and Richardson’s iteration and its applications , 2005 .

[12]  A. Hadjidimos,et al.  Is A∈Cn,n a general H-matrix? , 2012 .

[13]  S. A. Edalatpanah,et al.  On the Convergence Regions of Generalized Accelerated Overrelaxation Method for Linear Complementarity Problems , 2013, J. Optim. Theory Appl..

[14]  Apostolos Hadjidimos,et al.  A New Iterative Criterion for H-Matrices , 2006, SIAM J. Matrix Anal. Appl..

[15]  Apostolos Hadjidimos,et al.  Accelerated overrelaxation method , 1978 .

[16]  Miroslav Šisler.  Über die Optimierung eines zweiparametrigen Iterationsverfahrens , 1975 .

[17]  R. Varga.  On recurring theorems on diagonal dominance , 1976 .

[18]  A. Hadjidimos,et al.  Optimal p-cyclic SOR for complex spectra , 1997 .

[19]  J. Ortega,et al.  Extensions of the Ostrowski-Reich theorem for SOR iterations , 1979 .

[20]  On the convergence of the generalized AOR method , 1997 .

[21]  A. Hadjidimos,et al.  A Young-Eidson's Type Algorithm for Complex p-Cyclic SOR Spectra , 1999 .

[22]  Apostolos Hadjidimos,et al.  The principle of extrapolation in connection with the accelerated overrelaxation method , 1980 .

[23]  Ting-Zhu Huang,et al.  Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant matrices , 2008 .

[24]  Miroslav Šisler.  Bemerkungen zur Optimierung eines zweiparametrigen Iterationsverfahrens , 1976 .

[25]  Konvergenzkriterien für das Verallgemeinerte AOR‐Verfahren , 1992 .

[26]  A new generalized AOR iterative method for solving linear systems , 2013 .

[27]  Apostolos Hadjidimos,et al.  On the Solution of the Linear Complementarity Problem by the Generalized Accelerated Overrelaxation Iterative Method , 2015, J. Optim. Theory Appl..

[28]  Yaotang Li,et al.  Generalized AOR methods for linear complementarity problem , 2007, Appl. Math. Comput..