Some sufficient conditions for the convergence of the Jacobi and Gauss-Seidel methods for large systems of linear equations
Digital Object Identifier (DOI): 10.14708/ma.v3i5.1182
Abstract
This article contains three sufficient conditions for the convergence of Jacobi and Gauss-Seidel iterative solutions of
systems of linear equations of the form Ax=b. These conditions rely on a special property of the matrix A defined in this work as `the sum criterion'.
This property is not equivalent in the general case to the irreducibility of the matrix A. The results obtained are regarded primarily as a theoretical
aid towards understanding the Gauss-Seidel method. Practical usefulness of these methods is minimal since generally we can list other more effective
iterative methods. (MR0455316)
systems of linear equations of the form Ax=b. These conditions rely on a special property of the matrix A defined in this work as `the sum criterion'.
This property is not equivalent in the general case to the irreducibility of the matrix A. The results obtained are regarded primarily as a theoretical
aid towards understanding the Gauss-Seidel method. Practical usefulness of these methods is minimal since generally we can list other more effective
iterative methods. (MR0455316)
Keywords: 65F10
References
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[2] H. Geiringer, On the solution of systems of linear equations by certain iterative methods, Reissner Anniversary volume J. W. Edwards, Ann. Arbor. 1949, str. 365-393, 2,19, 25, 57, 58, 95, 96, 160.
[3] R. Misess und H. Pollaczek-Geiringer, Praktische Verfahren der Gleichungs Auflosung, Z. Angew. Math. Mech. 9 (1929), str. 58-77, 152-164; 94, 95, 96.
[4] R. S. Varga, Matrix Iterative Analysis, Prentice Hall 1962.
[5] D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, 1971.
Pages: 43-50
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