Extensions of the Ostrowski-Reich theorem for SOR iterations
Extensions of the Ostrowski-Reich theorem for SOR iterations
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DOI：
10.1016/0024-3795(79)90131-9
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发表时间：
1979-12
影响因子：
1.1
通讯作者：
J. Ortega;R. Plemmons
中科院分区：
文献类型：
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作者：
J. Ortega;R. Plemmons
摘要 Ostrowski-Reich 定理指出，对于具有 A 非奇异的线性方程组 Ax= b，如果 A 是 Hermitian 且 A 的对角线为正，则当且仅当 A 为正定时，SOR 方法对于 (0, 2) 中的每个松弛参数收敛。这实际上是 Householder-John 定理的一个特例，该定理指出，对于 A= M− N 且 A, M 非奇异，如果 A 是埃尔米特矩阵且 M*+ N 是正定矩阵，则 M− 1 N 是收敛矩阵当且仅当 A 是正定矩阵。我们的目的是推广 Householder-John 定理，并深入了解 SOR 方法如何以及为何能够收敛。结果，奥斯特洛夫斯基-赖希定理在两个方向上得到扩展：一种是当 A 是埃尔米特式的，但 A 的对角线不一定是正的，因此 A 不一定是正定的；另一种是当 A+ A* 是正定的，但 A 不一定是埃尔米特式的。在此过程中，对于A的一般分裂，还获得了其他几个收敛结果。但是，没有声明这里获得的收敛结果可以应用于实际情况。
Abstract The Ostrowski-Reich theorem states that for a system Ax= b of linear equations with A nonsingular, if A is hermitian and if the diagonal of A is positive, then the SOR method converges for each relaxation parameter in (0, 2) if and only if A is positive definite. This is actually a special case of the Householder-John theorem, which states that for A= M− N with A, M nonsingular, if A is hermitian and M∗+ N is positive definite, then M− 1 N is a convergent matrix if and only if A is positive definite. Our purposes here are to generalize the Householder-John theorem and to provide an insight into how and why the SOR method can converge. As a result the Ostrowski-Reich theorem is extended in two directions; one is when A is hermitian but the diagonal of A is not necessarily positive, so that A is not necessarily positive definite, and the other is when A+ A∗ is positive definite but A is not necessarily hermitian. In the process, several other convergence results are obtained for general splittings of A. However, no claims are made concerning the case in which the convergence results obtained here can be applied to practical situations.