Extreme eigenvalues of Toeplitz forms and applications to elliptic difference equations
Extreme eigenvalues of Toeplitz forms and applications to elliptic difference equations
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Toeplitz 形式的极值特征值及其在椭圆差分方程中的应用
DOI：
10.1090/s0002-9947-1961-0120492-5
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发表时间：
1961
影响因子：
1.3
通讯作者：
S. Parter
中科院分区：
文献类型：
--
作者：
S. Parter
并得到了部分结果。这些使我们能够获得矩形区域上的拉普拉斯和双调和差分方程的“双线”迭代法的收敛速度的估计（2）。对于拉普拉斯方程，我们得到了一个精确的渐近结果.然而，在双调和方程的情况下，我们只得到了一个“单侧”估计。本报告有两个目的。在？？在图2，3和4中，我们推广了Kac，Murdoch和Szego以及Widom的结果.我们将充分利用Widom的成果和技术。在哪？5.讨论了上述结果在块Toeplitz矩阵极值特征值问题中的应用。这些包括矩阵的椭圆差分方程提交给社会，1961年1月26日;收到的编辑1960年10月18日。（'）其中一些结果是作者1959年夏天在布鲁克海文国家实验室时获得的。(2)研究了拉普拉斯和双调和差分方程的“双线”迭代法。S. [18]与此同时，他的方法与我们在[14]中研究的方法完全不同。他的方法来解决迭代方程是更普遍和可能更可取的。瓦尔加还估计收敛速度在拉普拉斯的情况下使用理论的非负矩阵。这个理论不适用于双调和的情况。
and obtained some partial results. These enabled us to obtain estimates on the rates of convergence of the "two-line" iterative methods of the Laplace and biharmonic difference equations in rectangular domains(2). In the case of Laplace's equation we obtained an exact asymptotic result. However, in the case of the biharmonic equation we obtained only a "one-sided" estimate. The purpose of this report is two-fold. In ??2, 3, and 4 we extend the results of Kac, Murdoch and Szego, and Widom. We will make very strong use of Widom's results and technique. In ?5 we discuss the application of the preceding results to the general problem of the extreme eigenvalues of "block" Toeplitz matrices. These include the matrices of elliptic difference equations Presented to the Society, January 26, 1961; received by the editors October 18, 1960. (') Some of these results were obtained while the author was at the Brookhaven National Laboratory, summer 1959. (2) The "two-line" iterative methods for the Laplace and biharmonic difference equations were studied by R. S. Varga [18] at the same time. His approach is totally different from the one we investigated in [14]. His approach to the solution of the iteration equations is more general and probably preferable. Varga also estimated the rate of convergence in the Laplace case using the theory of non-negative matrices. That theory does not apply to the biharmonic case.