The Continuous Newton's Method, Inverse Functions, and Nash-Moser
The Continuous Newton's Method, Inverse Functions, and Nash-Moser
复制标题
连续牛顿法、反函数和 Nash-Moser
DOI：
10.1080/00029890.2007.11920431
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
J. Neuberger
中科院分区：
文献类型：
--
作者：
E. Scheinerman;J. Neuberger
希望\(z_1\)，\(z_2\)，……收敛到\(F\)的一个零点。什么会阻止这个过程找到\(F\)的一个零点呢？一方面，\(F\)可能没有零点。另一方面，如果对于某个整数\(k\)，\(F^\prime(z_k)\)不可逆，那么这个过程可能会终止。对应于\(F\)的一个给定根的吸引域由所有初始值\(z_0\)组成，这些初始值通过\(z_1\)，\(z_2\)，……的收敛能得到这个根。牛顿法可能会导致混沌的吸引域，即使对于\(F\)的简单选择也是如此（见[8]）。这可能会产生惊人的图像，但对数值分析人员来说却是一场噩梦。这个事实（如果没有其他原因的话）导致了阻尼牛顿法的产生，它包括
hoping that z\, z2, ... converges to a zero of F. What can stop this process from find ing a zero of F? For one thing, there might not be a zero of F. For another, the process might terminate for some integer k in the event that F'izk) does not have an inverse. A domain of attraction corresponding to a given root of F consists of the set of all starting values z0 that lead, through convergence of z\, z2, ..., to this root. Newton's method can lead to chaotic domains of attraction, even for simple choices of F (see [8]). This can lead to striking pictures but constitutes a nightmare for the numerical analyst. This fact, if nothing else, leads one to the damped Newton's method, which consists of