以线性化方法为基础，推导出松弛法和打靶法
The linearization methods as a basis to derive the relaxation and the shooting methods
摘要：
本章研究非线性两点边值问题的数值解。它建立了三个重要的，看似无关的，类迭代法之间的联系，即：线性化方法，松弛法（有限差分法），和打靶法。最近已经证明，使用有限差分离散序列的线性问题得到的准线性化，皮卡德线性化，或常数斜率线性化，导致各自的松弛法的通常的迭代公式。因此，线性化方法可用作推导松弛法的基础。在这项工作中，我们证明了打靶法也可以从线性化方法中推导出来。我们表明，放松射击轨迹，即一个初值问题的解决方案，实际上是一个投影变换。所得到的函数称为投影轨迹，可以用来修正初始条件。利用新的初始条件，我们可以找到一个新的射击轨迹，等等，所描述的过程被称为射击投影迭代（SPI）。结果表明，用拟线性化方程松弛（投影）射击轨迹，得到牛顿法的常规射击;用常斜率线性化方程松弛（投影）射击轨迹，得到常斜率法的常规射击;用Picard线性化方程松弛（投影）射击轨迹，得到最近提出的射击投影法。因此，后一种方法可以正确地称为皮卡德方法射击。新的理论结果的一个可能的应用建议和数值计算机实验。提供了MATLAB代码。
英文摘要：
This chapter investigates numerical solution of nonlinear two-point boundary value problems. It establishes a connection between three important, seemingly unrelated, classes of iterative methods, namely: the linearization methods, the relaxation methods (finite difference methods), and the shooting methods. It has recently been demonstrated that using finite differences to discretize the sequence of linear problems obtained by quasi-linearization, Picard linearization, or constant-slope linearization, leads to the usual iteration formula of the respective relaxation method. Thus, the linearization methods can be used as a basis to derive the relaxation methods. In this work we demonstrate that the shooting methods can be derived from the linearization methods, too. We show that relaxing a shooting trajectory, i.e. an initial value problem solution, is in fact a projection transformation. The obtained function, called projection trajectory, can be used to correct the initial condition. Using the new initial condition, we can find a new shooting trajectory, and so on. The described procedure is called shooting-projection iteration (SPI). We show that using the quasi-linearization equation to relax (project) the shooting trajectory leads to the usual shooting by Newton method, the constant-slope linearization leads to the usual shooting by constant-slope method, while the Picard linearization leads to the recently proposed shooting-projection method. Therefore, the latter method can rightfully be called shooting by Picard method. A possible application of the new theoretical results is suggested and numerical computer experiments are presented. MATLAB codes are provided.
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