Iterative Solution of Nonlinear Equations
Iterative Solution of Nonlinear Equations
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DOI：
10.1007/978-3-319-89575-8_5
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发表时间：
2018
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影响因子：
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通讯作者：
P. Turner;Thomas Arildsen;Kathleen Kavanagh
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文献类型：
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作者：
P. Turner;Thomas Arildsen;Kathleen Kavanagh
本章讨论了一个最基本的数学问题：标量非线性方程的解。所有的方法都是迭代的。我们开始，也许最简单的想法-使用二分法，以减少一个间隔，我们知道包含一个解决方案，以一个可接受的公差。接下来，我们介绍牛顿的方法，它是基于切线在一个特定的点将交叉轴。如果我们能得到一个“足够好”的起点，牛顿法将很快收敛到方程的理想解。在二分法和牛顿法的基础上产生了割线法。现在，我们不再简单地将区间减半，而是观察区间两端之间的图的弦与轴相交的点。割线法可以比单纯的二分法具有显著更快的收敛。它是对二分法的改进，也是对牛顿法的差分近似，不需要导数的知识。最后，我们提出了非线性方程组的设置。我们的两个标准建模示例从一个未知扩展到两个。我们提供了实现两个方程和两个未知数的牛顿法的细节（除了标量算法之外），这样你就可以准备好求解本章中的应用问题了。
This chapter addresses one of the most fundamental mathematical problems: the solution of a scalar nonlinear equation,. All the methods presented are iterative in nature. We begin with perhaps the simplest idea – using bisection to reduce an interval which we know contains a solution to an acceptable tolerance. Next, we then present Newton’s method which is based on where the tangent line at a particular point would cross the axis. Provided we can get a “good enough” starting point Newton’s method will converge very quickly to the desired solution of the equation. Building on both the bisection method and Newton’s method gives rise to the secant method. Now instead of simply halving the interval we look at the point where the chord of the graph between the two ends of the interval would cross the axis. The secant method can have significantly faster convergence than mere bisection. It is both an improvement on bisection and a difference approximation to Newton’s method that does not require knowledge of the derivative. Finally, we present the setting for systems of nonlinear equations. Two of our standard modeling examples are extended from one unknown to two. The details for implementing Newton’s method for two equations and two unknowns are provided (in addition to the scalar algorithms) so that you have the solvers all ready to go and tackle the applied problems in this chapter – and beyond.