Globalization concepts for Newton-type iteration schemes are widely used when solving nonlinear problems numerically. Most of these schemes are based on a predictor/corrector step size methodology with the aim of steering an initial guess to a zero of f without switching between different attractors. In doing so, one is typically able to reduce the chaotic behavior of the classical Newton-type iteration scheme. In this note we propose a globalization methodology for general Newton-type iteration concepts which changes into a simplified Newton iteration as soon as the transformed residual of the underlying function is small enough. Based on Banach’s fixed-point theorem, we show that there exists a neighborhood around a suitable iterate \(x_{n}\) such that we can steer the iterates—without any adaptive step size control but using a simplified Newton-type iteration within this neighborhood—arbitrarily close to an exact zero of f. We further exemplify the theoretical result within a global Newton-type iteration procedure and discuss further an algorithmic realization. Our proposed scheme will be demonstrated on a low-dimensional example thereby emphasizing the advantage of this new solution procedure.

牛顿型迭代方案的全球化概念在数值求解非线性问题时被广泛使用。这些方案大多数基于预测器/校正器步长方法，目的是将初始猜测值控制为f的零而不在不同的吸引子之间进行切换。这样做通常可以减少经典的牛顿型迭代方案的混沌行为。在本说明中，我们为一般的牛顿型迭代概念提出了一种全球化方法，只要基础函数的转换残差足够小，该方法便会简化为牛顿迭代。基于Banach的不动点定理，我们证明在适当的迭代\（x_ {n} \）周围存在一个邻域这样我们就可以控制迭代（无需任何自适应步长控制，而是在该邻域内使用简化的牛顿型迭代），任意接近f的精确零。我们将在全局牛顿型迭代过程中进一步举例说明理论结果，并进一步讨论算法实现。我们的建议方案将在一个低维示例上进行演示，从而强调了这种新解决方案程序的优势。