The objective of this publication is to reduce the sensitivity of iterative equation solvers on the initial value. To this end, at the hand of Newton's method, we exemplify how to reformulate the initial problem by means of a set of generalized moment generating functions. The approach allows to choose that very function, which is best approximated by a linear function and thus allows to set up an efficient iteration procedure. As a result of this, the number of iterations required to meet a given precision goal is significantly reduced in comparison to Newton's method especially for large deviations between the initial value and the actual root. At the hand of seven academic examples and three applications we demonstrate that the computing time of the discussed approach reveals a far lower susceptibility on the initial value when compared to results from Newton's method. This insensitivity offers the prospect to implement iterative equation solvers for applications with strict real-time requirements such as power system simulation or on-demand control algorithms on embedded systems with low computing power. We are confident that the devised methodology may be generalized to other well-established iteration algorithms.

翻译：本文旨在降低迭代方程解法对初值的敏感性。为此，我们以牛顿法为例，展示了如何利用一组广义矩生成函数对原始问题进行重构。该方法允许选择最接近线性函数逼近的特定函数，从而构建高效的迭代过程。结果表明，与牛顿法相比，该方法在达到指定精度目标所需的迭代次数上显著减少，尤其是在初值与实际根偏差较大的情况下。通过七个学术算例和三个实际应用，我们证明了所讨论方法的计算时间对初值的敏感性远低于牛顿法。这种低敏感性为在严格实时要求的应用（如电力系统仿真或低计算能力嵌入式系统的按需控制算法）中实现迭代方程解法提供了可能性。我们相信，所提出的方法可推广至其他成熟的迭代算法。