Using typical solution strategies to compute the solution curve of challenging problems often leads to the break down of the algorithm. To improve the solution process, numerical continuation methods have proved to be a very efficient tool. However, these methods can still lead to undesired results. In particular, near severe limit points and cusps, the solution process frequently encounters one of the following situations : divergence of the algorithm, a change in direction which makes the algorithm backtrack on a part of the solution curve that has already been obtained and omitting important regions of the solution curve by converging to a point that is much farther than the one anticipated. Detecting these situations is not an easy task when solving practical problems since the shape of the solution curve is not known in advance. This paper will therefore present a modified Moore-Penrose continuation method that will include two key aspects to solve challenging problems : detection of problematic regions during the solution process and additional steps to deal with them. The proposed approach can either be used as a basic continuation method or simply activated when difficulties occur. Numerical examples will be presented to show the efficiency of the new approach.

翻译：使用典型的解决方案策略来计算具有挑战性的问题的解决方案曲线,往往会导致算法的分解。为了改进解算过程,数字延续方法被证明是一个非常有效的工具。然而,这些方法仍然可以导致不理想的结果。特别是,在接近严格的限制点和临界点的情况下,解决方案进程经常遇到以下一种情况:算法的差异、使算法在已经获得的解决方案曲线的一部分上出现反向路径的改变,以及通过汇集到比预期的要远得多的点而忽略了重要的解算曲线区域。在解决实际问题时,发现这些情况并非易事,因为解决方案曲线的形状事先不为人们所知。因此,本文件将提出一个修改的摩尔-彭罗持续方法,其中将包括解决具有挑战性的问题的两个关键方面:在解算过程期间发现有问题的区域,并采取其他步骤处理这些问题。拟议的方法可以作为一种基本的延续方法,或者在发生困难时仅仅被激活。将提出数字示例,以显示新办法的效率。</s>