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：采用典型求解策略计算具有挑战性问题的解曲线时，常会导致算法失效。为改进求解过程，数值延拓方法已被证明是一种非常有效的工具。然而，这些方法仍可能导致不期望的结果。特别是在严重极限点和尖点附近，求解过程频繁遇到以下情况：算法发散、方向变化使得算法回溯到已获得的解曲线部分、以及因收敛到比预期远得多的点而遗漏解曲线的重要区域。在实际问题求解中，由于解曲线形状未知，检测这些情况并非易事。因此，本文将提出一种改进的Moore-Penrose延拓方法，该方法包含两个关键方面以解决具有挑战性的问题：求解过程中检测问题区域，以及处理这些区域的附加步骤。所提出方法既可用作基本延拓方法，也可仅在出现困难时激活。数值算例将展示新方法的有效性。