In this study, we conduct a thorough and meticulous examination of the Runge phenomenon. Initially, we engage in an extensive review of relevant literature, which aids in delineating the genesis and essence of the Runge phenomenon, along with an exploration of both conventional and contemporary algorithmic solutions. Subsequently, the paper delves into a diverse array of resolution methodologies, encompassing classical numerical approaches, regularization techniques, mock-Chebyshev interpolation, the TISI (Three-Interval Interpolation Strategy), external pseudo-constraint interpolation, and interpolation strategies predicated upon Singular Value Decomposition (SVD). For each method, we not only introduce but also innovate a novel algorithm to effectively address the phenomenon. This paper executes detailed numerical computations for each method, employing visualization techniques to vividly illustrate the efficacy of various strategies in mitigating the Runge phenomenon. Our findings reveal that although traditional methods exhibit commendable performance in certain instances, novel approaches such as mock-Chebyshev interpolation and regularization-centric methods demonstrate marked superiority in specific contexts. Moreover, the paper provides a critical analysis of these methodologies, specifically highlighting the constraints and potential avenues for enhancement in SVD decomposition-based interpolation strategies. In conclusion, we propose future research trajectories and underscore the imperative of further exploration into interpolation strategies, with an emphasis on their practical application validation. This article serves not only as a comprehensive resource on the Runge phenomenon for researchers but also offers pragmatic guidance for resolving real-world interpolation challenges.

翻译：本研究对龙格现象进行了全面而细致的考察。首先，我们广泛回顾了相关文献，这有助于阐明龙格现象的起源与本质，并探讨了传统及现代的算法解决方案。随后，本文深入研究了多样化的解决方法，包括经典数值方法、正则化技术、模拟切比雪夫插值、三区间插值策略（TISI）、外部伪约束插值以及基于奇异值分解（SVD）的插值策略。针对每种方法，我们不仅进行了介绍，还创新性地提出了新颖算法以有效应对该现象。本文对每种方法执行了详细的数值计算，并采用可视化技术生动展示了各种策略在缓解龙格现象方面的效果。我们的研究结果表明，尽管传统方法在某些情况下表现出色，但模拟切比雪夫插值和基于正则化的方法等新方法在特定情境中展现出显著优势。此外，本文对这些方法进行了批判性分析，特别指出了基于SVD分解的插值策略的局限性与潜在的改进方向。最后，我们提出了未来的研究方向，并强调了进一步探索插值策略的必要性，重点关注其实践应用验证。本文不仅为研究人员提供了关于龙格现象的全面资源，还为解决实际插值挑战提供了实用指导。