Very Classical Moving Least Squares Algorithm with Multi-Dimensional Implementations

In this article, the author presents the highly classical Moving Least Squares (MLS) algorithm, which demonstrates robust applicability across one-dimensional, two-dimensional, and three-dimensional problem domains. Recognizing that these multi-dimensional implementations could potentially occupy significant storage space due to matrix operations and spatial discretization requirements, the author has compressed all algorithm variants into a single downloadable package for user convenience. The MLS algorithm implementation typically involves local approximation techniques where weight functions play a crucial role in determining solution accuracy. Beyond the basic distribution, the author could provide enhanced technical documentation covering specific application scenarios (such as surface reconstruction and data approximation), algorithmic advantages (including adaptability to irregular node distributions), and limitations (such as computational complexity in higher dimensions). Additional technical notes about key implementation aspects - like the selection of basis functions, weight function optimization, and matrix conditioning techniques - would further assist readers in understanding both the theoretical foundations and practical deployment of these algorithms. Overall, this article serves as an excellent starting point for researchers and practitioners to explore Moving Least Squares methodology and its diverse applications in computational mathematics and engineering simulations.