The geometric iterative method (GIM) is widely used in data interpolation/fitting, but its slow convergence affects the computational efficiency. Recently, much work was done to guarantee the acceleration of GIM in the literature. In this work, we aim to further accelerate the rate of convergence by introducing a preconditioning technique. After constructing the preconditioner, we preprocess the progressive iterative approximation (PIA) and its variants, called the preconditioned GIMs. We show that the proposed preconditioned GIMs converge and the extra computation cost brought by the preconditioning technique is negligible. Several numerical experiments are given to demonstrate that our preconditioner can accelerate the convergence rate of PIA and its variants.

翻译：几何迭代方法（GIM）广泛应用于数据插值/拟合中，但其收敛速度较慢影响计算效率。近年来，文献中已有大量工作致力于加速GIM的收敛。本文旨在通过引入预处理技术进一步加快收敛速度。在构造预处理算子后，我们对渐进迭代逼近（PIA）及其变体进行预处理，提出预处理几何迭代方法（GIMs）。我们证明了所提出的预处理GIMs能够收敛，且预处理技术带来的额外计算代价可忽略不计。通过若干数值实验表明，我们的预处理算子能有效加速PIA及其变体的收敛速度。