Radial Basis Functions (RBFs) have shown the potential to be a universal mesh-free method for solving interpolation and differential equations with highly accurate results. However, the trade-off principle states that while deciding the shape parameter’s value for an RBF such as Multiquadric (MQ) or Gaussian (GA), a compromise must be made between achieving accuracy and stability because of the resultant ill-conditioned matrix.

This study focus on the behaviors between the maximum and residual errors for the RBF interpolation. Based on the error behaviors, we propose a new approach, Residual-Error Cross Validation (RECV), to quickly select a suitable $c$ value for an interpolant using an RBF containing a shape parameter. The numerical results showed that an RBF interpolant could yield high accuracy with the RECV $c$ and a sufficiently small fill distance. Combining the RECV method and LOOCV method, we can easily avoid the local optimum issue when applying an optimization algorithm.

径向基函数 (RBF) 已显示出成为一种通用无网格方法的潜力，可用于求解具有高精度结果的插值和微分方程。然而，权衡原则指出，在确定 RBF（例如 Multiquadric (MQ) 或 Gaussian (GA)）的形状参数值时，由于产生的病态矩阵，必须在实现准确性和稳定性之间做出折衷。

本研究的重点是 RBF 插值的最大误差和残差之间的行为。基于错误行为，我们提出了一种新方法Residual-Error Cross Validation (RECV)，以快速选择合适的$C$使用包含形状参数的 RBF 的插值值。数值结果表明，RBF插值可以产生较高的RECV精度 $C$和足够小的填充距离。结合 RECV 方法和 LOOCV 方法，我们可以在应用优化算法时轻松避免局部最优问题。