科研之友
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摘要
Gaussian radial basis functions (RBFs) on an infinite interval with uniform grid pacing h are defined by phi(x; alpha, h) equivalent to exp(-[alpha(2)/h(2)]x(2)). The only significant numerical parameter is alpha, the inverse width of the RBF functions relative to h. In the limit alpha -> 0, we demonstrate that the coefficients of the interpolant of a typical function f(x) grow proportionally to exp(pi(2)/[4 alpha(2)]). However, we also show that the approximation to the constant f(x) equivalent to 1 is a Jacobian theta function whose coefficients do not blow up as alpha -> 0. The subtle interplay between the complex-plane singularities of f(x) (the function being approximated) and the RBF inverse width parameter alpha are analyzed. For alpha approximate to 1/2, the size of the RBF coefficients and the condition number of the interpolation matrix are both no larger than O(10(4)) and the error saturation is smaller than machine epsilon, so this alpha is the center of a "safe operating range" for Gaussian RBFs.
- 出版日期2010-6-15
