Some issues on interpolation matrices of locally scaled radial basis functions

Mun Bae Lee, Yeon Ju Lee, Hasik Sunwoo, Jungho Yoon

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Radial basis function interpolation on a set of scattered data is constructed from the corresponding translates of a basis function, which is conditionally positive definite of order m ≥ 0, with the possible addition of a polynomial term. In many applications, the translates of a basis function are scaled differently, in order to match the local features of the data such as the flat region and the data density. Then, a fundamental question is the non-singularity of the perturbed interpolation (N × N) matrix. In this paper, we provide some counter examples of the matrices which become singular for N ≥ 3, although the matrix is always non-singular when N = 2. One interesting feature is that a perturbed matrix can be singular with rather small perturbation of the scaling parameter.

Original languageEnglish
Pages (from-to)5011-5014
Number of pages4
JournalApplied Mathematics and Computation
Issue number10
StatePublished - 15 Jan 2011


  • Conditionally positive definite function
  • Radial basis function
  • Scaling parameter
  • Singularity


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