TITLE:
A Numerical Method for Solving Ill-Conditioned Equation Systems Arising from Radial Basis Functions
AUTHORS:
Edward J. Kansa
KEYWORDS:
Continuously Differentiable Radial Basis Functions, Global Maxima and Minima, Solutions of Ill-Conditioned Linear Equations, Block Gaussian Elimination, Arbitrary Precision Arithmetic
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.13 No.2,
June
27,
2023
ABSTRACT: Continuously
differentiable radial basis functions (C∞-RBFs), while
being theoretically exponentially convergent are considered impractical
computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the
Hilbert and Vandermonde have full matrices and become ill-conditioned. The
difference between a coefficient matrix generated by C∞-RBFs for partial differential or integral equations and Hilbert and Vandermonde
systems is that C∞-RBFs are very sensitive to small changes in the
adjustable parameters. These parameters affect the condition number and
solution accuracy. The error terrain has many local and global maxima and
minima. To find stable and accurate numerical solutions for full linear
equation systems, this study proposes a hybrid combination of block Gaussian
elimination (BGE) combined with arbitrary precision arithmetic (APA) to
minimize the accumulation of rounding errors. In the future, this algorithm can
execute faster using preconditioners and implemented on massively parallel
computers.