On the ill-conditioned nature of C∞ RBF strong collocation

Abstract Continuously differentiable radial basis functions ( C ∞ -RBFs) are the best method to solve numerically higher dimensional partial differential equations (PDEs). Among the reasons are: 1. An n-dimensional problem becomes a one-dimensional radial distance problem, 2. The convergence rate increases with the dimensionality, 3. Such RBFs possess spectral convergence.Finitely supported polynomial methods only converge at polynomial rates. C ∞ -RBFs have global support; the systems of equations may become computationally singular if the condition number exceeds the inverse machine epsilon, e M . The solution to computational singularity is to decrease the effective e M by either hardware or software methods. Computer scientists developed rapidly executable multi-precision packages.