  One commonly finds in applications of smooth radial basis functions (RBFs)
that scaling the kernels so they are `flat' leads to smaller discretization
errors. However, the direct numerical approach for computing with flat RBFs
(RBF-Direct) is severely ill-conditioned. We present an algorithm for bypassing
this ill-conditioning that is based on a new method for rational approximation
(RA) of vector-valued analytic functions with the property that all components
of the vector share the same singularities. This new algorithm (RBF-RA) is more
accurate, robust, and easier to implement than the Contour-Pad\'e method, which
is similarly based on vector-valued rational approximation. In contrast to the
stable RBF-QR and RBF-GA algorithms, which are based on finding a better
conditioned base in the same RBF-space, the new algorithm can be used with any
type of smooth radial kernel, and it is also applicable to a wider range of
tasks (including calculating Hermite type implicit RBF-FD stencils). We present
a series of numerical experiments demonstrating the effectiveness of this new
method for computing RBF interpolants in the flat regime. We also demonstrate
the flexibility of the method by using it to compute implicit RBF-FD formulas
in the flat regime and then using these for solving Poisson's equation in a 3D
spherical shell.
