  The determination of exponentially stable equilibria and their basin of
attraction for a dynamical system given by a general autonomous ordinary
differential equation can be achieved by means of a contraction metric. A
contraction metric is a Riemannian metric with respect to which the distance
between adjacent solutions decreases as time increases. The Riemannian metric
can be expressed by a matrix-valued function on the phase space.
  The determination of a contraction metric can be achieved by approximately
solving a matrix-valued partial differential equation by mesh-free collocation
using Radial Basis Functions (RBF). However, so far no rigorous verification
that the computed metric is indeed a contraction metric has been provided.
  In this paper, we combine the RBF method to compute a contraction metric with
the CPA method to rigorously verify it. In particular, the computed contraction
metric is interpolated by a continuous piecewise affine (CPA) metric at the
vertices of a fixed triangulation, and by checking finitely many inequalities,
we can verify that the interpolation is a contraction metric. Moreover, we show
that, using sufficiently dense collocation points and a sufficiently fine
triangulation, we always succeed with the construction and verification. We
apply the method to two examples.
