  Extrapolation is a generic problem in physics and mathematics: how to use
asymptotic data in one parametric regime to learn about the behavior of a
function in another parametric regime. For example: extending weak coupling
expansions to strong coupling, or high temperature expansions to low
temperature, or vice versa. Such extrapolations are particularly interesting in
systems possessing dualities. Here we study numerical procedures for performing
such an extrapolation, combining ideas from resurgent asymptotics with
well-known techniques of Borel summation, Pade approximants and conformal
mapping. We illustrate the method with the concrete example of the Painleve I
equation, which has applications in many branches of physics and mathematics.
Starting solely with a finite number of coefficients from asymptotic data at
infinity on the positive real line, we obtain a high precision extrapolation of
the function throughout the complex plane, even across the phase transition
into the pole region. The precision far exceeds that of state-of-the-art
numerical integration methods along the real axis. The methods used are both
elementary and general, not relying on Painleve integrability properties, and
so are applicable to a wide class of extrapolation problems.
