  Let $ E \subset \mathbb{R}^2 $ be a finite set, and let $ f : E \to
[0,\infty) $. In this paper, we address the algorithmic aspects of nonnegative
$C^2$ interpolation in the plane. Specifically, we provide an efficient
algorithm to compute a nonnegative $C^2(\mathbb{R}^2)$ extension of $ f $ with
norm within a universal constant factor of the least possible. We also provide
an efficient algorithm to approximate the trace norm.
