  We consider two inverse problems for the multi-channel two-dimensional
Schr\"odinger equation at fixed positive energy, i.e. the equation $-\Delta
\psi + V(x)\psi = E \psi$ at fixed positive $E$, where $V$ is a matrix-valued
potential. The first is the Gel'fand inverse problem on a bounded domain $D$ at
fixed energy and the second is the inverse fixed-energy scattering problem on
the whole plane $\R^2$. We present in this paper two algorithms which give
efficient approximate solutions to these problems: in particular, in both cases
we show that the potential $V$ is reconstructed with Lipschitz stability by
these algorithms up to $O(E^{-(m-2)/2})$ in the uniform norm as $E \to
+\infty$, under the assumptions that $V$ is $m$-times differentiable in $L^1$,
for $m \geq 3$, and has sufficient boundary decay.
