  The metaplectic transform (MT), also known as the linear canonical transform,
is a unitary integral mapping which is widely used in signal processing and can
be viewed as a generalization of the Fourier transform. For a given function
$\psi$ on an $N$-dimensional continuous space $\textbf{q}$, the MT of $\psi$ is
parameterized by a rotation (or more generally, a linear symplectic
transformation) of the $2N$-dimensional phase space $(\textbf{q},\textbf{p})$,
where $\textbf{p}$ is the wavevector space dual to $\textbf{q}$. Here, we
derive a pseudo-differential form of the MT. For small-angle rotations, or
near-identity transformations of the phase space, it readily yields asymptotic
\textit{differential} representations of the MT, which are easy to compute
numerically. Rotations by larger angles are implemented as successive
applications of $K \gg 1$ small-angle MTs. The algorithm complexity scales as
$O(K N^3 N_p)$, where $N_p$ is the number of grid points. We present a
numerical implementation of this algorithm and discuss how to mitigate the
associated numerical instabilities.
