  We describe an algorithm for computing Macaulay dual spaces for multi-graded
ideals. For homogeneous ideals, the natural grading is inherited by the
Macaulay dual space which has been leveraged to develop algorithms to compute
the Macaulay dual space in each homogeneous degree. Our main theoretical result
extends this idea to multi-graded Macaulay dual spaces inherited from
multi-graded ideals. This natural duality allows ideal operations to be
translated from homogeneous ideals to their corresponding operations on the
multi-graded Macaulay dual spaces. In particular, we describe a linear operator
with a right inverse for computing quotients by a multi-graded polynomial. By
using a total ordering on the homogeneous components of the Macaulay dual
space, we also describe how to recursively construct a basis for each
component. Several examples are included to demonstrate this new approach.
