  We develop a numerical approach for computing the additive, multiplicative
and compressive convolution operations from free probability theory. We utilize
the regularity properties of free convolution to identify (pairs of)
`admissible' measures whose convolution results in a so-called `invertible
measure' which is either a smoothly-decaying measure supported on the entire
real line (such as the Gaussian) or square-root decaying measure supported on a
compact interval (such as the semi-circle). This class of measures is important
because these measures along with their Cauchy transforms can be accurately
represented via a Fourier or Chebyshev series expansion, respectively. Thus,
knowledge of the functional inverse of their Cauchy transform suffices for
numerically recovering the invertible measure via a non-standard yet
well-behaved Vandermonde system of equations. We describe explicit algorithms
for computing the inverse Cauchy transform alluded to and recovering the
associated measure with spectral accuracy. Convergence is guaranteed under
broad assumptions on the input measures.
