  This work proposes algorithms for computing additive and multiplicative free
convolutions of two given measures. We consider measures with compact support
whose free convolution results in a measure with a density function that
exhibits a square-root decay at the boundary (for example, the semicircle
distribution or the Marchenko-Pastur distribution). A key ingredient of our
method is rewriting the intermediate quantities of the free convolution using
the Cauchy integral formula and then discretizing these integrals using the
trapezoidal quadrature rule, which converges exponentially fast under suitable
analyticity properties of the functions to be integrated.
