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.

翻译：本文提出了一种计算两个给定测度的加性自由卷积与乘性自由卷积的算法。我们考虑具有紧支撑的测度，其自由卷积结果在边界处呈现平方根衰减的密度函数（例如半圆分布或马尔琴科-帕斯图尔分布）。该方法的核心在于利用柯西积分公式重写自由卷积的中间量，进而通过梯形求积法则离散化这些积分——在待积分函数满足适当解析性质的前提下，该数值方法可实现指数级收敛速度。