  We define a class of "algebraic" random matrices. These are random matrices
for which the Stieltjes transform of the limiting eigenvalue distribution
function is algebraic, i.e., it satisfies a (bivariate) polynomial equation.
The Wigner and Wishart matrices whose limiting eigenvalue distributions are
given by the semi-circle law and the Marcenko-Pastur law are special cases.
Algebraicity of a random matrix sequence is shown to act as a certificate of
the computability of the limiting eigenvalue density function. The limiting
moments of algebraic random matrix sequences, when they exist, are shown to
satisfy a finite depth linear recursion so that they may often be efficiently
enumerated in closed form.
  In this article, we develop the mathematics of the polynomial method which
allows us to describe the class of algebraic matrices by its generators and map
the constructive approach we employ when proving algebraicity into a software
implementation that is available for download in the form of the RMTool random
matrix "calculator" package. Our characterization of the closure of algebraic
probability distributions under free additive and multiplicative convolution
operations allows us to simultaneously establish a framework for computational
(non-commutative) "free probability" theory. We hope that the tools developed
allow researchers to finally harness the power of the infinite random matrix
theory.
