  The distributions of random matrix theory have seen an explosion of interest
in recent years, and have been found in various applied fields including
physics, high-dimensional statistics, wireless communications, finance, etc.
The Tracy-Widom distribution is one of the most important distributions in
random matrix theory, and its numerical evaluation is a subject of great
practical importance. One numerical method for evaluating the Tracy-Widom
distribution uses the fact that the distribution can be represented as a
Fredholm determinant of a certain integral operator. However, when the spectrum
of the integral operator is computed by discretizing it directly, the
eigenvalues are known to at most absolute precision. Remarkably, the integral
operator is an example of a so-called bispectral operator, which admits a
commuting differential operator that shares the same eigenfunctions. In this
manuscript, we develop an efficient numerical algorithm for evaluating the
eigendecomposition of the integral operator to full relative precision, using
the eigendecomposition of the differential operator. With our algorithm, the
Tracy-Widom distribution can be evaluated to full absolute precision everywhere
rapidly, and, furthermore, its right tail can be computed to full relative
precision.
