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.

翻译：随机矩阵理论的分布近年来引起了人们的极大兴趣,并且在物理、高维统计、无线通信、金融等各种应用领域都发现了随机矩阵理论的分布。 Tracy-Widom分布是随机矩阵理论中最重要的分布之一,其数字评价是一个非常重要的主题。 用于评价Tracy-Widom分布的一个数字方法是,该分布可以作为某种整体操作员的Fredholm决定因素来表示。然而,当整体操作员的频谱通过直接分解来计算时,其值是绝对精确的。 值得注意的是,整体操作员是所谓的双光谱操作员的一个例子,它承认一个具有相同功能的通勤差操作员。 在这份手稿中,我们开发了一个有效的数字算法,用不同操作员的eigendecom组合来评价整体操作员的完全相对精确性。 使用我们的算法, Tristi-Widom分布可以被评估为各地的完全绝对精确度,此外,其右尾部可以计算为完全精确度。