We compute the Tracy-Widom distribution describing the asymptotic distribution of the largest eigenvalue of a large random matrix by solving a boundary-value problem posed by Bloemendal in his Ph.D. Thesis (2011). The distribution is computed in two ways. The first method is a second-order finite-difference method and the second is a highly accurate Fourier spectral method. Since $\beta$ is simply a parameter in the boundary-value problem, any $\beta> 0$ can be used, in principle. The limiting distribution of the $n$th largest eigenvalue can also be computed. Our methods are available in the Julia package TracyWidomBeta.jl.

翻译：我们通过求解Bloemendal在其博士论文（2011）中提出的边值问题，计算了描述大随机矩阵最大特征值渐近分布的Tracy-Widom分布。该分布通过两种方法计算：第一种方法是二阶有限差分法，第二种是高精度的傅里叶谱方法。由于$β$仅是边值问题中的一个参数，原则上可使用任意$β>0$。此外，还可计算第$n$大特征值的极限分布。我们的方法已集成于Julia软件包TracyWidomBeta.jl中。