The Gaussian and Laguerre orthogonal ensembles are fundamental to random matrix theory, and the marginal eigenvalue distributions are basic observable quantities. Notwithstanding a long history, a formulation providing high precision numerical evaluations for $N$ large enough to probe asymptotic regimes, has not been provided. An exception is for the largest eigenvalue, where there is a formalism due to Chiani which uses a combination of the Pfaffian structure underlying the ensembles, and a recursive computation of the matrix elements. We augment this strategy by introducing a generating function for the conditioned gap probabilities. A finite Fourier series approach is then used to extract the sequence of marginal eigenvalue distributions as a linear combination of Pfaffians, with the latter then evaluated using an efficient numerical procedure available in the literature. Applications are given to illustrating various asymptotic formulas, local central limit theorems, and central limit theorems, as well as to probing finite size corrections. Further, our data indicates that the mean values of the marginal distributions interlace with the zeros of the Hermite polynomial (Gaussian ensemble) and a Laguerre polynomial (Laguerre ensemble).

翻译：高斯与拉盖尔正交系综是随机矩阵理论的基础体系，其边际特征值分布是基本可观测物理量。尽管研究历史久远，针对足够大$N$值以探测渐近区域的、能提供高精度数值计算的数学表述尚未建立。最大特征值的情形是个例外，这归功于Chiani提出的形式体系——该体系结合了系综底层Pfaffian结构与矩阵元素的递归计算。我们通过引入条件间隔概率的生成函数来扩展此策略，进而采用有限傅里叶级数方法将边际特征值分布序列提取为Pfaffian的线性组合，并利用文献中的高效数值程序进行求值。本方法被应用于阐释各类渐近公式、局部中心极限定理及中心极限定理，同时用于探测有限尺寸修正效应。此外，我们的数据表明：边际分布均值与埃尔米特多项式（高斯系综）及拉盖尔多项式（拉盖尔系综）的零点呈现交错分布规律。