We establish a new perturbation theory for orthogonal polynomials using a Riemann--Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the orthogonal polynomials with respect to two measures can be effectively compared using the difference of their Stieltjes transforms on a suitably chosen contour. Moreover, when two measures are close and satisfy some regularity conditions, we use the theta functions of a hyperelliptic Riemann surface to derive explicit and accurate expansion formulae for the perturbed orthogonal polynomials. In contrast to other approaches, a key strength of the methodology is that estimates can remain valid as the degree of the polynomial grows. The results are applied to analyze several numerical algorithms from linear algebra, including the Lanczos tridiagonalization procedure, the Cholesky factorization and the conjugate gradient algorithm. As a case study, we investigate these algorithms applied to a general spiked sample covariance matrix model by considering the eigenvector empirical spectral distribution and its limits. For the first time, we give precise estimates on the output of the algorithms, applied to this wide class of random matrices, as the number of iterations diverges. In this setting, beyond the first order expansion, we also derive a new mesoscopic central limit theorem for the associated orthogonal polynomials and other quantities relevant to numerical algorithms.
翻译:我们使用 Riemann- Hilbert 方法为正弦多面体建立新的扰动理论, 并考虑数值线性代数和随机矩阵理论的应用。 这一新方法显示, 两种计量的正弦多面体可以有效地比较, 使用其 Stieltjes 变异在适当选择的等距上的差异。 此外, 当两个计量接近并符合某些规律性条件时, 我们使用超伸缩性里叶曼表面的函数, 来为过弯或超直线性多面体论理论得出明确和准确的扩展公式。 与其他方法相比, 方法的关键优点是, 估计值随着多元性增长程度的提高, 可以继续有效比较。 其结果用于分析线性平面变异性变异性的若干数值算法, 包括 兰科佐斯 三角方位化程序、 空基因数 和 conjugate 梯度算法。 作为案例研究, 我们用这些算算法应用到一个通用的缩略图性矩阵矩阵矩阵矩阵模型模型模型。 考虑这个精确的分数的分数 。 在我们测的分数矩阵中, 的分数矩阵中, 也提供了这个分数的分数的分数的分数 。