Related papers: New Methods of Integration in Matrix Models

We show how to calculate correlation functions of two matrix models. Our method consists in making full use of the integrable hierarchies and their reductions, which were shown in previous papers to naturally appear in multi--matrix models.…

We compute the correlation functions mixing the powers of two non-commuting random matrices within the same trace. The angular part of the integration was partially known in the literature: we pursue the calculation and carry out the…

Many physical systems can be described in terms of matrix models that we often cannot solve analytically. Fortunately, they can be studied numerically in a straightforward way. Many commonly used algorithms follow the Monte Carlo method,…

Matrix transformations in terms of triangular matrices is the easiest method of evaluating matrix-variate gamma and beta integrals in the real and complex cases. Here we give several procedures of explicit evaluation of gamma and beta…

We consider the simplest gauge theories given by one- and two- matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric…

This article proposes novel estimation methods for the Matrix Autoregressive (MAR) model, specifically adaptations of the Yule-Walker equations and Burg's method, addressing limitations in existing techniques. The MAR model, by maintaining…

Random matrix theory allows one to deduce the eigenvalue spectrum of a large matrix given only statistical information about its elements. Such results provide insight into what factors contribute to the stability of complex dynamical…

We apply matrix methods to arithmetic functions by associating matrices to the functions in a manner drawn from the theory of symmetric functions. Then we study the characteristic polynomials of the associated matrices.

A matrix-based approach to numerical integration of the DGLAP evolution equations is presented. The method arises naturally on discretisation of the Bjorken x variable, a necessary procedure for numerical integration. Owing to peculiar…

We propose a simple technique that, if combined with algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a specific scaling similarity transformation that reduces…

We consider solvable matrix models. We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals (Gross-Witten integral and integrals over complex matrices) using the notion of tau function of matrix argument. In this case one…

Contour integral methods for nonlinear eigenvalue problems seek to compute a subset of the spectrum in a bounded region of the complex plane. We briefly survey this class of algorithms, establishing a relationship to system realization…

Using random matrix technique we determine an exact relation between the eigenvalue spectrum of the covariance matrix and of its estimator. This relation can be used in practice to compute eigenvalue invariants of the covariance…

The paper introduces a generalization for known probabilistic models such as log-linear and graphical models, called here multiplicative models. These models, that express probabilities via product of parameters are shown to capture…

The techniques for polynomial interpolation and Gaussian quadrature are generalized to matrix-valued functions. It is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the…

We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters;…

A novel orthogonalization-free method together with two specific algorithms are proposed to solve extreme eigenvalue problems. On top of gradient-based algorithms, the proposed algorithms modify the multi-column gradient such that earlier…

We propose a novel multi-dimensional integration algorithm using a machine learning (ML) technique. After training a ML regression model to mimic a target integrand, the regression model is used to evaluate an approximation of the integral.…

An algorithm for computing an analytic function of a matrix $A$ is described. The algorithm is intended for the case where $A$ has some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This…

We present a new power method to obtain solutions of eigenvalue problems. The method can determine not only the dominant or lowest eigenvalues but also all eigenvalues without the need for a deflation procedure. The method uses a functional…