Related papers: Evaluating Non-Analytic Functions of Matrices
The purpose of the paper is to provide a characterization of the error of the best polynomial approximation of composite functions in weighted spaces. Such a characterization is essential for the convergence analysis of numerical methods…
We study a general class of recurrence relations that appear in the application of a matrix diagonalization procedure. We find general closed formula and determine analytical properties of the solutions. We finally apply these findings in…
For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=\frac{1}{N}\sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $\ell^{p}$-improving…
There are two fundamental problems motivated by Silverman's conversations over the years concerning the nature of the exact values of canonical heights of $f(z)\in\bar{\mathbb{Q}}(z)$ where $f$ has degree $d\geq 2$. The first problem is the…
We analyze matrix convex functions of a fixed order defined on a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus. We obtain for each order conditions for matrix…
We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several…
We develop several efficient algorithms for the classical \emph{Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input $n\times n$ matrix $A$, this…
Given a matrix of distribution functions and a quasi-stochastic matrix, i.e. an irreducible nonnegative matrix with maximal eigenvalue one and associated unique positive left and right eigenvectors, the article studies the properties of an…
We give some experimental observations on the growth of the norm of certain matrices related to the Mertens function. The results obtained in these experiments convince us that linear algebra may help in the study of Mertens function and…
The paper proves sum-of-square-of-rational-function based representations (shortly, sosrf-based representations) of polynomial matrices that are positive semidefinite on some special sets: $\mathbb{R}^n;$ $\mathbb{R}$ and its intervals…
Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…
Interior eigenvalue problems for large-scale sparse Hermitian matrices are fundamental in computational science. We propose an adaptive polynomial filtering strategy based on Chebyshev expansion of a step function, integrated into a…
Estimating the Domain of Attraction (DA) of non-polynomial systems is a challenging problem. Taylor expansion is widely adopted for transforming a nonlinear analytic function into a polynomial function, but the performance of Taylor…
Chebyshev expansion coefficients can be computed efficiently by using the FFT, and for smooth functions the resulting approximation is close to optimal, with computations that are numerically stable. Given sufficiently accurate function…
Sparse interpolation} refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now…
Recent years have witnessed the introduction and development of extremely fast rational function algorithms. Many ideas in this realm arose from polynomial-based linear-algebraic algorithms. However, polynomial approximation is occasionally…
Let $M$ be a 3-connected matroid and let $\mathbb F$ be a field. Let $A$ be a matrix over $\mathbb F$ representing $M$ and let $(G,\mathcal B)$ be a biased graph representing $M$. We characterize the relationship between $A$ and…
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 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…
We present in this paper the motivation and theory of nonlinear spectral representations, based on convex regularizing functionals. Some comparisons and analogies are drawn to the fields of signal processing, harmonic analysis and sparse…