Related papers: On root mean square approximation by exponential f…
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…
We consider the problem of solving a large-scale Quadratically Constrained Quadratic Program. Such problems occur naturally in many scientific and web applications. Although there are efficient methods which tackle this problem, they are…
The geometric mean of two matrices is considered and analyzed from a computational viewpoint. Some useful theoretical properties are derived and an analysis of the conditioning is performed. Several numerical algorithms based on different…
In this article we study the arithmetic mean value $\Sigma(n)$ of the square roots of the first $n$ integers. For this quantity, we develop an asymptotic expression, and derive a formula for its integer part which has been conjectured…
In this paper we develop an optimisation based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalised rational approximation. In the…
This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…
Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis…
The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for…
We study certain step functions whose nonnegativity is related to the integrality of sequences of ratios of factorial products. In particular, we obtain a lower bound for the mean square of such step functions which allows us to give a…
We study a random sampling technique to approximate integrals $\int_{[0,1]^s}f(\mathbf{x})\,\mathrm{d}\mathbf{x}$ by averaging the function at some sampling points. We focus on cases where the integrand is smooth, which is a problem which…
The Sinc approximation is a function approximation formula that attains exponential convergence for rapidly decaying functions defined on the whole real axis. Even for other functions, the Sinc approximation works accurately when combined…
A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…
These notes are about ridge functions. Recent years have witnessed a flurry of interest in these functions. Ridge functions appear in various fields and under various guises. They appear in fields as diverse as partial differential…
Various approaches to iterative refinement (IR) for least-squares problems have been proposed in the literature and it may not be clear which approach is suitable for a given problem. We consider three approaches to IR for least-squares…
We consider the solution of nonlinear equations in one real variable, the problem usually called by root finding. Although this is an old problem, we believe that some aspects of its solution using interval arithmetic are not well…
We describe a strategy for solving nonlinear eigenproblems numerically. Our approach is based on the approximation of a vector-valued function, defined as solution of a non-homogeneous version of the eigenproblem. This approximation step is…
A new sampling methodology based on incomplete cosine expansion series is presented as an alternative to the traditional sinc function approach. Numerical integration shows that this methodology is efficient and practical. Applying the…
We consider a linear partial integro-differential equation that arises in the modeling of various physical and biological processes. We study the problem in a spatial periodic domain. We analyze numerical stability and numerical convergence…
Our main interest in this paper is to study some approximation problems for classes of functions with mixed smoothness. We use technique, based on a combination of results from hyperbolic cross approximation, which were obtained in 1980s --…
A non trivial problem that arises in several applications is the estimation of the mean of a truncated normal distribution. In this paper, an iterative deterministic scheme for approximating this mean is proposed. It has been inspired from…