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We present fully polynomial approximation schemes for a broad class of Holant problems with complex edge weights, which we call Holant polynomials. We transform these problems into partition functions of abstract combinatorial structures…

Given the importance of floating-point~(FP) performance in numerous domains, several new variants of FP and its alternatives have been proposed (e.g., Bfloat16, TensorFloat32, and Posits). These representations do not have correctly rounded…

We present a new algorithm for reconstructing an exact algebraic number from its approximate value using an improved parameterized integer relation construction method. Our result is consistent with the existence of error controlling on…

We present effective numerical algorithms for locally recovering unknown governing differential equations from measurement data. We employ a set of standard basis functions, e.g., polynomials, to approximate the governing equation with high…

We consider in this paper the Hahn polynomials and their application in numerical methods. The Hahn polynomials are classical discrete orthogonal polynomials. We analyse the behaviour of these polynomials in the context of spectral…

Generalized Fourier series with orthogonal polynomial bases have useful applications in several fields, including differential equations, pattern recognition, and image and signal processing. However, computing the generalized Fourier…

Unitary best approximation to the exponential function on an interval on the imaginary axis has been introduced recently. In the present work two algorithms are considered to compute this best approximant: an algorithm based on rational…

Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…

A subalgebraic approximation algorithm is proposed to estimate from a set of time series the parameters of the observer representation of a discrete-time polynomial system without inputs which can generate an approximation of the observed…

In many applications (hupergeometric-type) special functions like orthogonal polynomials are needed. For example in more than 50% of the published solutions for the (application-oriented) questions in the "Problems Section" of SIAM Review…

To facilitate the numerical analysis of particle methods, we derive truncation error estimates for the approximate operators in a generalized particle method. Here, a generalized particle method is defined as a meshfree numerical method…

In this paper we introduce a family of rational approximations of the reciprocal of a $\phi$-function involved in the explicit solutions of certain linear differential equations, as well as in integration schemes evolving on manifolds. The…

Using geometric considerations, we provide a clear derivation of the integral representation for the error function, known as the Craig formula. We calculate the corresponding power series expansion and prove the convergence. The same…

A blend of two Taylor series for the same smooth real- or complex-valued function of a single variable can be useful for approximation. We use an explicit formula for a two-point Hermite interpolational polynomial to construct such blends.…

Based on tensor neural network, we propose an interpolation method for high dimensional non-tensor-product-type functions. This interpolation scheme is designed by using the tensor neural network based machine learning method. This means…

In this master's thesis, we introduce expansion systems as a general framework to describe a large variety of approximation algorithms, such as Taylor approximation, decimal expansion and continued fraction. We consider some basic…

We study the approximation of functions which are invariant with respect to certain permutations of the input indices using flow maps of dynamical systems. Such invariant functions includes the much studied translation-invariant ones…

We investigate training and using Gaussian kernel SVMs by approximating the kernel with an explicit finite- dimensional polynomial feature representation based on the Taylor expansion of the exponential. Although not as efficient as the…

Physically relevant field-theoretic quantities are usually derived from perturbation techniques. These quantities are solved in the form of an asymptotic series in powers of small perturbation parameters related to the physical system, and…

In this paper we consider the approximation of a function by its interpolating multilinear spline and the approximation of its derivatives by the derivatives of the corresponding spline. We derive formulas for the uniform approximation…