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The existing doubling algorithms have been proven efficient for several important nonlinear matrix equations arising from real-world engineering applications. In a nutshell, the algorithms iteratively compute a basis matrix, in one of the…
In this paper, we collect the basic theory and the most important applications of a novel technique that has shown to be suitable for scattered data interpolation, quadrature, bio-imaging reconstruction. The method relies on polynomial…
We describe an algorithm for fast multiplication of skew polynomials. It is based on fast modular multiplication of such skew polynomials, for which we give an algorithm relying on evaluation and interpolation on normal bases. Our…
Multiplying matrices is among the most fundamental and compute-intensive operations in machine learning. Consequently, there has been significant work on efficiently approximating matrix multiplies. We introduce a learning-based algorithm…
In the paper, we introduce a matrix method to constructively determine spaces of polynomial solutions (in general, multiplied by exponentials) to a system of constant coefficient linear PDE's with polynomial (multiplied by exponentials)…
In this paper we develop a new method which is a generalization of the Obreshkoff -Ehrlich method for the cases of algebraic, trigonometric and exponential polynomials. This method has a cubic rate of convergence. It is efficient from the…
The goal of the paper is to analyze a Gaudin model for a polynomial representation of the Kohno-Drinfeld Lie algebra associated with the multinomial distribution. The main result is the construction of an explicit basis of the space of…
It is shown in "SIAM J. Sci. Comput. 39 (2017):B424-B441" that free-form curves used in computer aided geometric design can usually be represented as the solutions of linear differential systems and points and derivatives on the curves can…
Zernike polynomials are a basis of orthogonal polynomials on the unit disk that are a natural basis for representing smooth functions. They arise in a number of applications including optics and atmospheric sciences. In this paper, we…
We consider ideals involving the maximal minors of a polynomial matrix. For example, those arising in the computation of the critical values of a polynomial restricted to a variety for polynomial optimisation. Gr\"obner bases are a…
We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular…
Subresultant is a powerful tool for developing various algorithms in computer algebra. Subresultants for polynomials in standard basis (i.e., power basis) have been well studied so far. With the popularity of basis-preserving algorithms,…
Symbolic Mathematical tasks such as integration often require multiple well-defined steps and understanding of sub-tasks to reach a solution. To understand Transformers' abilities in such tasks in a fine-grained manner, we deviate from…
We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalisation of multiple polylogarithms, we construct our set of integration kernels ensuring…
We present an algebraic method for constructing a highly effective coarse grid correction to accelerate domain decomposition. The coarse problem is constructed from the original matrix and a small set of input vectors that span a low-degree…
The radial polynomials of the 2D (circular) and 3D (spherical) Zernike functions are tabulated as powers of the radial distance. The reciprocal tabulation of powers of the radial distance in series of radial polynomials is also given, based…
We give formulas for the multiplicity of any affine isolated zero of a generic polynomial system of n equations in n unknowns with prescribed sets of monomials. First, we consider sets of supports such that the origin is an isolated root of…
In~\cite{algorithmic} was given an algorithm that computes arithmetical structures on matrices. We use some of the ideas contained there to get an algorithm that computes arithmetical structures over dominated polynomials. A dominated…
We extend the algorithms of Robinson, Smyth, and McKee--Smyth to enumerate all real-rooted integer polynomials of a fixed degree, where the first few (at least three) leading coefficients are specified. Additionally, we introduce new linear…
Factorization machines and polynomial networks are supervised polynomial models based on an efficient low-rank decomposition. We extend these models to the multi-output setting, i.e., for learning vector-valued functions, with application…