We describe a simple analytical method for effective summation of series, including divergent series. The method is based on self-similar approximation theory resulting in self-similar root approximants. The method is shown to be general and applicable to different problems, as is illustrated by a number of examples. The accuracy of the method is not worse, and in many cases better, than that o...

A novel non-paraxial Time-Domain technique based on Pade approximant for modeling ultra short optical pulses has been formulated and tested. The new method is an extension to the parabolic Time-Domain Beam Propagation Method (TDBPM) that solves the time domain wave equation along one direction that allows the time window to follow the evolution of the pulse. In this work the technique was appli...

The main idea of the present article is to introduce a new computational technique, explicitly, the modified Laplace Padé decomposition method (MLPDM) which is a recipe of Laplace transformation, decomposition technique, and rational polynomial to offer new solution mechanism of magnetohydrodynamics (MHD) flow of an steady viscous, incompressible nonlinear stretching sheet. A good harmony among...

We study a constructive method of rational approximation of analytic functions based on ideas of the theory of Hankel operators. Some properties of the corresponding Hankel operator are investigated. We also consider questions related to the convergence of rational approximants. Analogues of Montessus de Ballore’s and Gonchars’s theorems on the convergence of rows of Padé approximants are proved.

We determine all integers n such that n2 has at most three base-q digits for q ∈ {2, 3, 4, 5, 8, 16}. More generally, we show that all solutions to equations of the shape Y 2 = t + M · q + N · q, where q is an odd prime, n > m > 0 and t2, |M |, N < q, either arise from “obvious” polynomial families or satisfy m ≤ 3. Our arguments rely upon Padé approximants to the binomial function, considered ...

Abstract The inverse scaling and squaring algorithm computes the logarithm of a square matrix $A$ by evaluating rational approximant to at $B:=A^{2^{-s}}$ for suitable choice $s$. We introduce dual approach approximate $B$ solving equation $r(X)=B$, where $r$ is diagonal Padé exponential $0$. This solved substitution technique in style those developed Fasi &amp; Iannazzo (2020, Substitution alg...

We design convergent multipoint Padé interpolation schemes to Cauchy transforms of non-vanishing complex densities with respect to Jacobi-type weights on analytic arcs, under mild smoothness assumptions on the density. We rely on the work [10] for the choice of the interpolation points, and dwell on the Riemann-Hilbert approach to asymptotics of orthogonal polynomials introduced in [33] in the ...

Given function f holomorphic at infinity, the n-th diagonal Padé approximant to f , say [n/n]f , is a rational function of type (n,n) that has the highest order of contact with f at infinity. Equivalently, [n/n]f is the n-th convergent of the continued fraction representing f at infinity. BernsteinSzegő theorem provides an explicit non-asymptotic formula for [n/n]f and all n large enough in the...

In this article, homotopy perturbation method coupled with Laplace transform and Padé 0approximants is applied on the re-formulated nonlinear oscillatory systems. Numerical results and graphical represenations explicitly reveal the complete reliability and effeicincy of the suggested algorithm.

We classify all integer squares (and, more generally, q-th powers for certain values of q) whose ternary expansions contain at most three digits. Our results follow from Padé approximants to the binomial function, considered 3-adically.