Searcharxiv⌕ Search

arXiv subjects

Sergey P. Suetin

Publications and source records attributed to Sergey P. Suetin.

At least 19 recordsLinked to original sources
arXiv2026

On Convergence of Rational Hermite-Padé Approximants

The main purpose of this paper is to compare the convergence properties of Padé approximants and rational Hermite-Padé approximants for some model class of multivalued analytic functions based of the inverse Zhoukovsky transform. We prove that in the class of analytic functions under consideration the rational Hermite-Padé approximants converge faster than the corresponding Padé approximants. In contrast to the classical vector potential-theoretic approach, which was introduced by A. A. Gonchar and E. A. Rakhmanov in 1981 and developed later by A. I. Aptekarev, V. N. Sorokin and others, the proofs here are based on some scalar mixed Green-logarithmic potential problems.

math.CV↗
arXiv2023

Rational Hermite-Padé Approximants vs Padé Approximants. Numerical Results

The main purpose of the paper is to present some powerful data on the advantage of the rational approximation procedure based on Hermite-Padé polynomials over the Padé approximation procedure. The first part of the paper is devoted to some numerical examples in this direction. The second part will be devoted to some theoretical results. In particular, we demonstrate our ideas about the advantage of rational Hermite-Padé approximants over Padé approximants analyzing the analytical structure of the frequency function $ν$ of the free Van der Pol equation.

math.CV↗
arXiv2022

On Some Algebraic Properties of Hermite--Padé Polynomials

Let $[f_0,\dots,f_m]$ be a tuple of series in nonnegative powers of $1/z$, $f_j(\infty)\neq0$. It is supposed that the tuple is in "general position". We give a construction of type I and type II Hermite--Padé polynomials to the given tuple of degrees $\leq{n}$ and $\leq{mn}$ respectively and the corresponding $(m+1)$-multi-indexes with the following property. Let $M_1(z)$ and $M_2(z)$ be two $(m+1)\times(m+1)$ polynomial matrices, $M_1(z),M_2(z)\in\operatorname{GL}(m+1,\mathbb C[z])$, generated by type I and type II Hermite--Padé polynomials respectively. Then we have $M_1(z)M_2(z)\equiv I_{m+1}$, where $I_{m+1}$ is the identity $(m+1)\times(m+1)$-matrix. The result is motivated by some novel applications of Hermite--Padé polynomials to the investigation of monodromy properties of Fuchsian systems of differential equations.

math.CV↗
arXiv2021

Maximum Principle and Asymptotic Properties of Hermite--Padé Polynomials

In the paper, we discuss how it would be possible to succeed in Stahl's novel approach, 1987--1988, to explore Hermite--Padé polynomials based on Riemann surface properties. In particular, we explore the limit zero distribution of type I Hermite--Padé polynomials $Q_{n,0},Q_{n,1},Q_{n,2}$, $\operatorname{deg}{Q_{n,j}}\leq{n}$, for a collection of three analytic elements $[1,f_\infty,f^2_\infty]$. The element $f_\infty$ is an element of a function $f$ from the class $\mathbb C(z,w)$ where $w$ is supposed to be from the class $Z_{\pm1/2}([-1,1])$ of multivalued analytic functions generated by the inverse Zhukovskii function with the exponents from the set $\{\pm1/2\}$. The Riemann surface corresponding to $f\in\mathbb C(z,w)$ is a four-sheeted Riemann surface $\mathfrak R_4(w)$ and all branch points of $f$ are of the first order (i.e., all branch points are of square root type). Since the algebraic function $f\in\mathbb C(z,w)$ is of fourth order and we consider the triple of the analytic elements $[1,f_\infty,f^2_\infty]$ but not the quadruple $[1,f_\infty,f^2_\infty,f^3_\infty]$ ones, the result is new and does not follow from the known results. As in previous paper arXiv: 2108.00339 and following to Stahl's ideas, 1987--1988, we do not use the orthogonality relations at all. The proof is based on the maximum principle only.

math.CV↗
arXiv2021

A direct proof of Stahl's theorem for a generic class of algebraic functions

Under the assumption of the existence of Stahl's $S$-compact set we give a short proof of the limit zeros distribution of Padé polynomials and convergence in capacity of diagonal Padé approximants for a generic class of algebraic functions. The proof is direct but not from the opposite as Stahl's original proof is. The generic class means in particular that all branch points of the multi-sheeted Riemann surface of the algebraic function are of the first order (i.e., we assume the surface is such that all branch points are of square root type). We do not use the relations of orthogonality at all. The proof is based on the maximum principle only.

math.CV↗
arXiv2021

Two examples based on the properties of discrete measures

In the paper we represent two examples which are based on the properties of discrete measures. In the first part of the paper we prove that for each probability measure $μ$, $\operatorname{supp}μ=[-1,1]$, which logarithmic potential is a continuous function on $[-1,1]$ there exists a (discrete) measure $σ=σ(μ)$, $\operatorname{supp}σ=[-1,1]$, with the following property. Let $\{P_n(x;σ)\}$ be the sequence of polynomials orthogonal with respect to $σ$. Then $\dfrac1nχ(P_n(\cdot;σ))\overset{*}\toμ$, $n\to\infty$, where $χ(\cdot)$ is zero counting measure for the corresponding polynomial. The construction of the measure $σ$ is based of the properties of weighted Leja points. In the second part we give an example of a compact set and a sequence of discrete measures supported on that compact set with the following property. The sequence of measures converges in weak-$*$ topology to the equilibrium measure for the compact set but the corresponding sequence of the logarithmic potentials does not converge to the equilibrium potential in any neighbourhood of the compact set.

math.CV↗
arXiv2021

Tschebyshev-Padé approximations for multivalued functions

We discuss the relation between the linear Tschebyshev-Padé approximations to analytic function $f$ and the diagonal type I Hermite-Padé polynomials for the tuple of functions $[1,f_1,f_2]$ where the pair of functions $f_1,f_2$ forms certain Nikishin system. An approach is proposed of how to extend the seminal Stahl's Theory for Padé approximations for multivalued analytic functions to the Tschebyshev-Padé approximations. The approach is based on the relation between Tschebyshev-Padé approximations and Hermite-Padé polynomials and also on a connection of Hermite-Padé polynomials and multipoint Padé approximants.

math.CV↗
arXiv2019

Scalar equilibrium problem and the limit distribution of the zeros of Hermite-Padé polynomials of type II

The existence of the limit distribution of the zeros of Hermite-Padé polynomials of type II for a pair of functions forming a Nikishin system is proved using the scalar equilibrium problem posed on the two-sheeted Riemann surface. The relation of the results obtained here to some results of H. Stahl (1988) is discussed. Results of numerical experiments are presented. The results of the present paper and those obtained in the earlier paper of the second author [28], [32], [33] are shown to be in good accordance with both H. Stahl's results and with results of numerical experiments.

math.CV↗
arXiv2018

On one example of a Nikishin system

The paper puts forward an example of a~Markov function $f=\operatorname{const}+\widehatσ$ such that the three functions $f,f^2$ and $f^3$ form a Nikishin system. A conjecture is proposed that there exists a~Markov function $f$ such that, for each $n\in\mathbb N$, the system $f,f^2,\dots,f^n$ constitutes a~Nikishin system. Bibliography:~20~titles.

math.CV↗
arXiv2016

Embedding AC Power Flow in the Complex Plane Part II: A Reliable Framework for Voltage Collapse Analysis

Part II of this paper elaborates on the unique capability of the proposed power flow analysis framework to obtain the true solution corresponding to the stable operating point of a network. It explains the significance of obtaining the true solution for an accurate assessment of the voltage collapse margin. This feature distinguishes the framework from all iterative and non-iterative heuristic approaches as demonstrated in the context of a 7-bus network with Newton-Raphson, its variants and semidefinite and moment-based relaxations of power flow. Another important feature of this framework is that it obtains the true solution when it exists and declares its non-existence otherwise. This is demonstrated in the context of small networks and in comparison with heuristic approaches. This paper also explores how the proposed framework detects a limit-induced bifurcation where a network controller reaching its limit can initiate voltage collapse.

eess.SY↗
arXiv2015

On the limit zero distribution of type I Hermite-Padé polynomials

In this paper are discussed the results of new numerical experiments on zero distribution of type I Hermite-Padé polynomials of order $n=200$ for three different collections of three functions $[1,f_1,f_2]$. These results are obtained by the authors numerically and do not match any of the theoretical results that were proven so far. We consider three simple cases of multivalued analytic functions $f_1$ and $f_2$, with separated pairs of branch points belonging to the real line. In the first case both functions have two logarithmic branch points, in the second case they both have branch points of second order, and finally, in the third case they both have branch points of third order. All three cases may be considered as representative of the asymptotic theory of Hermite-Padé polynomials. In the first two cases the numerical zero distribution of type I Hermite-Padé polynomials are similar to each other, despite the different kind of branching. But neither the logarithmic case, nor the square root case can be explained from the asymptotic point of view of the theory of type I Hermite-Padé polynomials. The numerical results of the current paper might be considered as a challenge for the community of all experts on Hermite-Padé polynomials theory.

math.CV↗
arXiv2015

On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

For an interval $E=[a,b]$ on the real line, let $μ$ be either the equilibrium measure, or the normalized Lebesgue measure of $E$, and let $V^μ$ denote the associated logarithmic potential. In the present paper, we construct a function $f$ which is analytic on $E$ and possesses four branch points of second order outside of $E$ such that the family of the admissible compacta of $f$ has no minimizing elements with regard to the extremal theoretic-potential problem, in the external field equals $V^{-μ}$.

math.CV↗
arXiv2015

Embedding AC Power Flow with Voltage Control in the Complex Plane : The Case of Analytic Continuation via Padé Approximants

This paper proposes a method to embed the AC power flow problem with voltage magnitude constraints in the complex plane. Modeling the action of network controllers that regulate the magnitude of voltage phasors is a challenging task in the complex plane as it has to preserve the framework of holomorphicity for obtention of these complex variables with fixed magnitude. Hence this paper presents a significant step in the development of the idea of Holomorphic Embedding Load Flow Method (HELM), introduced in 2012, that exploits the theory of analytic continuation, especially the monodromy theorem for resolving issues that have plagued conventional numerical methods for decades. This paper also illustrates the indispensable role of Padé approximants for analytic continuation of complex functions, expressed as power series, beyond the boundary of convergence of the series. Later the paper demonstrates the superiority of the proposed method over the well-established Newton-Raphson as well as the recently developed semidefinite and moment relaxation of power flow problems.

eess.SY↗
arXiv2011

Heine, Hilbert, Pade, Riemann, and Stieltjes: a John Nuttall's work 25 years later

In 1986 J. Nuttall published in Constructive Approximation the paper "Asymptotics of generalized Jacobi polynomials", where with his usual insight he studied the behavior of the denominators ("generalized Jacobi polynomials") and the remainders of the Pade approximants to a special class of algebraic functions with 3 branch points. 25 years later we try to look at this problem from a modern perspective. On one hand, the generalized Jacobi polynomials constitute an instance of the so-called Heine-Stieltjes polynomials, i.e. they are solutions of linear ODE with polynomial coefficients. On the other, they satisfy complex orthogonality relations, and thus are suitable for the Riemann-Hilbert asymptotic analysis. Along with the names mentioned in the title, this paper features also a special appearance by Riemann surfaces, quadratic differentials, compact sets of minimal capacity, special functions and other characters.

math.CA↗