Defects and the Convergence of Padé Approximants

A selective survey is given of convergence results for sequences of Padé approximants. Various approaches for dealing with the convergence problems due to `defects" are discussed. Attention is drawn to the close relationship between analyticity properties of a function and the `smoothness" of its Taylor series coefficients. A new theorem on the convergence of horizontal sequences of Padé approximants to functions in the Baker–Gammel–Wills conjecture function class is presented.     

Hermite—Padé Approximants of the Mittag-Leffler Functions

The convergence rate of type II Hermite–Padé approximants for a system of degenerate hypergeometric functions {₁F₁(1, γ; λ_jz)} _j=1 ^k is found in the case when the numbers {λ_j} _j=1 ^k are the roots of the equation λ^k = 1 or real numbers and \(\gamma\in\mathbb{C}\;\backslash\left\{0,-1,-2,...\right\}\). More general statements are obtained for approximants of this type (including nondiagonal ones) in the case of k = 2. The theorems proved in the paper complement and generalize the results obtained earlier by other authors.     

Matrix Padé Approximation of rational functions

In many applications it is of major interest to decide whether a given formal power series with matrix-valued coefficients of arbitrary dimensions results from a matrix-valued rational function. As the main result of this paper we provide an answer to this question in terms of Matrix Padé Approximants of the given power series. Furthermore, given a matrix rational function, the smallest degrees of the matrix polynomials which represent it are not necessarily unique. Therefore we study a certain minimality-type, that is, minimum degrees. We aim to obtain all the minimum degrees for the polynomials which represent the function as equivalents. In addition, given that the rational representation of the function for the same pair of degrees need not be unique, we have obtained conditions to study the uniqueness of said representation. All the results obtained are presented graphically in tables setting out the above information. They lead to a number of properties concerning special structures, staired blocks, in the Padé Table.     

Square Matrix Padé Approximation and Convergence Acceleration of Sequences

This paper provides a new method for approximating matrix-valued functions — square Padéapproximation. Some computational methods of the approximants are given. For accelerating matrix sequences, a family of nonlinear extrapolation formulas based on the square Padé approximation is given, a convergence acceleration theorem is proved and numerical examples are presented.     

Effective Conductivity of Nonlinear Two-Phase Media: Homogenization and Two-Point Padé Approximants

The aim of this paper is a study of the quasilinear transport equation, for instance the stationary heat equation. For periodically microheterogeneous media asymptotic homogenization has been performed with the local problem formulated as a minimization problem. The Golden–Papanicolaou integral representation theorem and some bounds developed for the linear equation have been extended. Two-point Padé approximants have been used to calculate bounds. Examples are also provided.     

Direct and Inverse Results on Row Sequences of Hermite–Padé Approximants

We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of simultaneous rational interpolants with a bounded number of poles. The conditions are expressed in terms of intrinsic properties of the system of functions used to build the approximants. Exact rates of convergence for these denominators and the simultaneous rational approximants are provided.     

Fourier–Padé Approximants for Nikishin Systems

We study type I Fourier–Padé approximation for certain systems of functions formed by the Cauchy transform of finite Borel measures supported on bounded intervals of the real line. This construction is similar to type I Hermite–Padé approximation. Instead of power series expansions of the functions in the system, we take their development in a series of orthogonal polynomials. We give the exact rate of convergence of the corresponding approximants. The answer is expressed in terms of the extremal solution of an associated vector-valued equilibrium problem for the logarithmic potential.      

Matrix orthogonal Laurent polynomials and two-point Padé approximants

This paper is concerned with double sequencesC={C _n} _n ^=–/ of Hermitian matrices with complex entriesC _n M _s×s ) and formal Laurent seriesL ₀(z)=– _k=1 C _–k z ^k andL (z)= _k=0 C _k z ^–k . Making use of a Favard-type theorem for certain sequences of matrix Laurent polynomials which was obtained previously in [1] we can establish the relation between the matrix counterpart of the so-calledT-fractions and matrix orthogonal Laurent polynomials. The connection with two-point Padé approximants to the pair (L ₀,L ) is also exhibited proving that such approximants are Hermitian too. Finally, error formulas are also given.     

Approximants de Padé simultanés de logarithmes

Simultaneous Padé-approximants of logarithms give simultaneous diophantine approximations for logarithms of rational numbers close to 1. Lower bounds for linear forms of logarithms with integer coefficients are derived.     

Matrix Padé Approximation: Recursive Computations

In this paper we consider computational aspect of the matrix Padé approximants whose definitions and properties were considered in an accompanying paper. A three-term recursive approach for the computation is established.     

The convergence of diagonal Padé approximants and the Padé Conjecture

Questions related to the convergence problem of diagonal Padé approximants are discussed. A central place is taken by the Padé Conjecture (also known as the Baker-Gammel-Wills Conjecture). Partial results concerning this conjecture are reviewed and weaker and more special versions of the conjecture are formulated and their plausibility is investigated. Great emphasis is given to the role of spurious poles of the approximants. A conjecture by Nuttall (1970) about the number and distribution of such poles is stated and its importance for the Padé Conjecture is analyzed.     

The A-Stability of Methods with Padé and Generalized Padé Stability Functions

This paper surveys some stability results and suggests the use of order arrows as an alternative to order stars in studying questions about the possible A-stability of a numerical method. A discussion of the so-called Butcher–Chipman conjecture includes a proof of a partial result.     

Existence and Uniqueness of the Solution of Nonlinear Population Evolution Equations

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Existence and Uniqueness of Weak Solutions for Quasilinear Parabolic Systems

Consider the following IBVP for quasilinear parabolic systemsWhereΩis a bounded domain in R~n with a sufficiently smooth boundary(?)Ω,u=     

Existence and Uniqueness of Solutions to Random Impulsive Differential Systems

The existence and uniqueness in mean square of solutions to certain random impulsive differentialsystems is discussed in this paper.Cauchy-Schwarz inequality,Lipschtiz condition and techniques in stochasticanalysis are employed in achieve the desired results.     

Convergence of Row Sequences of Simultaneous Padé–Faber Approximants

For meromorphic circumferentially mean p-valent functions, an analog of the classical distortion theorem is proved. It is shown that the existence of connected lemniscates of the function and a constraint on a cover of two given points lead to an inequality involving the Green energy of a discrete signedmeasure concentrated at the zeros of the given function and the absolute values of its derivatives at these zeros. This inequality is an equality for the superposition of a certain univalent function and an appropriate Zolotarev fraction.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part II: Clenshaw–Lord Type Approximants

Laurent–Padé (Chebyshev) rational approximants P _m (w,w ^–1)/Q _n (w,w ^–1) of Clenshaw–Lord type [2,1] are defined, such that the Laurent series of P _m /Q _n matches that of a given function f(w,w ^–1) up to terms of order w ^±(m+n), based only on knowledge of the Laurent series coefficients of f up to terms in w ^±(m+n). This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent series of P _m matches that of Q _n f up to terms of order w ^±(m+n), but based on knowledge of the series coefficients of f up to terms in w ^±(m+2n). The Clenshaw–Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of the Padé–Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations are obtainable for all m0, n0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé–Chebyshev approximants. Remarkably similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw–Lord type methods, thus validating the use of either.     

Padé Approximations of and preservation of quadratic Lyapunov functions

We investigate whether or not quadratic Lyapunov functions are preserved under Padé approximations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)