A new approach to vector-valued rational interpolation

In this work we propose three different procedures for vector-valued rational interpolation of a function F(z), where , and develop algorithms for constructing the resulting rational functions. We show that these procedures also cover the general case in which some or all points of interpolation coalesce. In particular, we show that, when all the points of interpolation collapse to the same point, the procedures reduce to those presented and analyzed in an earlier paper (J. Approx. Theory 77 (1994) 89) by the author, for vector-valued rational approximations from Maclaurin series of F(z). Determinant representations for the relevant interpolants are also derived.     

A de Montessus type convergence study of a least-squares vector-valued rational interpolation procedure

In a recent paper of the author [A. Sidi, A new approach to vector-valued rational interpolation, J. Approx. Theory 130 (2004) 177–187], three new interpolation procedures for vector-valued functions F(z), where F:C→C^N, were proposed, and some of their algebraic properties were studied. One of these procedures, denoted IMPE, was defined via the solution of a linear least-squares problem. In the present work, we concentrate on IMPE, and study its convergence properties when it is applied to meromorphic functions with simple poles and orthogonal vector residues. We prove de Montessus and Koenig type theorems when the points of interpolation are chosen appropriately.     

A superfast algorithm for multi-dimensional Padé systems

For a vector ofk+1 matrix power series, a superfast algorithm is given for the computation of multi-dimensional Padé systems. The algorithm provides a method for obtaining matrix Padé, matrix Hermite Padé and matrix simultaneous Padé approximants. When the matrix power series is normal or perfect, the algorithm is shown to calculate multi-dimensional matrix Padé systems of type (n ₀,...,n _k ) inO(n · log²n) block-matrix operations, where n=n ₀+...+n _k . Whenk=1 and the power series is scalar, this is the same complexity as that of other superfast algorithms for computing Padé systems. Whenk>1, the fastest methods presently compute these matrix Padé approximants with a complexity ofO(n²). The algorithm succeeds also in the non-normal and non-perfect case, but with a possibility of an increase in the cost complexity.Supported in part by NSERC grant No. A8035.Partially supported by NSERC operating grant No. 6194.     

Clifford algebras and vector-valued rational forms II

We demonstrate how the use of Clifford algebras in the theory of vector-valued rational forms leads to practical recurrence relations which do not involve representations of the algebras     

A Newton type rational interpolation formula

We give a Newton type rational interpolation formula (Theorem 2.2). It contains as a special case the original Newton interpolation, as well as the interpolation formula of Liu, which allows to recover many important classical q-series identities. We show in particular that some bibasic identities are a consequence of our formula.     

On some aspects of multivariate polynomial interpolation

The purpose of this paper is to present some aspects of multivariate Hermite polynomial interpolation. We do not focus on algebraic considerations, combinatoric and geometric aspects, but on explicitation of formulas for uniform and non-uniform bivariate interpolation and some higher dimensional problems. The concepts of similar and equivalent interpolation schemes are introduced and some differential aspects related to them are also investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

A new method for computing the matrix exponential operation based on vector valued rational approximations

In this paper a new method for computing the action of the matrix exponential on a vector e^Atb, where A is a complex matrix and t is a positive real number, is proposed. Our approach is based on vector valued rational approximation where the approximants are determined by the denominator polynomials whose coefficients are obtained by solving an inexpensive linear least-squares problem. No matrix multiplications or divisions but matrix-vector products are required in the whole process. A technique of scaling and recurrence enables our method to be more effective when the problem is for fixed A,b and many values of t. We also give a backward error analysis in exact arithmetic for the truncation errors to derive our new algorithm. Preliminary numerical results illustrate that the new algorithm performs well.     

On the divided differences of the remainder in polynomial interpolation

We present formulas for the divided differences of the remainder of the interpolation polynomial that include some recent interesting formulas as special cases.     

A rational Lanczos algorithm for model reduction

This paper presents a model reduction method for large-scale linear systems that is based on a Lanczos-type approach. A variant of the nonsymmetric Lanczos method, rational Lanczos, is shown to yield a rational interpolant (multi-point Padé approximant) for the large-scale system. An exact expression for the error in the interpolant is derived. Examples are utilized to demonstrate that the rational Lanczos method provides opportunities for significant improvements in the rate of convergence over single-point Lanczos approaches.     

Normal indices in Nikishin systems

We improve the class of indices for which normality takes place in a Nikishin system and apply this in Hermite–Padé approximation of such systems of functions.     

On the size of multivariate polynomial lemniscates and the convergence of rational approximants

In a previous paper, the author introduced a class of multivariate rational interpolants, which are called optimal Padé-type approximants (OPTA). The main goal of this paper is to extend classical results on convergence both in measure and in capacity of sequences of Padé approximants to the multivariate case using OPTA. To this end, we obtain some estimations of the size of multivariate polynomial lemniscates in terms of the Hausdorff content, which we also think are of some interest.     

Scalar and matrix Riemann–Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight

We describe methods for the derivation of strong asymptotics for the denominator polynomials and the remainder of Padé approximants for a Markov function with a complex and varying weight. Two approaches, both based on a Riemann–Hilbert problem, are presented. The first method uses a scalar Riemann–Hilbert boundary value problem on a two-sheeted Riemann surface, the second approach uses a matrix Riemann–Hilbert problem. The result for a varying weight is not with the most general conditions possible, but the loss of generality is compensated by an easier and transparent proof.     

Hermite–Padé approximation and simultaneous quadrature formulas

We study Hermite–Padé approximation of the so-called Nikishin systems of functions. In particular, the set of multi-indices for which normality is known to take place is considerably enlarged as well as the sequences of multi-indices for which convergence of the corresponding simultaneous rational approximants takes place. These results are applied to the study of the convergence properties of simultaneous quadrature rules of a given function with respect to different weights.     

二元有理插值的迭加算法

在已有基础上给出一种新的算法,即迭加插值算法,并给出相应的插值有理函数的具体表达式,以及与已有算法比较,该算法具有较大的灵活性,更便于实际应用.     

A matrix for determining lower complexity barycentric representations of rational interpolants

Among the representations of rational interpolants, the barycentric form has several advantages, for example, with respect to stability of interpolation, location of unattainable points and poles, and differentiation. But it also has some drawbacks, in particular the more costly evaluation than the canonical representation. In the present work we address this difficulty by diminishing the number of interpolation nodes embedded in the barycentric form. This leads to a structured matrix, made of two (modified) Vandermonde and one Löwner, whose kernel is the set of weights of the interpolant (if the latter exists). We accordingly modify the algorithm presented in former work for computing the barycentric weights and discuss its efficiency with several examples.     

向量值切触有理插值存在性的判别方法

在用广义Vandermonde行列式给出Hermite插值多项式的表达式的基础上,针对a<,i>=2(i=1,2,…,s)的情形给出向量值切触有理插值存在性问题有解的条件及表达式.     

Gaussian quadrature formulae on the unit circle

Let μ be a probability measure on [0,2π]. In this paper we shall be concerned with the estimation of integrals of the form

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For this purpose we will construct quadrature formulae which are exact in a certain linear subspace of Laurent polynomials. The zeros of Szegö polynomials are chosen as nodes of the corresponding quadratures. We will study this quadrature formula in terms of error expressions and convergence, as well as, its relation with certain two-point Padé approximants for the Herglotz–Riesz transform of μ. Furthermore, a comparison with the so-called Szegö quadrature formulae is presented through some illustrative numerical examples.     

On the distribution of poles of Padé approximants to the Z-transform of complex Gaussian white noise

In the application of Padé methods to signal processing a basic problem is to take into account the effect of measurement noise on the computed approximants. Qualitative deterministic noise models have been proposed which are consistent with experimental results. In this paper the Padé approximants to the Z-transform of a complex Gaussian discrete white noise process are considered. Properties of the condensed density of the Padé poles such as circular symmetry, asymptotic concentration on the unit circle and independence on the noise variance are proved. An analytic model of the condensed density of the Padé poles for all orders of the approximants is also computed. Some Monte Carlo simulations are provided.     

一元切触有理插值存在性的判别方法

在用广义Vandermonde行列式给出Hermite插值多项式的表达式的基础上,分别针对iα=2,iα=3(i=1,2,…,s)的情形给出切触有理插值问题有解的条件及解的表达式.     

A reliable study for extensions of the Bratu problem with boundary conditions

In this paper, we present a reliable study on extensions of the Bratu problem with boundary conditions. The work rests mainly on Adomian decomposition method and Padé approximants. The study shows a variety of approximations, one for each extension. The work highlights the effect of the extensions on the structure of the approximate solutions. Copyright © 2012 John Wiley & Sons, Ltd.