Some explicit banded Hessenberg matrices appearing in the theory of multiple orthogonal polynomials
Walter Van Assche
Department of Mathematics
Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001 Leuven, Belgium
e-mail: Walter.VanAssche@wis.kuleuven.ac.be

First we give a number of examples of multiple orthogonal polynomials on the real line for which one can compute in full detail the recurrence relation of the polynomials on the diagonal and stepline. This recurrence relation gives rise to a banded Hessenberg operator. The examples consist of several Angelesco systems (multiple orthogonality on disjoint intervals) and AT systems (multiple orthogonality on one interval but with weights that form a Chebyshev system). It turns out that these operators (or scaled versions of the operators) are all perturbations of banded Hessenberg matrices with a simple structure, which need to be studied in more detail. We analyse one of these and formulate some open problems.

Vector orthogonal polynomials and hamiltonian lattices
Alexandre A. Aptekarev, Valeri A. Kaliaguine
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences
e-mail: aptekaa@rfbr.ru
Nizhny Novgorod State Technical University
Laboratoire ANO, Universite de Sciences et Technologie de Lille
e-mail: kalia@waise.nntu.sci-nnov.ru, kalia@ano.univ-lille1.fr

It is well known since J. Moser (1976), M. Adler and P. Van Moerbeke (1995) how orthogonal polynomials appear in the context of non-linear dynamical systems. In this talk we will present some recent developments concerning the connection of vector orthogonal polynomials with hamiltonian lattices. In particular we will establish a fruitful link between vector orthogonality in the Stieltjes case and the hierarchy of discrete Kadomtsev-Petviashvili equations. This leads to the appropriate formulation of the Riemann-Hilbert problem for asymptotics of vector orthogonal polynomials. Some related problems will be also discussed.

Non Hermitian orthogonality and pole behavior in meromorphic approximation
Laurent Baratchart
INRIA, 2004 route des Lucioles, BP 93, Sophia-Antipolis 06902 Cedex, France
e-mail: Laurent.Baratchart@sophia.inria.fr

As a general rule, non-Hermitian orthogonality relations pop up in rational or meromorphic approximation when computing derivatives of the criterion and equating them to zero. The precise form of the orthogonality relation one gets reflects the nature of the approximation problem. A typical example is the relation:

$$\int_C\frac{q_n}{{\tilde q_n}^2}(\xi)\xi^kw_n(\xi)d\nu(\xi)=0,~~k=0,\ldots n-1,$$

expressing the first order optimality conditions in best $L^p$ approximation on the unit circle, for $2\leq p\leq\infty$, to the function

$$f(z)=\int_C\frac{d\nu(\xi)}{\xi-z}$$

where $d\nu$ is a complex measure supported on the arc $C$ contained in the unit disk and $f\in \bar{H}_0^p$, the Hardy-space of exponent $p$ of the complement of the disk, by a meromorphic function of the form $g/q_n$ where $g\in H^p$ lies in the Hardy space of the disk and $q_n$ is a polynomial of degree $n$. This type of approximation interpolates between rational $L^2$ approximation on the circle (when $p=2$) and AAK-approximation (when $p=\infty$). Here, $w_n$ is the outer factor of a $n$-th singular vector associated to the Hankel operator with symbol $f$, and $\tilde q_n$ is the reciprocal of the polynomial $q_n$. The nonlinear term $q_n/{\tilde q_n}^2$ in the orthogonality relation is characteristic of the fact that we approximate on the unit circle.

The non-Hermitian orthogonality appears to be the main tool to assess the asymptotic behaviour of the zeros of $q_n$, namely the poles of the approximant, and subsequently error rates and uniqueness issues. This asymptotic behaviour is governed by equilibrium distributions on arcs minimizing the capacity of the condenser made of the unit circle, on the one hand, and the arc in question on the other hand. We shall illustrate this in the case where $C$ is a segment and the measure $\nu$ is rather irregular, and in the case where $f$ is a function with two branchpoints in which case $\nu$ is rather smooth.

Diophantine approximations using Padé approximation
Marc Prévost
LMPA Joseph Liouville
Université du Littoral, B.P. 699, F-62228 Calais, France
e-mail: prevost@lma.univ-littoral.fr

We show how Padé approximations (PA) are used to get Diophantine approximations (DA) of real or complex numbers, and so to prove irrationality. We present, first, some types of series for which PA provide exactly DA. Then, we show how PA to the asymptotic expansion of the remainder term of a value of a series also leads to DA.

Optimal Rational Functions for the Generalized Zolotarev Problem in the Complex Plane
B. Le Bailly and J.P. Thiran
Department of Mathematics
Facultés Universitaires Notre-Dame de la Paix, B - 5000 Namur, Belgium

It has been long recognized that the determination of optimal parameters for the classical ADI method leads to the Zolotarev problem

$$\min_{r \in R_{nn}} \frac{\max \{\vert r(z)\vert, z \in E \}}{\min \{ \vert r(z)\vert, z \in F\}} \,\,\,,$$

for disjoint compact sets $E, F \subseteq {\hbox{\it I\hskip -5pt C}}$, where $R_{nn}$ is the collection of rational functions of order $n$. In case where $E$ and $F$ are real intervals, it was more recently pointed out that, if they have different lengths, it is of interest to generalize the foregoing problem to the set $R_{mn}$ with unequal numerator degree $m$ and denominator degree $n$. The object of the talk is to present some results on the generalized Zolotarev problem in the complex plane. A method is proposed to construct the optimal rational function, which is then applied to the particular example of two line segments, $E$ on the real axis, $F$ parallel to the imaginary axis. Numerical experiments show that the improvement on the classical solution may be amazingly great. Explicit expressions are also provided for special values of data.

Recursive Computation of Padé-Legendre Approximants and Some Acceleration Properties
Ana C. Matos
Laboratoire d'Analyse Numérique et d'Optimisation
UFR IEEA - M3, Université des Sciences et Technologies de Lille, 59655 Villeneuve d'Ascq Cedex, France
e-mail: matos@ano.univ-lille1.fr

In this talk, we will begin by recalling the two definitions of the generalizations of the Padé approximants to orthogonal series and by defining the Padé-Legendre approximants of a Legendre series.

We will propose two algorithms for the recursive computation of some sequences of these approximants - a Frobenius-type algorithm and a Kronecker-type algorithm.

We will obtain some estimations of the speed of convergence of the columns of the Padé-Legendre table from the asymptotic behaviour of the coefficients of the Legendre series and deduce some acceleration results.

Finally we will illustrate these results with some numerical examples.

Sobolev orthogonal polynomials and Markov-Bernstein inequalities
André Draux
Génie Mathematique, INSA de Rouen
BP-08, Place Emile Blondel, 76131 Mont Saint Aignan, Cedex, France
e-mail: draux@lmi.insa-rouen.fr

We present an overview on the research of the three past years in the study of some cases of orthogonality in Sobolev spaces in connection with Markov-Bernstein type inequalities. As a consequence we give lower and upper bounds for the Markov-Bernstein constant for weighted $L^2$ norms with a classical weight.

The convergence of the Lanczos method and a minimal energy problem with constraint
Arno Kuijlaars
Department of Mathematics
Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001 Leuven, Belgium
e-mail: Arno.KUIJLAARS@wis.kuleuven.ac.be

The Lanczos method for the symmetric eigenvalue problem is an efficient algorithm to reduce a symmetric matrix to tridiagonal form. The entries of the tridiagonal matrix generate through a three-term recurrence a finite sequence of polynomials, which are orthogonal on the spectrum of the matrix. The zeros of these polynomials are called Ritz values, and it turns out that some of the Ritz values are good approximations to some of the eigenvalues.

In this talk, the convergence of Ritz values is considered in an asymptotic setting. We assume that we have a sequence of matrices of increasing dimensions N, such that their eigenvalues have an asymptotic distribution in the weak* sense. Furthermore, the degree n of the polynomial, or equivalently the number of iterations, is assumed to increase with N in such a way that the ratio n/N tends to a non-zero limit. In this situation we will describe the region where the eigenvalues are well-approximated by the Ritz values. This region is determined by a minimal energy problem in potential theory with an upper constraint. The constraint comes from the asymptotic distribution of eigenvalues and the limit of the ratios n/N determines the normalization.

The predicted region is quite accurate when compared to the results of actual numerical calculations, provided that the eigenvalues follow the asymptotic distribution closely.

Superlinear convergence of Conjugate Gradients
Bernhard Beckermann
Laboratoire d'Analyse Numérique et d'Optimisation
UFR IEEA - M3, Université des Sciences et Technologies de Lille, 59655 Villeneuve d'Ascq Cedex, France
e-mail: bbecker@ano.univ-lille1.fr

We give a theoretical explanation for superlinear convergence behavior observed while solving large symmetric systems of equations using the Conjugate Gradient method, or other Krylov subspace methods. We present a new bound on the relative error after $n$ iterations. This bound is valid in an asymptotic sense, when the size $N$ of the system grows together with the number of iterations. The bound depends on the asymptotic eigenvalue distribution and on the ratio $n/N$. An important tool in our investigations is a constrained energy problem in logarithmic potential theory.

The new asymptotic bounds for the rate of convergence are illustrated by discussing Toeplitz systems, as well as a model problem stemming from the discretization of the 2D-Poisson equation.