Annie Cuyt

Nonlinear methods in numerical analysis

I. Continued Fractions. Since these play an important role, the first chapter introduces their basic properties, evaluation algorithms and convergence theorems. From the section dealing with convergence it can be seen that in certain situations nonlinear approximations are more powerful than linear approximations. The recent notion of branched continued fraction is introduced in the multivariate section and is later used for the construction of multivariate rational interpolants. II. Pade Approximants. A survey of the theory of these local rational approximants for a given function is presented, including the problems of existence, unicity and computation. Also considered are the convergence of sequences of Pade approximants and the continuity of the Pade operator which associates with a function its Pade approximant of a certain order. A special section is devoted to the multivariate case. III. Rational Interpolants. These rational functions fit a given function at some given points. Many results of the previous chapter remain valid for this more general case where the interpolation conditions are spread over several points. In between the rational interpolation case and Pade approximation case lies the theory of rational Hermite interpolation where each interpolation point can be assigned more than one interpolation condition. Some results on the convergence of sequences of rational Hermite interpolants are mentioned and multivariate rational interpolants are introduced in two different ways. IV. Applications. The previous types of rational approximants are used here to develop several numerical methods for the solution of classical problems such as convergence acceleration, nonlinear equations, ordinary differential equations, numerical quadrature, partial differential equations and integral equations. Many numerical examples illustrate the different techniques, and it is seen that nonlinear methods are very useful in situations involving singularities. Subject Index.

How well can the concept of Padé approximant be generalized to the multivariate case

What we know about multivariate Pade approximation has been developed in the last 25 years. In the next sections we compare and discuss many of these results. It will become clear that simple properties or requirements, such as the uniqueness of the Pade approximant and consequently its consistency property, can play a crucial role in the development of the multivariate theory. A separate section is devoted to a discussion of the convergence properties. At the end we include an extensive reference list on the topic.

Padé Approximants for Operators: Theory and Applications

Abstract pade approximants in operator theory.- Multivariate pade approximants.- The solution of nonlinear operator equations.

Multivariate rational interpolation

Many papers have already been published on the subject of multivariate polynomial interpolation and also on the subject of multivariate Padé approximation. But the problem of multivariate rational interpolation has only very recently been considered; we refer among others to [8] and [3].The computation of a univariate rational interpolant can be done in various equivalent ways: one can calculate the explicit solution of the system of interpolatory conditions, or start a recursive algorithm, or calculate the convergent of a continued fraction.In this paper we will generalize each of those methods from the univariate to the multivariate case. Although the generalization is simple, the equivalence of the computational methods is completely lost in the multivariate case. This was to be expected since various authors have already remarked [2,7] that there is no link between multivariate Padé approximants calculated by matching the Taylor series and those obtained as convergents of a continued fraction.ZusammenfassungDas multivariate polynomiale Interpolationsproblem sowie die multivariate Padé-Approximation sind schon einige Jahre alt, aber das multivariate rationale Interpolationsproblem ist noch verhältnismäßig jung [3,8].Für univariate Funktionen gibt es verschiedene äquivalente Algorithmen zur Berechnung vom rationalen Interpolant: die Lösung eines Gleichungssystems, die rekursive Berechnung oder die Berechnung eines Kettenbruchs.Diese Algorithmen werden hier verallgemeinert auf multivariate Funktionen. Wir bemerken, daß sie nun nicht mehr equivalent sind. Diese Beobachtung ist auch schon von anderen Mathematikern gemacht worden für das multivariate Padé-Approximationsproblem [2,7], das man auch auf verschiedene Weisen lösen kann.

General order Newton-Padé approximants for multivariate functions

SummaryPadé approximants are a frequently used tool for the solution of mathematical problems. One of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off(x1, ...,xp). Therefore multivariate Newton-Padé approximants are introduced; their computation will only use the value off at some points. In Sect. 1 we shall repeat the univariate Newton-Padé approximation problem which is a rational Hermite interpolation problem. In Sect. 2 we sketch some problems that can arise when dealing with multivariate interpolation. In Sect. 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate Newton-Padé approximants in Sect. 4. A numerical example is given in Sect. 5, together with the proof that forp=1 the classical Newton-Padé approximants for a univariate function are obtained.

Multivariate Padé-approximants

Abstract For an operator F: R n → R , analytic in the origin, the notion of (abstract multivariate Pade-approximant (APA) is introduced, by making use of abstract polynomials. The classical Pade-approximant (n = 1) is a special case of the multivariate theory and many interesting properties of classical Pade-approximants remain valid such as covariance properties and the block-structure [Annie A. M. Cuyt, J. Oper. Theory6 (2) (1981), 207–209] of the Pade-table. Also a projection-property for multivariate Pade-approximants is proved.

Multivariate Pade´ approximants revisited

Several definitions of multivariate Padé approximants have been introduced during the last decade. We will here consider all types of definitions based on the choice that the coefficients in numerator and denominator of the multivariate Padé approximant are defined by means of a linear system of equations. In this case a determinant representation for the multivariate Padé approximant exists. We will show that a general recursive algorithm can be formulated to compute a multivariate Padé approximant given by any definition of this type. Here intermediate results in the recursive computation scheme will also be multivariate Padé approximants. Up to now such a recursive computation of multivariate Padé approximants only seemed possible in some special cases.

A review of branched continued fraction theory for the construction of multivariate rational approximations

While the history of continued fractions goes back to Euclid’s algorithm, branched continued fractions are only twenty years old. The idea to construct them was born in Lvov (U.S.S.R.) in the early sixties. The first and most general form of these fractions was introduced by Skorobogatko in [14] together with Droniuk, Bobyk and Ptashnik. An ordinary continued frution (CF) is an expression of the form &I + a1 Q2 , b, + b2+ b 3 +“’ . . .

A recursive computation scheme for multivariate rational interpolants

We derive here a recursive computation scheme for the rational interpolation method introduced in [7]. Explicit formulas for these multivariate rational interpolants are repeated in § 2 while the recursive algorithm is described in § 3. A number of interesting special cases such as the univariate rational interpolation problem and the multivariate Pade approximants introduced in [6] and [10] are dealt with in § 4. For some of these rational approximants other recursive schemes were described previously. Finally § 5 contains the numerical results: the multivariate rational interpolants described here are compared with multivariate polynomial interpolants, interpolating branched continued fractions introduced by Cuyt and Verdonk [8], interpolating branched continued fractions introduced by Siemaszko [12] and several multivariate Pade approximants [6], [3], [10]. Besides the fact that our multivariate rational interpolants allow a large degree of freedom in the choice for the numerator and denominator in ord...

Rational BRDF

Over the last two decades, much effort has been devoted to accurately measuring Bidirectional Reflectance Distribution Functions (BRDFs) of real-world materials and to use efficiently the resulting data for rendering. Because of their large size, it is difficult to use directly measured BRDFs for real-time applications, and fitting the most sophisticated analytical BRDF models is still a complex task. In this paper, we introduce Rational BRDF, a general-purpose and efficient representation for arbitrary BRDFs, based on Rational Functions (RFs). Using an adapted parametrization, we demonstrate how Rational BRDFs offer 1) a more compact and efficient representation using low-degree RFs, 2) an accurate fitting of measured materials with guaranteed control of the residual error, and 3) efficient importance sampling by applying the same fitting process to determine the inverse of the Cumulative Distribution Function (CDF) generated from the BRDF for use in Monte-Carlo rendering.