A nodal spline interpolant for the Gregory rule of even order

Summary The Gregory rule is a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order. In the literature, the methods of constructing the Gregory rule have, in contrast to Newton-Cotes quadrature,not been based on the integration of an interpolant. In this paper, after first characterizing an even-order Gregory interpolant by means of a generalized Lagrange interpolation operator, we proceed to explicitly construct such an interpolant by employing results from nodal spline interpolation, as established in recent work by the author and C.H. Rohwer. Nonoptimal order error estimates for the Gregory rule of even order are then easily obtained.     

Interpolation lattices in several variables

Principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in n dimensions. A principal lattice can be described as the set of intersection points of n + 1 pencils of parallel hyperplanes. Using a projective point of view, Lee and Phillips extended this situation to n + 1 linear pencils of hyperplanes. In two recent papers, two of us have introduced generalized principal lattices in the plane using cubic pencils. In this paper we analyze the problem in n dimensions, considering polynomial, exponential and trigonometric pencils, which can be combined in different ways to obtain generalized principal lattices.We also consider the case of coincident pencils. An error formula for generalized principal lattices is discussed. Partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo.     

The numerical evaluation of cauchy principal value integrals with non-standard weight functions

The authors develop an algorithm for the numerical evaluation of the finite Hilbert transform, with respect to non-standard weight functions, by a product quadrature rule. In particular, this algorithm allows us to deal with the weight functions with algebraic and/or logarithmic singularities in the interval [−1, 1], by using the Chebyshev points as quadrature nodes. The practical application of the rule is shown to be straightforward and to yield satisfactory numerical results. Convergence theorems are also given, when the nodes are the zeros of certain classical Jacobi polynomials and the weight is defined as a generalized Ditzian-Totik weight. This work was supported by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica (first author) and by the Italian Research Council (second author).     

A recurrence relation for generalized divided differences with respect to ECT-systems

It is well known that ordinary divided differences can be computed recursively. This holds true also for generalized divided differences with respect to complete Chebyshev-systems. In this note for extended complete Chebyshev-systems and possibly repeated nodes for the recurrence relation a simple proof is given which also covers the case of complex valued functions. As an application, interpolation by linear combinations of certain complex exponential functions is considered. Moreover, it is shown that generalized divided differences are also continuous functions of their nodes. This revised version was published online in August 2006 with corrections to the Cover Date.     

Rational interpolation through the optimal attachment of poles to the interpolating polynomial

After recalling some pitfalls of polynomial interpolation (in particular, slopes limited by Markov's inequality) and rational interpolation (e.g., unattainable points, poles in the interpolation interval, erratic behavior of the error for small numbers of nodes), we suggest an alternative for the case when the function to be interpolated is known everywhere, not just at the nodes. The method consists in replacing the interpolating polynomial with a rational interpolant whose poles are all prescribed, written in its barycentric form as in [4], and optimizing the placement of the poles in such a way as to minimize a chosen norm of the error. This revised version was published online in June 2006 with corrections to the Cover Date.     

Minimally supported error representations and approximation by the constants

Summary. Distribution theory is used to construct minimally supported Peano kernel type representations for linear functionals such as the error in multivariate Hermite interpolation. The simplest case is that of representing the error in approximation to f by the constant polynomial f(a) in terms of integrals of the first order derivatives of f. This is discussed in detail. Here it is shown that suprisingly there exist many representations which are not minimally supported, and involve the integration of first order derivatives over multidimensional regions. The distance of smooth functions from the constants in the uniform norm is estimated using our representations for the error. Received June 30, 1997 / Revised version received April 6, 1999 / Published online February 17, 2000     

An error term and uniqueness for Hermite–Birkhoff interpolation involving only function values and/or first derivative values

This paper discusses various aspects of Hermite–Birkhoff interpolation that involve prescribed values of a function and/or its first derivative. An algorithm is given that finds the unique polynomial satisfying the given conditions if it exists. A mean value type error term is developed which illustrates the ill-conditioning present when trying to find a solution to a problem that is close to a problem that does not have a unique solution. The interpolants we consider and the associated error term may be useful in the development of continuous approximations for ordinary differential equations that allow asymptotically correct defect control. Expressions in the algorithm are also useful in determining whether certain specific types of problems have unique solutions. This is useful, for example, in strategies involving approximations to solutions of boundary value problems by collocation.     

Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals

Summary. We show that, if (), the error term of every modified positive interpolatory quadrature rule for Cauchy principal value integrals of the type , , fulfills uniformly for all , and hence it is of optimal order of magnitude in the classes (). Here, is a weight function with the property . We give explicit upper bounds for the Peano-type error constants of such rules. This improves and completes earlier results by Criscuolo and Mastroianni (Calcolo 22 (1985), 391–441 and Numer. Math. 54 (1989), 445–461) and Ioakimidis (Math. Comp. 44 (1985), 191–198). For the special case of the Gaussian rule, we show that the restriction can be dropped. The results are based on a new representation of the Peano kernels of these formulae via the Peano kernels of the underlying classical quadrature formulae. This representation may also be useful in connection with some different problems. Received November 21, 1994     

Multipoint Taylor formulæ

Summary. The main result of this paper is an abstract version of the Kowalewski–Ciarlet–Wagschal multipoint Taylor formula for representing the pointwise error in multivariate Lagrange interpolation. Several applications of this result are given in the paper. The most important of these is the construction of a multipoint Taylor error formula for a general finite element, together with the corresponding –error bounds. Another application is the construction of a family of error formul? for linear interpolation (indexed by real measures of unit mass) which includes some recently obtained formul?. It is also shown how the problem of constructing an error formula for Lagrange interpolation from a D–invariant space of polynomials with the property that it involves only derivatives which annihilate the interpolating space can be reduced to the problem of finding such a formula for a ‘simpler’ one–point interpolation map. Received March 29, 1996 / Revised version received November 22, 1996     

A multivariate QD-like algorithm

The problem of constructing a univariate rational interpolant or Padé approximant for given data can be solved in various equivalent ways: one can compute the explicit solution of the system of interpolation or approximation conditions, or one can start a recursive algorithm, or one can obtain the rational function as the convergent of an interpolating or corresponding continued fraction.In case of multivariate functions general order systems of interpolation conditions for a multivariate rational interpolant and general order systems of approximation conditions for a multivariate Padé approximant were respectively solved in [6] and [9]. Equivalent recursive computation schemes were given in [3] for the rational interpolation case and in [5] for the Padé approximation case. At that moment we stated that the next step was to write the general order rational interpolants and Padé approximants as the convergent of a multivariate continued fraction so that the univariate equivalence of the three main defining techniques was also established for the multivariate case: algebraic relations, recurrence relations, continued fractions. In this paper a multivariate qd-like algorithm is developed that serves this purpose.     

Convergence and computation of simultaneous rational quadrature formulas

We discuss the convergence and numerical evaluation of simultaneous quadrature formulas which are exact for rational functions. The problem consists in integrating a single function with respect to different measures using a common set of quadrature nodes. Given a multi-index n, the nodes of the integration rule are the zeros of the multi-orthogonal Hermite–Padé polynomial with respect to (S, α, n), where S is a collection of measures, and α is a polynomial which modifies the measures in S. The theory is based on the connection between Gauss-type simultaneous quadrature formulas of rational type and multipoint Hermite–Padé approximation. The numerical treatment relies on the technique of modifying the integrand by means of a change of variable when it has real poles close to the integration interval. The output of some tests show the power of this approach in comparison with other ones in use.     

Polynomial interpolation of minimal degree

Summary. Minimal degree interpolation spaces with respect to a finite set of points are subspaces of multivariate polynomials of least possible degree for which Lagrange interpolation with respect to the given points is uniquely solvable and degree reducing. This is a generalization of the concept of least interpolation introduced by de Boor and Ron. This paper investigates the behavior of Lagrange interpolation with respect to these spaces, giving a Newton interpolation method and a remainder formula for the error of interpolation. Moreover, a special minimal degree interpolation space will be introduced which is particularly beneficial from the numerical point of view. Received June 9, 1995 / Revised version received June 26, 1996     

High dimensional integration of smooth functions over cubes

Summary. We construct a new algorithm for the numerical integration of functions that are defined on a -dimensional cube. It is based on the Clenshaw-Curtis rule for and on Smolyak's construction. This way we make the best use of the smoothness properties of any (nonperiodic) function. We prove error bounds showing that our algorithm is almost optimal (up to logarithmic factors) for different classes of functions with bounded mixed derivative. Numerical results show that the new method is very competitive, in particular for smooth integrands and . Received April 3, 1995 / Revised version received November 27, 1995     

Error saturation in Gaussian radial basis functions on a finite interval

Radial basis function (RBF) interpolation is a “meshless” strategy with great promise for adaptive approximation. One restriction is “error saturation” which occurs for many types of RBFs including Gaussian RBFs of the form ?(x;α,h)=exp(−α²(x/h)²): in the limit h→0 for fixed α, the error does not converge to zero, but rather to E_S(α). Previous studies have theoretically determined the saturation error for Gaussian RBF on an infinite, uniform interval and for the same with a single point omitted. (The gap enormously increases E_S(α).) We show experimentally that the saturation error on the unit interval, x∈[−1,1], is about 0.06exp(−0.47/α²)‖f‖_∞ — huge compared to the O(2π/α²)exp(−π²/[4α²]) saturation error for a grid with one point omitted. We show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points. The saturation error can be avoided by choosing α?1, the “flat limit”, but the condition number of the interpolation matrix explodes as O(exp(π²/[4α²])). The best strategy is to choose the largest α which yields an acceptably small saturation error: If the user chooses an error tolerance δ, then .     

Adaptive integration for multi-factor portfolio credit loss models

We propose algorithms of adaptive integration for calculation of the tail probability in multi-factor credit portfolio loss models. We first modify the classical Genz-Malik rule, a deterministic multiple integration rule suitable for portfolio credit models with number of factors less than 8. Later on we arrive at the adaptive Monte Carlo integration, which essentially replaces the deterministic integration rule by antithetic random numbers. The latter can not only handle higher-dimensional models but is also able to provide reliable probabilistic error bounds. Both algorithms are asymptotic convergent and consistently outperform the plain Monte Carlo method.     

Lagrange interpolation for continuous piecewise smooth functions

This note is devoted to Lagrange interpolation for continuous piecewise smooth functions. A new family of interpolatory functions with explicit approximation error bounds is obtained. We apply the theory to the classical Lagrange interpolation.     

On orders of approximation of plane curves by parametric cubic splines

The aim of the present paper is to show that the convergence rate of the parametric cubic spline approximation of a plane curve is of order four instead of order three. For the first and second derivatives, the rates are of order three and two, respectively. Finally some numerical examples are given to illustrate the predicted error behaviour.     

Exponentials Reproducing Subdivision Schemes

This paper is concerned with a family of nonstationary, interpolatory subdivision schemes that have the capability of reproducing functions in a finite-dimensional subspace of exponential polynomials. We give conditions for the existence and uniqueness of such schemes, and analyze their convergence and smoothness. It is shown that the refinement rules of an even-order exponentials reproducing scheme converge to the Dubuc—Deslauriers interpolatory scheme of the same order, and that both schemes have the same smoothness. Unlike the stationary case, the application of a nonstationary scheme requires the computation of a different rule for each refinement level. We show that the rules of an exponentials reproducing scheme can be efficiently derived by means of an auxiliary orthogonal scheme , using only linear operations. The orthogonal schemes are also very useful tools in fitting an appropriate space of exponential polynomials to a given data sequence.