Computation of Gauss-Kronrod quadrature rules

Recently Laurie presented a new algorithm for the computation of -point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order from certain mixed moments, and then computes a partial spectral factorization. We describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined, and thereby avoids computations that can be sensitive to perturbations. Our algorithm uses the consolidation phase of a divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. We also discuss how the algorithm can be applied to compute Kronrod extensions of Gauss-Radau and Gauss-Lobatto quadrature rules. Throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible. Numerical examples illustrate the performance of the algorithm.

     

Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature

Geodetic and meteorological data, collected via satellites for example, are genuinely scattered and not confined to any special set of points. Even so, known quadrature formulas used in numerically computing integrals involving such data have had restrictions either on the sites (points) used or, more significantly, on the number of sites required. Here, for the unit sphere embedded in , we obtain quadrature formulas that are exact for spherical harmonics of a fixed order, have nonnegative weights, and are based on function values at scattered sites. To be exact, these formulas require only a number of sites comparable to the dimension of the space. As a part of the proof, we derive -Marcinkiewicz-Zygmund inequalities for such sites.

     

On statistical limit points

The set of all statistical limit points of a given sequence is characterized as an -set. It is also characterized in terms of discontinuity points of distribution functions of .

     

Sieving for rational points on hyperelliptic curves

We give a new and efficient method of sieving for rational points on hyperelliptic curves. This method is often successful in proving that a given hyperelliptic curve, suspected to have no rational points, does in fact have no rational points; we have often found this to be the case even when our curve has points over all localizations . We illustrate the practicality of the method with some examples of hyperelliptic curves of genus .

     

Periodic points and normal families

Let be the family of all functions which are holomorphic in some domain and do not have periodic points of some period greater than one there. It is shown that is quasinormal, and the sequences in which do not have convergent subsequences are characterized. The method also yields a new proof of the result that transcendental entire functions have infinitely many periodic points of all periods greater than one.

     

Linear systems of plane curves with a composite number of base points of equal multiplicity

In this article we study linear systems of plane curves of degree passing through general base points with the same multiplicity at each of them. These systems are known as homogeneous linear systems. We especially investigate for which of these systems, the base points, with their multiplicities, impose independent conditions and which homogeneous systems are empty. Such systems are called non-special. We extend the range of homogeneous linear systems that are known to be non-special. A theorem of Evain states that the systems of curves of degree with base points with equal multiplicity are non-special. The analogous result for points was conjectured. Both of these will follow, as corollaries, from the main theorem proved in this paper. Also, the case of points will follow from our result. The proof uses a degeneration technique developed by C. Ciliberto and R. Miranda.

     

Strong tractability of integration using scrambled Niederreiter points

We study the randomized worst-case error and the randomized error of scrambled quasi-Monte Carlo (QMC) quadrature as proposed by Owen. The function spaces considered in this article are the weighted Hilbert spaces generated by Haar-like wavelets and the weighted Sobolev-Hilbert spaces. Conditions are found under which multivariate integration is strongly tractable in the randomized worst-case setting and the randomized setting, respectively. The -exponents of strong tractability are found for the scrambled Niederreiter nets and sequences. The sufficient conditions for strong tractability for Sobolev spaces are more lenient for scrambled QMC quadratures than those for deterministic QMC net quadratures.

     

Isolated spectral points

The paper studies isolated spectral points of elements of Banach algebras and of bounded linear operators in terms of the existence of idempotents, and gives an elementary characterization of spectral idempotents. It is shown that is isolated in the spectrum of a bounded linear operator if the (not necessarily closed) space is nonzero and complemented by a closed subspace satisfying .

     

A computational approach to Hilbert modular group fixed points

Some useful information is known about the fundamental domain for certain Hilbert modular groups. The six nonequivalent points with nontrivial isotropy in the fundamental domains under the action of the modular group for , , and have been determined previously by Gundlach. In finding these points, use was made of the exact size of the isotropy groups. Here we show that the fixed points and the isotropy groups can be found without such knowledge by use of a computer scan. We consider the cases and . A computer algebra system and a C compiler were essential in perfoming the computations.

     

Group automorphisms with few and with many periodic points

For each a compact group automorphism is constructed with the property that

This may be interpreted as a combinatorial analogue of the (still open) problem of whether compact group automorphisms with any given topological entropy exist.

     

Anti-Szego quadrature rules

Szego quadrature rules are discretization methods for approximating integrals of the form . This paper presents a new class of discretization methods, which we refer to as anti-Szego quadrature rules. Anti-Szego rules can be used to estimate the error in Szego quadrature rules: under suitable conditions, pairs of associated Szego and anti-Szego quadrature rules provide upper and lower bounds for the value of the given integral. The construction of anti-Szego quadrature rules is almost identical to that of Szego quadrature rules in that pairs of associated Szego and anti-Szego rules differ only in the choice of a parameter of unit modulus. Several examples of Szego and anti-Szego quadrature rule pairs are presented.

     

Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations

This paper concerns a class of deferred correction methods recently developed for initial value ordinary differential equations; such methods are based on a Picard integral form of the correction equation. These methods divide a given timestep [t_n,t_n+1] into substeps, and use function values computed at these substeps to approximate the Picard integral by means of a numerical quadrature. The main purpose of this paper is to present a detailed analysis of the implications of the location of quadrature nodes on the accuracy and stability of the overall method. Comparisons between Gauss-Legendre, Gauss-Lobatto, Gauss-Radau, and uniformly spaced points are presented. Also, for a given set of quadrature nodes, quadrature rules may be formulated that include or exclude function values computed at the left-hand endpoint t_n. Quadrature rules that do not depend on the left-hand endpoint (which are referred to as right-hand quadrature rules) are shown to lead to L(α)-stable implicit methods with α≈π/2. The semi-implicit analog of this property is also discussed. Numerical results suggest that the use of uniform quadrature nodes, as opposed to nodes based on Gaussian quadratures, does not significantly affect the stability or accuracy of these methods for orders less than ten. In contrast, a study of the reduction of order for stiff equations shows that when uniform quadrature nodes are used in conjunction with a right-hand quadrature rule, the form and extent of order-reduction changes considerably. Specifically, a reduction of order to is observed for uniform nodes as opposed to for non-uniform nodes, where Δt denotes the time step and ε a stiffness parameter such that ε→0 corresponds to the problem becoming increasingly stiff. AMS subject classification (2000) 65B05     

Extreme points, exposed points, differentiability points in CL-spaces

This paper presents a property of geometric and topological nature of Gateaux differentiability points and Fréchet differentiability points of almost CL-spaces. More precisely, if we denote by a maximal convex set of the unit sphere of a CL-space , and by the cone generated by , then all Gateaux differentiability points of are just n-s, and all Fréchet differentiability points of are (where n-s denotes the non-support points set of ).

     

Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points

Let be a positive measure whose support is an interval plus a denumerable set of mass points which accumulate at the boundary points of only. Under the assumptions that the mass points satisfy Blaschke's condition and that the absolutely continuous part of satisfies Szegö's condition, asymptotics for the orthonormal polynomials on and off the support are given. So far asymptotics were only available if the set of mass points is finite.

     

Universally optimal distribution of points on spheres

We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are distances between distinct points in it and it is a spherical -design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the and Leech lattices. We also prove the same result for the vertices of the -cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact two-point homogeneous spaces, and we conclude with an extension to Euclidean space.

     

On the set of points with a dense orbit

Under certain conditions on the topological space we prove that for every continuous map the set of all points with a dense orbit has empty interior in . This result implies a negative answer to two problems proposed by M. Barge and J. Kennedy.

     

On generalized averaged Gaussian formulas

We present a simple numerical method for constructing the optimal (generalized) averaged Gaussian quadrature formulas which are the optimal stratified extensions of Gauss quadrature formulas. These extensions exist in many cases in which real positive Kronrod formulas do not exist. For the Jacobi weight functions ( ) we give a necessary and sufficient condition on the parameters and such that the optimal averaged Gaussian quadrature formulas are internal.

     

On singular critical points of positive operators in Krein spaces

We give an example of a positive operator in a Krein space with the following properties: the nonzero spectrum of consists of isolated simple eigenvalues, the norms of the orthogonal spectral projections in the Krein space onto the eigenspaces of are uniformly bounded and the point is a singular critical point of

     

Affine curves with infinitely many integral points

Let be an irreducible affine curve of (geometric) genus 0 defined by a finite family of polynomials having integer coefficients. In this note we give a necessary and sufficient condition for to possess infinitely many integer points, correcting a statement of J. H. Silverman (Theoret. Comput. Sci., 2000).

     

Harmonic calculus on fractals--A measure geometric approach II

Riesz potentials of fractal measures in metric spaces and their inverses are introduced. They define self-adjoint operators in the Hilbert space and the former are shown to be compact.

In the Euclidean case the corresponding spectral asymptotics are derived with Besov space methods. The inverses of the Riesz potentials are fractal pseudodifferential operators. For the order two operator the spectral dimension coincides with the Hausdorff dimension of the underlying fractal.