On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals

Summary In a previous paper the authors proposed a modified Gaussian rule _* ^m (wf;t)to compute the integral (wf;t) in the Cauchy principal value sense associated with the weightw, and they proved the convergence in closed sets contained in the integration interval. The main purpose of the present work is to prove uniform convergence of the sequence { _* ^m (wf;t)} on the whole integration interval and to give estimates for the remainder term. The same results are shown for particular subsequences of the Gaussian rules _m (wf;t) for the evaluation of Cauchy principal value integrals. A result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made.     

Numerical integration by means of adapted Euler summation formulas

Summary The purpose of the paper is the study of formulas and methods for numerical integration based on Euler summation formulas. Cubature formulas are developed from multidimensional generalizations. Estimates of the truncation error are given in adaptation to specific properties of the integrand.     

A Monte Carlo method for high dimensional integration

Summary A new method for the numerical integration of very high dimensional functions is introduced and implemented based on the Metropolis' Monte Carlo algorithm. The logarithm of the high dimensional integral is reduced to a 1-dimensional integration of a certain statistical function with respect to a scale parameter over the range of the unit interval. The improvement in accuracy is found to be substantial comparing to the conventional crude Monte Carlo integration. Several numerical demonstrations are made, and variability of the estimates are shown.     

Monotonicity of quadrature formulae of Gauss type and comparison theorems for monosplines

Summary This paper deals with quadrature formulae of Gauss type corresponding to subspaces of spline functionsS _{m–1, k} of degreem–1 withk fixed knots. We shall show monotonicity of the quadrature formulae for functions which are contained in the so-called convexity cone ofS _m–1,k Moreover, we apply these results to monosplines and establish comparison theorems for these splines.     

Eine Fehlerabschätzung für positive Quadraturformeln

Summary Some well-known error estimates for the quadrature formulas of Lobatto and Radau are shown to be applicable to all positive quadrature formulae of the same degree.Herrn Prof. Dr. L. Collatz zum 75. Geburtstag gewidmet     

Product integration of piecewise continuous integrands based on cubic splineinterpolation at equally spaced nodes

Summary In this paper we consider the approximate evaluation of , whereK(x) is a fixed Lebesgue integrable function, by product formulas of the form based on cubic spline interpolation of the functionf.Generally, whenever it is possible, product quadratures incorporate the bad behaviour of the integrand in the kernelK. Here, however, we allowf to have a finite number of jump discontinuities in [a, b]. Convergence results are established and some numerical applications are given for a logarithmic singularity structure in the kernel.Work sponsored by the Ministero della Pubblica Istruzione of Italy     

Product integration of singular integrands based on cubic spline interpolation at equally spaced nodes

Summary In this paper the convergence of product integration rules, based on cubic spline interpolation at equally spaced nodes, with not-a-knot end condition, is investigated for integrand functions with a interior or endpoint singularity in the integration interval.Work sponsored by the Ministero della Pubblica Istruzione of Italy     

Tables of good lattices in four and five dimensions

Summary The good lattice theory yields a powerful method of computing approximations for the integral of functions defined on [0,1]^s through averaged sums ofm evaluations. We present a continuation of the only existing table of best lattices fors=4 up tom=3298, and the first table fors=5 up tom=772.From the Ecole Polytechnique de Montréal, département de Mathématiques appliquées, C.P. 6079 Station A, Montréal QC, Canada H3C 3A7. This research was supported by Grant A3087 from the National Science and Engineering Research Council of Canada     

Is Gauss quadrature optimal for analytic functions?

Summary We consider the problem of optimal quadratures for integrandsf: [–1,1] which have an analytic extension to an open diskD _r of radiusr about the origin such that 1 on . Ifr=1, we show that the penalty for sampling the integrand at zeros of the Legendre polynomial of degreen rather than at optimal points, tends to infinity withn. In particular there is an infinite penalty for using Gauss quadrature. On the other hand, ifr>1, Gauss quadrature is almost optimal. These results hold for both the worst-case and asymptotic settings.This research was supported in part by the National Science Foundation under Grants MCS-8203271 and MCS-8303111This research was supported in part by the National Science Foundation under Grant MCS-8923676     

Fehlerabschätzungen für nichtsymmetrische Gauß-Quadraturformeln

Summary We consider Gaussian quadrature formulaeQ _n , n, approximating the integral , wherew is a weight function. In certain spaces of analytic functions the error functionalR _n :=I–Q _n is continuous. Previously one of the authors deduced estimates for R _n for symmetric Gaussian quadrature formulae. In this paper we extend these results to nonsymmetric Gaussian formulae using a recent result of Gautschi concerning the sign ofR _n (q _K ),q _K (x):=x ^K , for a wide class of weight functions including the Jacobi weights.

     

Suboptimal Kronrod extension formulae for numerical quadrature

Summary We consider cases where the Stieltjes polynomial associated with a Gaussian quadrature formula has complex zeros. In such cases a Kronrod extension of the Gaussian rule does not exist. A method is described for modifying the Stieltjes polynomial so that the resulting polynomial has no complex zeros. The modification is performed in such a way that the Kronrod-type extension rule resulting from the addition of then+1 zeros of the modified Stieltjes polynomial to the original knots of the Gaussian rule has only slightly lower degree of precision than normally achieved when the Kronrod extension rule exists. As examples of the use of the method, we present some new formulae extending the classical Gauss-Hermite quadrature rules. We comment on the limited success of the method in extending Gauss-Laguerre rules and show that several modified extensions of the Gauss Gegenbauer formulae exist in cases where the standard Kronrod extension does not.     

Stable convergence acceleration using Laplace transforms

Summary We consider the general class of power series where the terms may be expressed as the Laplace transforms of known functions. The sum of the series can then be evaluated efficiently and accurately by means of quadrature schemes, recently published by Frank Stenger. The method works also far outside the region of convergence as will be illustrated by numerical examples.Dedicated to Professor Lothar Collatz on the occasion of his 75th birthday     

A contour integral representation of linear extrapolation methods

Summary Extrapolation methods are well-known to be very efficient tools for the acceleration of the convergence of certain sequences of numbers or functions, cf. [2, 3, 4, 8, 9]. In this note we present a representation of linear extrapolation procedures in terms of complex contour integrals. For the proof we make use of a complete characterization of these procedures as linear functionals, which is itself of some interest and can be found, for example, in [2] or [8].     

Comparison theorems for compound quadrature formulas

Summary The aim of this note is to compare the rates of convergence of quadrature processes. To this end, representation formulas for the remainders of one process in terms of a second one are developed. The main tool is the Möbius inversion. It turns out that for a large class of compound quadrature processes the rates of convergence are essentially the same.     

A sinc quadrature rule for Hadamard finite-part integrals

Summary A Sinc quadrature rule is presented for the evaluation of Hadamard finite-part integrals of analytic functions. Integration over a general are in the complex plane is considered. Special treatment is given to integrals over the interval (–1,1). Theoretical error estimates are derived and numerical examples are included.     

A probabilistic theory for error estimation in automatic integration

Summary A probabilistic theory for derivation and analysis of error criteria for automatic quadrature is presented. In particular, conditional average error criteria are derived for quadratures which have derivative-bound error estimates. These probabilistic error criteria are compared to variations of heuristic error criteria derived by discretizing the derivative in the original error bound. It is shown that the theory provides a mathematical foundation and a quantitative model for these discrete error criteria. It is also shown that estimating the conditional average error is equivalent to testing error with the spline interpolation as a sample integrand, and that this process can be made implicit by using appropriate error criteria with local error-checks.This paper is based on the author's Ph.D. thesis in computational complexity and numerical analysis, completed at the University of California, Berkeley     

Regularity theorems and optimal error estimates for linear parabolic Cauchy problems

Summary We prove some regularity results for the solution of a linear abstract Cauchy problem of parabolic type. As an application, we study the approximation of the solution by means of an implicit-Euler discretization in time, which is stable with respect to a wide class of Galerkin approximation methods in space. The error is evaluated in norms of typeL ²(0, ,L ²) andL ²(0, ,V)(H ₀₀ ^1/2 (0, ,H)+H ¹(0, ,V)), whereVHV are Hilbert spaces (the embeddings are supposed to be dense and continuous). We prove error estimates which are optimal with respect to the regularity assumptions on the right-hand side of the equation.The author was supported by G.N.A.F.A. and I.A.N. of C.N.R. and by M.P.I.     

An analysis of Ralston's quadrature

Summary Ralston's quadrature achieves higher accuracy in composite rules than analogous Newton-Cotes or Gaussian formulas. His rules are analyzed, computable expressions for the weights and knots are given, and a more suitable form of the remainder is derived.     

A numerical algorithm for the Stokes problem based on an integral equation for the pressure via conformal mappings

Summary In this paper we present an algorithm for solving numerically the Stokes problem in the plane. The known algorithms are all based on certain discretization schemes for the analytic equations. In contrast to this recent work our algorithm uses an explicit analytic solution of a certain approximating problem, which can easily be solved numerically up to machine accuracy. On the one hand this analytic formula is based on a complex representation of all solutions of the Stokes differential equations, and on the other hand it is based on the conformal mapping of the given domain on the unit disc. Therefore, a central prerequisite of our corresponding program is a program for computing this conformal mapping.     

Continuous time collocation methods for Volterra-Fredholm integral equations

Summary Integral equations of mixed Volterra-Fredholm type arise in various physical and biological problems. In the present paper we study continuous time collocation, time discretization and their global and discrete convergence properties.