Generalized Stieltjes Polynomials and Rational Gauss–Kronrod Quadrature

Generalized Stieltjes polynomials are introduced and their asymptotic properties outside the support of the measure are studied. As applications, we prove the convergence of sequences of interpolating rational functions, whose poles are partially fixed, to Markov functions and give an asymptotic estimate of the error of rational Gauss–Kronrod quadrature formulas when functions which are analytic on some neighborhood of the set of integration are considered.     

Some remarks on the numerical computation of integrals on an unbounded interval

An account of the error and the convergence theory is given for Gauss–Laguerre and Gauss–Radau–Laguerre quadrature formulae. We develop also truncated models of the original Gauss rules to compute integrals extended over the positive real axis. Numerical examples confirming the theoretical results are given comparing these rules among themselves and with different quadrature formulae proposed by other authors (Evans, Int. J. Comput. Math. 82:721–730, 2005; Gautschi, BIT 31:438–446, 1991).      

Numerical Quadrature Computation of the Macdonald Function for Complex Orders

The use of Gaussian quadrature formulae is explored for the computation of the Macdonald function (modified Bessel function) of complex orders and positive arguments. It is shown that for arguments larger than one, Gaussian quadrature applied to the integral representation of this function is a viable approach, provided the (nonclassical) weight function is suitably chosen. In combination with Gauss–Legendre quadrature the approach works also for arguments smaller than one. For very small arguments, power series can be used. A Matlab routine is provided that implements this approach. AMS subject classification (2000) 33-04, 33C10, 65D15, 65D32     

Stopping functionals for Gaussian quadrature formulas

Gaussian formulas are among the most often used quadrature formulas in practice. In this survey, an overview is given on stopping functionals for Gaussian formulas which are of the same type as quadrature formulas, i.e., linear combinations of function evaluations. In particular, methods based on extended formulas like the important Gauss–Kronrod and Patterson schemes, and methods which are based on Gaussian nodes, are presented and compared.     

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.

     

A historical note on Gauss–Kronrod quadrature

Summary The idea of Gauss–Kronrod quadrature, in a germinal form, is traced back to an 1894 paper of R. Skutsch.     

AN EXTREMAL APPROACH TO BIRKHOFF QUADRATURE FORMULAS

1. Introduction and Main ResultsIn tfor paPer we shaJl use the ddstions and notations of [3l. Let E = (e'k)7t' kt. be anincidence matrir with entries consisting of zeros and ones and satisfying lEl:= Z.,* ei* = n + 1(here we allow a zero row ). Furthermore, in wha follOws we assume that(A) E satisfies the P6lya condition(B) all sequences of E in the interior rows, 0 < i < m + 1, are even.Let Sm denote the set of poiats X = (xo, z1 l "') xm, x.+1) fOr whichand Sm its clOusure. If some O…     

High-order generalized Gauss–Radau and Gauss–Lobatto formulae for Jacobi and Laguerre weight functions

The generation of generalized Gauss–Radau and Gauss–Lobatto quadrature formulae by methods developed by us earlier breaks down in the case of Jacobi and Laguerre measures when the order of the quadrature rules becomes very large. The reason for this is underflow resp. overflow of the respective monic orthogonal polynomials. By rescaling of the polynomials, and other corrective measures, the problem can be circumvented, and formulae can be generated of orders as high as 1,000. In memoriam Gene H. Golub.     

Convergence of Gaussian quadrature formulas on infinite intervals

More general and stronger estimations of bounds for the fundamental functions of Hermite interpolation of high order on an arbitrary system of nodes on infinite intervals are given. Based on this result, convergence of Gaussian quadrature formulas for Riemann–Stieltjes integrable functions on an arbitrary system of nodes on infinite intervals is discussed.     

On families of quadrature formulas based on Euler identities

A family consisting of quadrature formulas which are exact for all polynomials of order ?5 is studied. Changing the coefficients, a second family of quadrature formulas, with the degree of exactness higher than that of the formulas from the first family, is produced. These formulas contain values of the first derivative at the end points of the interval and are sometimes called “corrected”.     

Symmetric Gauss–Lobatto and Modified Anti-Gauss Rules

The present paper is concerned with symmetric Gauss–Lobatto quadrature rules, i.e., with Gauss–Lobatto rules associated with a nonnegative symmetric measure on the real axis. We propose a modification of the anti-Gauss quadrature rules recently introduced by Laurie, and show that the symmetric Gauss–Lobatto rules are modified anti-Gauss rules. It follows that for many integrands, symmetric Gauss quadrature rules and symmetric Gauss–Lobatto rules give quadrature errors of opposite sign.     

Fibonacci求积公式对具有有界混合差分的光滑函数类的误差估计

考虑对具有有界混合差分的二元光滑函数类Ｂ＾γ，ｐ，θ的求积公式，本文证明了Ｆｉｂｏｎａｃｃｉ求积公式是渐近最优的，并求出了春误差的渐近最优价。     

Gaussian quadrature in Ramanujan’s Second Notebook

Ramanujan’s notebooks contain many approximations, usually without explanations. Some of his approximations to series are explained as quadrature formulas, usually of Gaussian type. Dedicated to the memory of Professor K G Ramanathan     

Adaptive Quadrature—Revisited

First, the basic principles of adaptive quadrature are reviewed. Adaptive quadrature programs being recursive by nature, the choice of a good termination criterion is given particular attention. Two Matlab quadrature programs are presented. The first is an implementation of the well-known adaptive recursive Simpson rule; the second is new and is based on a four-point Gauss-Lobatto formula and two successive Kronrod extensions. Comparative test results are described and attention is drawn to serious deficiencies in the adaptive routines quad and quad8 provided by Matlab.     

Efficiency of numerical integration algorithms related to divisor theory in cyclotomic fields

For function classes with dominant mixed derivative and bounded mixed difference in the metric ofL ^q (1<q≤2), quadrature formulas are constructed so that the following properties are achieved simultaneously: the grid is simple, the algorithm is efficient and close to the optimal algorithm for constructing the grid, and the order of the error on the power scale cannot be further improved. The caseq=2 was studied earlier. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 297–301, February, 1997. Translated by N. K. Kulman     

Szegő–Lobatto quadrature rules

Gauss-type quadrature rules with one or two prescribed nodes are well known and are commonly referred to as Gauss–Radau and Gauss–Lobatto quadrature rules, respectively. Efficient algorithms are available for their computation. Szeg? quadrature rules are analogs of Gauss quadrature rules for the integration of periodic functions; they integrate exactly trigonometric polynomials of as high degree as possible. Szeg? quadrature rules have a free parameter, which can be used to prescribe one node. This paper discusses an analog of Gauss–Lobatto rules, i.e., Szeg? quadrature rules with two prescribed nodes. We refer to these rules as Szeg?–Lobatto rules. Their properties as well as numerical methods for their computation are discussed.     

An extended doubly-adaptive quadrature method based on the combination of the Ninomiya and the FLR schemes

An improvement is made to an automatic quadrature due to Ninomiya (J. Inf. Process. 3:162–170, 1980) of adaptive type based on the Newton–Cotes rule by incorporating a doubly-adaptive algorithm due to Favati, Lotti and Romani (ACM Trans. Math. Softw. 17:207–217, 1991; ACM Trans. Math. Softw. 17:218–232, 1991). We compare the present method in performance with some others by using various test problems including Kahaner’s ones (Computation of numerical quadrature formulas. In: Rice, J.R. (ed.) Mathematical Software, 229–259. Academic, Orlando, FL, 1971).      

On sensitivity of Gauss–Christoffel quadrature

In numerical computations the question how much does a function change under perturbations of its arguments is of central importance. In this work, we investigate sensitivity of Gauss–Christoffel quadrature with respect to small perturbations of the distribution function. In numerical quadrature, a definite integral is approximated by a finite sum of functional values evaluated at given quadrature nodes and multiplied by given weights. Consider a sufficiently smooth integrated function uncorrelated with the perturbation of the distribution function. Then it seems natural that given the same number of function evaluations, the difference between the quadrature approximations is of the same order as the difference between the (original and perturbed) approximated integrals. That is perhaps one of the reasons why, to our knowledge, the sensitivity question has not been formulated and addressed in the literature, though several other sensitivity problems, motivated, in particular, by computation of the quadrature nodes and weights from moments, have been thoroughly studied by many authors. We survey existing particular results and show that even a small perturbation of a distribution function can cause large differences in Gauss–Christoffel quadrature estimates. We then discuss conditions under which the Gauss–Christoffel quadrature is insensitive under perturbation of the distribution function, present illustrative examples, and relate our observations to known conjectures on some sensitivity problems. The work of the first author was supported by the National Science Foundation under Grants CCR-0204084 and CCF-0514213. The work of the other two authors was supported by the Program Information Society under project 1ET400300415 and by the Institutional Research Plan AV0Z100300504. P. Tichy in the years 2003–2006 on leave at the Institute of Mathematics, TU Berlin, Germany.     

Numerical Quadratures for Hadamard Hypersingular Integrals

In this paper,we develop Gaussian quadrature formulas for the Hadamard fi- nite part integrals.In our formulas,the classical orthogonal polynomials such as Legendre and Chebyshev polynomials are used to approximate the density function f(x)so that the Gaussian quadrature formulas have degree n-1.The error estimates of the formulas are obtained.It is found from the numerical examples that the convergence rate and the accu- racy of the approximation results are satisfactory.Moreover,the rate and the accuracy can be improved by choosing appropriate weight functions.     

On the Fine and Rough Theory of Lagrange Type Interpolation of Higher Order

Convergence of a general Gaussian quadrature formula is shown and its rate of convergence is also given.