Gaussian quadrature is a well-known technique for numerical integration. Recently Gaussian quadrature with respect to discrete measures corresponding to finite sums has found some new interest. In this paper we apply these ideas to infinite sums in general and give an explicit construction for the weights and abscissae of Gaussian formulas. The abscissae of the Gaussian summation have a very in...

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 bas...

A high-order accurate numerical quadrature algorithm is presented for the evaluation of integrals over curved surfaces and volumes which are defined implicitly via a fixed isosurface of a given function restricted to a given hyperrectangle. By converting the implicitly defined geometry into the graph of an implicitly defined height function, the approach leads to a recursive algorithm on the nu...

After studying Gaussian type quadrature formulae with mixed boundary conditions, we suggest a fast algorithm for computing their nodes and weights. It is shown that the latter are computed in the same manner as in the theory of the classical Gauss quadrature formulae. In fact, all nodes and weights are again computed as eigenvalues and eigenvectors of a real symmetric tridiagonal matrix. Hence,...

Konrad Zuses frühe vollautomatische Rechenmaschinen Z1-Z4 zeichnen sich nicht nur durch das Prinzip des minimalen Entwurfs, Boolescher Logik, binärer Gleitkommazahlen, usw. aus, sondern auch durch ergonomisch konstruierte Schnittstellen zwischen dem Nutzer und der Rechenmaschine. Diese Aspekte sind bisher nur wenig beachtet worden.

When Cavalieri found his theory in 1635, Chinese mathematicians had used the theory for more than one millennium. 2003 Elsevier Inc. All rights reserved.

Let X ≤ 1. Recently, there has been much interest in the classification of combinatorially Desargues, Cavalieri, naturally right-geometric subgroups. We show that n is not smaller than c. A useful survey of the subject can be found in [3, 27]. Next, in this context, the results of [11] are highly relevant.