The Jacobi matrix of the (2n+1)-point Gauss-Kronrod quadrature rule for a given measure is calculated efficiently by a five-term recurrence relation. The algorithm uses only rational operations and is therefore also useful for obtaining the Jacobi-Kronrod matrix analytically. The nodes and weights can then be computed directly by standard software for… (More)
In this article, software for the numerical approximation ofdouble integrals over a variety of regions is described. Thesoftware was written in C++. Classes for a large number of shapesare provided. A global adaptive integration algorithm is used basedon transformations and subdivisions of regions.
The classic Coupon-Collector Problem (CCP) is generalized. Only basic probability theory is used. Centerpiece rather is an algorithm that efficiently counts all k-element transversals of a set system.
A refinable linear functional is one that can be expressed as a convex combination and defined by a finite number of mask coefficients of certain stretched and shifted replicas of itself. The notion generalizes an integral weighted by a refinable function. The key to calculating a Gaussian quadrature formula for such a functional is to find the three-term… (More)