We give a survey on packages for multiple precision interval arithmetic, with the main focus on three specific packages. One is within a Maple environment, intpakX, and two are C/C++ libraries, GMP-XSC and MPFI. We discuss their different features, present timing results and show several applications from various fields, where high precision intervals are… (More)
We consider error estimates for optimal and Gaussian quadrature for-mulae if the integrand is analytic and bounded in a certain complex region. First, a simple technique for the derivation of lower bounds for the optimal error constants is presented. This method is applied to Szegg o-type weight functions and ellipses as regions of analyticity. In this… (More)
We study optimal quadrature formulas for convex functions in several variables. In particular, we answer the following two questions: Are adaptive methods better than nonadaptive ones? And: Are randomized (or Monte Carlo) methods better than deterministic methods?
We study the intrinsic difficulty of solving linear parabolic initial value problems numerically at a single point. We present a worst case analysis for determin-istic as well as for randomized (or Monte Carlo) algorithms, assuming that the drift coefficients and the potential vary in given function spaces. We use fundamental solutions (parametrix method)… (More)