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For many numerical problems involving smooth multivariate functions on d-cubes, the so-called Smolyak algorithm (or Boolean method, sparse grid method, etc.) has proved to be very useful. The final form of the algorithm (see equation (12) below) requires functional evaluation as well as the computation of coefficients. The latter can be done in different(More)
Explicit bounds for the quadrature error of thenth Gauss-Legendre quadrature rule applied to themth Chebyshev polynomial are derived. They are precise up to the orderO(m 4 n −6). As an application, error constants for classes of functions which are analytic in the interior of an ellipse are estimated. The location of the maxima of the corresponding kernel(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 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)