Multiple/arbitrary precision interval computations in C-XSC

Abstract

As a new feature, C-XSC provides so-called wrapper classes to some external arbitrary precision real and interval packages. Operator and function name overloading is used to give the user easy access to the arithmetic operations and mathematical functions provided by the underlying Ansi C packages. We will discuss briefly so-called staggered precision arithmetics based on exact scalar products. Such an arithmetic is available in C-XSC e.g. for multiple precision complex intervals. We also discuss the usage of the arbitrary precision arithmetic packages MPFR and MPFI, which are now accessible conveniently from within C-XSC via class interfaces. As a typical application, we will present an arbitrary precision interval Newton method to find the root(s) of a continuously differentiable function in a prescribed domain. The user only has to supply the expression for the function in the usual mathematical notation. The derivative needed in the interval Newton operator is computed using automatic differentiation based on the arbitrary precision interval operations. To demonstrate the power of the package we compute an enclosure of the zero of a model problem with guaranteed accuracy of more than 10 million decimal digits.