Extrapolation methods and sequence transformations for computing slowly convergent integrals

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It is well known that in applied mathematics and in the numerical treatment of scientific and engineering problems, slowly convergent integrals and series occur very frequently. They are produced by approximation procedures depending on a parameter, perturbation techniques and reliable evaluation of functions that are defined by integrals. These slowly convergent integrals and series present severe numerical and computational difficulties. Traditional quadrature rules and summation techniques fail to provide accurate approximations to these integrals and series. An effective remedy for these problems is to use extrapolation methods and convergence acceleration. These methods have proved very useful for improving convergence of infinite series and infinite-range integrals but many challenges occur when dealing with complicated integrals. In this research program, we will introduce a new method based on a generalization of the extremely powerful S. This generalization will eliminate the boundary conditions required by the original S transformation and will expand applicability beyond spherical Bessel integrals. This new method will represent a slowly convergent integral by a divergent series as boundary terms and a transformed integral asymptotically more favorable than the initial one. For the implementation of the boundary terms, we propose the use of sequence transformations for the summation of divergent series. In the case of the transformed integral, we will use extrapolation methods, nonlinear transformations as well as the state of the arts techniques for computing oscillatory integrals, namely numerical steepest descent, Filon-type and Levin-type methods. A part of this research program is concerned with challenging applications of the developed methods and algorithms. These applications include the Sommerfield-type integrals and incomplete Bessel functions. The most challenging application will concern the computation of the so-called molecular multi-center integrals and integrals of nuclear magnetic resonance (NMR) parameters. The analytical treatment of these NMR integrals, which is also a part of the research program, will be obtained using the Fourier transformation.