Horvat Martin - Computational methods for physicists: compendium for students

Simon irca 1 and Martin Horvat 1

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Abstract

After reviewing the basic properties of floating-point representation, examples of typical use of expressions in finite-precision arithmetic are given, along with illustrations of common programming pitfalls. Selected classes of function approximation are shown next: optimal (minimax) and rational (Pad) approximations, as well as approximations of evolution operators for Hamiltonian systems. Emphasis is given to the efficient calculation of the Pad approximants and preservation of unitarity. Fundamental techniques of power and asymptotic expansions and asymptotic analysis are discussed: Taylor and Laurent series, treatment of divergent asymptotic series, asymptotic analysis of integrals by the Laplace method and the stationary-phase approximation. The treatment of differential equations involving large parameters is elucidated on the example by the WKB method in quantum mechanics. Tests of convergence of series are listed, followed by a presentation of efficient acceleration techniques for their summation. Examples and Problems include the system of interacting electric dipoles, integral of the Gaussian function, computation of Airy and Bessel functions, and calculation of the Coulomb scattering amplitude.

An ever increasing amount of computational work is being relegated to computers, and often we almost blindly assume that the obtained results are correct. At the same time, we wish to accelerate individual computation steps and improve their accuracy. Numerical computations should therefore be approached with a good measure of skepticism. Above all, we should try to understand the meaning of the results and the precision of operations between numerical data.

A prudent choice of appropriate algorithms is essential (see, for example, []). In their implementation, we should be aware that the compiler may have its own will and has no clue about mathematical physics. In order to learn more about the essence of the computation and its natural limitations, we strive to simplify complex operations and restrict the tasks of functions to smaller, well-defined domains. It also makes sense to measure the execution time of programs (see Appendix J): large fluctuations in these well measurable quantities without modifications in running conditions typically point to a poorly designed program or a lack of understanding of the underlying problem.