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Computational Methods for Physicists pp 1–55Cite as

Basics of Numerical Analysis

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Part of the book series: Graduate Texts in Physics ((GTP))

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 (Padé) approximations, as well as approximations of evolution operators for Hamiltonian systems. Emphasis is given to the efficient calculation of the Padé 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.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Simon Širca & Martin Horvat

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  1. Simon Širca
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  2. Martin Horvat
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Širca, S., Horvat, M. (2012). Basics of Numerical Analysis. In: Computational Methods for Physicists. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32478-9_1

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