Computational Techniques

Essential to atomic, molecular, and optical physics is the ability to perform numerical computations accurately and efficiently. Whether the specific approach involves perturbation theory, close coupling expansion, solution of classical equations of motion, or fitting and smoothing of data, basic computational techniques such as integration, differentiation, interpolation, matrix and eigenvalue manipulation, Monte Carlo sampling, and solution of differential equations must be among the standard tool kit. This chapter outlines a portion of this tool kit with the aim of giving guidance and organization to a wide array of computational techniques. After having digested the present overview, the reader is then referred to detailed treatments given in many of the large number of texts existing on numerical analysis and computational techniques 1 ; 2 ; 3 ; 4 ; 5 ; 6 , mathematical functions 7 ; 8 ; 9 , and mathematical physics 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18 . In addition to these excellent general references, in the age of the internet, many resources are also available through free publishing projects or research laboratory resources made public. Many of these resources seek to provide techniques and computer codes of high accuracy, portability, robustness, and efficiency, and often take advantage of modern structured programming and computational parallelism, going beyond the highly accessible, broadly applicable, but simple numerical recipes and codes described in the classic texts. A list of such numerical analysis software is given on the Wikipedia, providing very brief descriptions of the packages available 19 , and the journal Computer Physics Communications (CPC) publishes computational physics research and applications software with many codes applicable to atomic, molecular, and optical physics (see the CPC program library maintained at Queen's University Belfast 20 ). Especially in the sections that follow on differential equations and computational linear algebra, mention is made of the role of software packages readily available to aid in implementing practical solutions. Finally, in this brief introduction to computational techniques, we note the existence of commercial packages for mathematics, including those for computer algebra, performing numerical calculations and visualizing results through proprietary programming languages, and even performing simulations through such tools as finite-element analysis, including Mathematica, Maple, MATLAB, Mathcad, and COMSOL, for example.