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Practical Guide to Spectral Computational Methods (Paperback) (George Rawitscher)
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This book disseminates basic aspects of modern spectral computational methods, which are not generally taught in traditional courses. The main advantage of these methods is that they take into account all available information at the same time, rather only the information available at a limited number of meshpoints. That leads to more complicated matrix equations, but the elegance, speed, and accuracy of the method more than compensates for this drawback. The method consists in expanding the function to be calculated into a set of appropriate basis functions (generally orthogonal polynomials), and the respective expansion coefficients are obtained via collocation equations.
In the various chapters, the authors examine the usually rapid convergence of the spectral expansions and the improved accuracy that results when nonequispaced support points are used, in contrast to the equispaced points used in finite difference methods, and, in particular, they demonstrate the enhanced accuracy obtained in the solution of integral equations.
It includes an informative introduction to the old and new computational methods with numerous practical examples, while at the same time raising awareness of the errors that each of the available algorithms introduces into the specific solution. It is a valuable resource for graduate students wishing to compare the available computational methods (canned or not canned) and judge which is the most suitable to solve the particular scientific problem they are confronting.