Spectral Methods in MATLAB (Software, Environments, Tools): Lloyd N. Trefethen

Description

This is the only book on spectral methods built around MATLAB programs. Along with finite differences and finite elements, spectral methods are one of the three main technologies for solving partial differential equations on computers. Since spectral methods involve significant linear algebra and graphics they are very suitable for the high level programming of MATLAB. This hands-on introduction is built around forty short and powerful MATLAB programs, which the reader can download from the World Wide Web. This book presents the key ideas along with many figures, examples, and short, elegant MATLAB programs for readers to adapt to their own needs. It covers ODE and PDE boundary value problems, eigenvalues and pseudospectra, linear and nonlinear waves, and numerical quadrature.
Summary: setting a precedent in numerical methods books
Rating: 5
i would have given it a 4-stars, but i feel that i need to counter the negative comment by a reader below who complains that the book is loaded with errors.
first i should say i am not a mathematician although i used mathematics and numerical methods frequently. so there are certainly many details that i overlook or ignore.
in my opinion, most of the errors in the book (inc those not listed on trefethen's website but which i have noticed) are insignificant and inconsequential. they are things like writing 'n' instead of 'N'; the confusions and errors in the indices (this is due to matlab must start from '1', while frequently we are used to start from '0' when writing the mathematics); calling something obvious an inappropriate name, etc. to me, these are trivial errors. (if you are a numerical analysis theorist, i accept that you don't agree with me). only a handful of errors are actually serious in terms of the mathematics, and i believe the number of them are acceptable. every book has errors.
at the level i am reading this book, i just need an introduction to the theory, and then start computing. as far as i am concerned, if the mathematics is seriously wrong, you will notice when you do the computing.
what i like about the book is that it starts you computing right from the first chapter. although mathematical softwares such as matlab have been around for one to two decades, until recently not many books on numerical methods give you an opportunity to do the computations yourself. if the matlab scripts were not provided, i doubt that many would know how to code them (in any language you are familiar with); or even if you do, you may be doing it in an inelegant or inefficient way.
i used to wonder what is the use of a book on advanced numerical methods. for example, i would not know how to write a fluid dynamics code using spectral method to do direct numerical simulation of turbulent flows after reading and even mastering, say, Canuto et al's Spectral Dynamics in Fluid Dynamics. on the other hand, if i am familiar with spectral methods programming, it is also likely that i am familiar with the theory. so what these books are for is just as a store of knowledge and convenient references. you may know that the kind of fluid dynamics code i mentioned above can require a team effort over many years. so most numerical methods books which claim to cover applications are in fact useless, except you are already a practitioner, when come to developing codes for modern advanced simulations.
however, i think introductory books should have examples of how the computer codes to some numerical methods actually look like. before these past few years, i didn't see any such book. only recently, there are at least a dozen books on numerical methods or differential equations with C/C++, Matlab, etc. and i think trefethen's book is setting an example on how to do this without much fuss and long discourse. who knows, you may also learn the basic skills require to work on a much more complicated piece of computational code.
also, trefethen never claims that you can learn matlab with this book. he suggested, for example, higham & higham's matlab guide.
lastly, i should say one should not judge a book by the personality of the author. he may (or may not) be arrogant, but that does not necessarily make his book bad. (it appears to me that the reader below is the arrogant one. just because trefethen probably didn't give him responsive and 'grateful' replies, he wrote this review here).

Summary: A useful introduction to spectral methods
Rating: 4
The author achieves the hard task of explaining the subject in a very simple and, at the same time, intelligent way. The MATLAB m-files are short, educative and full of useful tricks. The book is well motivated, focused in the essentials and easy to read. I am very fond of most of Prof. Trefethen writings and this book is no disappoitment. It contrasts starkly with the lack of mathematical insight in most of the engineering collocation literature, but at the same time has a hands-on approach to the problems. This book should be useful to anybody interested in solving PDE in regular geometries.

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