The emphasis in this book has been on solving problems. My overriding aim has been to showyou that elementary tools of spectral methods can be applied easily to get high-accuracy results in all kinds of areas.

Of course, such a style deemphasizes certain things, and one, regrettably, is mathematical depth. In twoways, this book gives too little indication of the mathematical maturity that the field of spectral methods has achieved today. One is that there has been little mention of theoretical developments, even though by now, a quarter century into the "modern" era, some remarkable mathematics has been worked out to establish the foundations of spectral methods. Numerous papers and books by Bernardi, Canuto, Funaro, Gottlieb, Maday, Quarteroni, and Tadmor, to mention just a few, represent this strong tradition. The crucial contribution of this theory, for method after method applied to problem after problem, is a proof of rapid convergence as N ? ?. Spectral methods are delicate, and plausible methods frequently fail, so results of this kind are much more than confirmations of the obvious.

At the algorithmic level, this book has also oversimplified. For example, we have squared differentiation matrices by explicit matrix multiplication, so that everything can be based on the simple code cheb, but this is not the most stable method, nor the most efficient; the use of recurrences reduces operation counts from O( N ³) to O( N ²) [Wel97, WeRe00]. Similarly, we have not hinted at various more...