Customer Reviews

Applied Analysis, A Philosophical and Theoretical Analysis of the Tools Of Mathematics Used in Numerical Solutions of Physical and Engineering Problems

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3 star:		(1)
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While this booked is dated because it was written for the days of mechanical calculators, it contains a great deal of very useful material. His discussion of Chebyeshev Polynomials one of the best I seen. His discussion on telescoping of power series is one of the few available. He gives great insight into a host of numerical methods. A very valuable work for the computer age as well.

My dissertation advisor introduced me to this book over thirty years ago. I have since read it in its entirety twice and it is still the first book I consult when confronted with a new mathematical problem.

Lanczos's understanding of applied mathematics is very deep and he has a rare way of explaining things clearly yet concisely. I find his description of linear systems in terms of multidimensional coordinate systems, both orthogonal and skewed, to be the best anywhere. Also, his understanding and explanation of harmonic analysis (he invented the FFT after all) is worth the price of the book by itself.

Buy it, read it (at least once) then see if really need any other book on applied mathematics.

Then this is the best book. Well, Hamming's is also so good! For Fourier analysis, and the taming of the Gibbs phenomenon, go straight to Lanczos. He knew it all, and was one of the inventors of the fast Fourier transform. This book is in the class of Sommerfeld's "Partial Differential Equations of Physics" and Lighthill's "Fourier Analysis and Generalizaed Functions". This is a very high compliment. Did you know he was also a first rate physicist, and a pioneer of quantum mechanics?

Applied Analysis, by Cornelius Lanczos is, in the author's words in the Preface, that branch of analysis devoted to the analysis of finite algorithms, or "workable mathematics". Today it would be called numerical analysis.

Written in 1956,foloowing assignments with then North american Aviation and the Boeing Airplane Company, the book is a compendium of compuational techniques for the solution of cubic, quartic and higher order algebraic equations, matrices and eigenvalue problems, large scale linear systems, harmonic analyis. An entire chapter of 65 pages is devoted to data analysis, the problems associates with processing large amounts of data, and some of the hidden dangers of straightforward(equidistant) interpolation. A chapter is devoted to quadrature methods, and the book concludes with a chapter on power expansions.

Most of the computational methods are devoted to hand calculation, or electro mechanical calculation. Today, algorithms developed specifically for high speed digital processors make most of these methods obsolete.

I bought the book because of an interest in Legendre polynomials, and their use in fitting to data. However, fascinating tidbits (at least to me) pop up unexpectedly. One tidbit that caught my eye was in the chapter on Data Analysis(Chapter V) in which the Sturm-Liouville equation suddenly appears, is solved by an application of Green's identity, and hence the result is a proof of the orthoganility of a class of equations (Legendre's being just one of several) in just two steps. Perhaps I missed something when I studied Series and Special Functions, but I've not seen this anywhere else.

A very pleasant characteristic of the book is the almost seamless movement back and forth between theory and computationl application. On the other hand, be prepared to spend time with a pencil and paper following not only the derivations and proofs, but the computational algorithms. For someone doing, or planning to do, a massive amount of number crunching of large collections of physical date, this is probably a good book to have.

It's an excellent book. The best parts for

we were the chapters on Matrices and on

Harmonic Analysis. An outstanding aspect

of the latter chapter is Lanczos's exposition

of the motivation behind the Fourier integral

(transform) and its basic theory. The quality

of the writing is superb, very classical

and lucid.

It cannot, of course, serve as a textbook.

But if you're taking a Fourier theory

course using Stein and Shakarchi's book, say,

as I am currently, then it's a very handy

book that can complement abstract theory

with physical intuition.

Lanczos' work is a fine, thorough text that covers most areas of advanced analysis in a readable style. His derivations are clear, his tangential explorations are absorbing, and he cites practical examples. The one area in which I find the book weak is harmonic functions, potential theory, and the applications of these to the calculus of resides. Consequently, the book is not "one-shop stopping" for all the mathematical techniques that an electrical engineer or physicist might require in his bag of tricks....