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Applied Analysis (Dover Books on Mathematics), by Cornelius Lanczos
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This is a basic text for graduate and advanced undergraduate study in those areas of mathematical analysis that are of primary concern to the engineer and the physicist, most particularly analysis and design of finite processes that approximate the solution of an analytical problem. The work comprises seven chapters:
Chapter I (Algebraic Equations) deals with the search for roots of algebraic equations encountered in vibration and flutter problems and in those of static and dynamic stability. Useful computing techniques are discussed, in particular the Bernoulli method and its ramifications.
Chapter II (Matrices and Eigenvalue Problems) is devoted to a systematic development of the properties of matrices, especially in the context of industrial research.
Chapter III (Large-Scale Linear Systems) discusses the "spectroscopic method" of finding the real eigenvalues of large matrices and the corresponding method of solving large-scale linear equations as well as an additional treatment of a perturbation problem and other topics.
Chapter IV (Harmonic Analysis) deals primarily with the interpolation aspects of the Fourier series and its flexibility in representing empirically given equidistant data.
Chapter V (Data Analysis) deals with the problem of reduction of data and of obtaining the first and even second derivatives of an empirically given function — constantly encountered in tracking problems in curve-fitting problems. Two methods of smoothing are discussed: smoothing in the small and smoothing in the large.
Chapter VI (Quadrature Methods) surveys a variety of quadrature methods with particular emphasis on Gaussian quadrature and its use in solving boundary value problems and eignenvalue problems associated with ordinary differential equations.
Chapter VII (Power Expansions) discusses the theory of orthogonal function systems, in particular the "Chebyshev polynomials."
This unique work, perennially in demand, belongs in the library of every engineer, physicist, or scientist interested in the application of mathematical analysis to engineering, physical, and other practical problems.
- Sales Rank: #857397 in Books
- Published on: 2010-07-21
- Released on: 2010-06-23
- Original language: English
- Number of items: 1
- Dimensions: 8.32" h x 1.17" w x 5.42" l, 1.10 pounds
- Binding: Paperback
- 576 pages
Most helpful customer reviews
40 of 40 people found the following review helpful.
A Gem of an Applied Math Book
By reader 1001
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.
36 of 37 people found the following review helpful.
Simply the best book on numerical analysis
By A Customer
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.
16 of 17 people found the following review helpful.
An interesting if some what dated book
By Eugene Mcmanus
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.
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