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190 of 193 found the following review helpful:
The Purpose of Computing is Insight, Not NumbersMar 24, 1999
Throughout the book, that motto is repeated.
By reading and absorbing the material in this book, the reader is left with the tools and the insights necessary to derive their own numerical methods.
No longer will numerical methods be memorized as textbook formulas -- now the reader can adapt and derive a formula to solve a specific problem, instead of trying to fit one of a small number of textbook formulas to a problem.
The distinction is made between numerical analysis and numerical methods, with emphasis on the latter.
The book is roughly divided into two parts. The first part covers classical numerical methods, using classical error analysis (truncation error, roundoff error). The second part reexamines these methods under the frequency domain, analyzing how numerical methods affect various frequencies (the "transfer function" approach).
Numerical methods are derived under an information theory model, such as by finding a quadrature formula of the highest polynomial degree of accuracy, given limited information about the function and its derivatives.
Matrices and linear systems are not discussed as much as one might expect, although one chapter convincingly leads the reader to question some classical methods.
The content is well-rounded, introducing many readers to topics such as random number generators, difference equations and summation formulas, digital filters and quantization, discrete fourier transforms and the FFT, and orthogonal polynomials. A background in calculus is all that is needed.
Many real-world examples and anecdotes are cited, but without too much detail or too many illustrations given.
This book encourages the reader to ask: "What information is available about the problem? How can it be used to solve the problem? What are the limits of this information?" The approach is practical, not merely analytical.
This book teaches what most other numerical books fail to teach: How to derive your own formulas, and thus your own solutions to problems. And that is perhaps the most important lesson of all.
34 of 34 found the following review helpful:
Gives an intuitive feel for numerical methodsMar 10, 2006
Chances are, if you have a degree in engineering of any kind, you have seen all of the numerical methods outlined here before. However, the purpose of this book is not just to detail how to perform different kinds of calculations. Instead, the author is attempting to give you an intuitive feel for the mathematics as well.
The book starts with an essay on numerical methods that discusses the book's five main ideas starting with its motto, which is the first idea - "The purpose of computing is insight, not numbers." The second idea is that it is necessary to study families of numbers and algorithms and to relate one family to another. The third and fourth major ideas are that of roundoff and truncation error, each of which is an effect of computing on finite machines. The final main idea is that of feedback and stability, where numbers produced at one stage are looped back to feed other stages of computations, and the result may or may not be stable.
The remainder of the book then studies many families of calculations and numbers based on these insights. The book is divided into five parts - Fundamentals and Algorithms, Polynomial Approximation, Fourier Approximation, Exponential Approximation, and finally a miscellaneous section which talks about approximations to singularities, optimization, linear independence, and eigenvalues and eigenvectors of Hermitian matrices. As you can see, the idea throughout this book is that since numerical methods are the use of numbers to simulate mathematical processes, then all of these algorithms are actually approximations. Throughout the book there are clearly worked out examples with plenty of illustrations and also many exercises, some with solutions. Highly recommended.
33 of 34 found the following review helpful:
Like no other book on numerical methods I have read.Aug 24, 2003
I sympathise with the reviewer who said this is one of the few
books on numerical methods he could stand. I will go further
and say this is a book that can be enjoyed. Example: section 2.8
"The Frequency Distribution of Mantissas" explains why the
leading digits of of decimal numbers are not uniformly
distributed, a result that is surely counterintuitive. There is
much more material of interest in this book too. It does
contain standard material too but is more readable than many
books. The author offers much practical advice and insight.
(Hamming is a famous name in applied mathematics and electical
83 of 93 found the following review helpful:
Can numerical analysis be fun?Jul 23, 1998
This is the only book on numerical methods I can stand. But, not only can I stand it: now I love it. It's one of the cleverest books I ever met. Hamming must be a genius of insight. Even if you wrote your thesis on differential equations, I bet you will be enriched by reading his considerations on them, from the numerical precision viewpoint. The same is true for Fourier methods, only much more, as this is the main topic of this surprising and wonderful book. Since then I bought every book written by Hamming during his lifespan, which is unfortunately over.
14 of 15 found the following review helpful:
A beautiful book.Jun 09, 2010
R.W. Hamming instantly became one of my favorite authors after I received this book in the mail. He wasn't afraid to inject his hard-earned experience into a field which even today really is in its infancy. While some of the material is dated in that if you ever are writing serious numerical code, you are very unlikely to implement many of the details in this book yourself, it is a great place to go to understand what your library is doing (or at least provide a general idea).
Hamming wrote this book during a very special period of time for mathematics and computer science. Algorithms and (numerical)mathematics were very much seen as a means to an end, and not an end themselves. Many mathematicians now will provide a numerical analysis of an algorithm, or derive a numerical algorithm only because it's the fashionable thing to do now, and it somehow enbiggens them by portraying a false sense of "application."
There is a sort of purity in this book, everything in it is a solution to what was a serious engineering problem in the implementation of numerics on computers. On top of this, it is written in an absolutely lucid style which I have found to be very characteristic of Hamming.
My favorite chapters are of course chapter 1: "An Essay on Numerical Methods", and chapter 'N+1' "The Art of Computing for Scientists and Engineers" If you bought the book and only read these two chapters, I would say you got your money's worth.
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