Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms

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Product Description

Numerical Methods provides a clear and concise exploration of standard numerical analysis topics, as well as nontraditional ones, including mathematical modeling, Monte Carlo methods, Markov chains, and fractals. Filled with appealing examples that will motivate students, the textbook considers modern application areas, such as information retrieval and animation, and classical topics from physics and engineering. Exercises use MATLAB and promote understanding of computational results.

The book gives instructors the flexibility to emphasize different aspects--design, analysis, or computer implementation--of numerical algorithms, depending on the background and interests of students. Designed for upper-division undergraduates in mathematics or computer science classes, the textbook assumes that students have prior knowledge of linear algebra and calculus, although these topics are reviewed in the text. Short discussions of the history of numerical methods are interspersed throughout the chapters. The book also includes polynomial interpolation at Chebyshev points, use of the MATLAB package Chebfun, and a section on the fast Fourier transform. Supplementary materials are available online.

• Clear and concise exposition of standard numerical analysis topics
• Explores nontraditional topics, such as mathematical modeling and Monte Carlo methods
• Covers modern applications, including information retrieval and animation, and classical applications from physics and engineering
• Promotes understanding of computational results through MATLAB exercises
• Provides flexibility so instructors can emphasize mathematical or applied/computational aspects of numerical methods or a combination
• Includes recent results on polynomial interpolation at Chebyshev points and use of the MATLAB package Chebfun
• Short discussions of the history of numerical methods interspersed throughout
• Supplementary materials available online

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Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms Review

I teach computers to do math, so-- disclaimer-- I'm on the applied, not pure math side of NA. I came across this text while compiling a Body of Knowledge entry on Spectral, Fourier and Chebyshev methods for IEEE and the International Association of Bodies of Knowledge (The 9bok dot org people who certify math BoKs). The usual "track" for advanced undergrads is Calc up to PDE's, some linear algebra, a little computer arithmetic (and maybe some of my field, Computer Algebra), then on to Engineering or Physics.

Along the way, most of us will touch Numerical Analysis. There are two distinct sides to NA-- pure, as a way of defining formal proofs with "results" as much as methods, and applied-- solving problems, especially using algorithms, via close approximation, guessing, brute force, iteration, and other "cheats." The problem with many of the classic NA texts is that "applied" usually means, you guessed it, physics and engineering. Today, however, NA is as much at home with digital artists, game programmers creating physics engines, animators, Maya programmers, etc. as with physicists!

You'd think with that going on, there would be some rocking texts that are also fun. Not the case. Sadly, most of the "better" (read understandable) texts in NA date back to the late 1980s, when there was no internet (there were 50 websites in 1992 when Clinton took office). In fact, this author's book on Iterative Linear methods dates back to 1987, and John Boyd's classic on Fourier Spectrals to 1989.

This text changes a lot of that! The authors use a LOT more current examples you're likely to find in many other fields, from protein folding to NASCAR. Who uses computers to "guess" at difficult PDE solutions other than astrophysicists? Try Neurologists modeling cognition as Dynamic Systems! Yes, the applications today are way beyond what they were in 1987, and we finally have an NA text that covers not only the basics, but MANY cutting edge areas-- like fractals-- that weren't even taken seriously back then.

To be fair, some of the examples just give a "taste" of the field, and were filled in by experts, but not really used in the text, and apparently not really understood by the authors. For example, Dan Goldman was tapped to give some fun examples of collision detection for Yoda in Star Wars, but if you look up Inverse Kinematics or Kinematics in the Index, there is no mention. Lorentz transforms and dynamic analysis are not really covered, and when their NA engines are mentioned (Newton's Method, for example), they are in the context of Julia sets and fractals, not Kinematics.

If you haven't taken linear algebra and aren't very familiar with matrices, there is "some" review here, but not enough to make this text fun and painless. You really need to brush up on LA before tackling this. Look at it this way: you can think of many NA techniques as you would visual basic behind Excel. Your computer is using a lot of "spreadsheet" type crunching, only of functions, to "guesstimate" things like zeros/roots, parameters, etc. So what are these "spreadsheets" called by mathematicians? Right, Linear Algebra (on roids). Back in ancient times (the 90's) NA was considered the playground of mathematicians, engineers and physicists. Today, game programmers, animators, digital artists and even programmers like yours truly need to understand it to know what's going on underneath the calculations-- namely, crunching, right down to the stacks and registers.

Another group that will like this text are the embedded circuit folks-- instead of nail biting about on or off chip memory limits, many of the newer memory limit "work arounds" are in NA functions, algorithms and shortcuts. Sure, we'll eventually have to SOLVE the memory issues, but for now, the real world IS about working around with "close enough" solutions. You won't find any of these applications in most NA texts-- the present work is a gem, and unique in being up to date on many NOW applications, including several beyond the traditional physics and engineering examples. Oh, and yes, you do learn analysis here too, including the proofs and pure math sides if your track is math. I'm not qualified to opine in that "pure proof" track, but if you're on the applied side of NA, and want to go deeper, you'll love this text.

History note for a few emailers: Thanks for reminding me that 1987 is "recent" compared to many NA techniques that adapt Euler to algorithmic form-- by "recent" I also mean that these authors use examples like web surfing and Google's (secret sauce) analytics. I'm talking examples too, not just the fact --which I'll gladly grant-- that much of NA stands on the shoulders of giants going back to the 1700s. I challenge anyone to show me an NA text that is this relevant to today's applications, however! If you're a student, you also won't feel like you're being forced to study stuff that will have no relevance to your future. If you're a prof-- don't you want to orient your students via examples that are being used right now? They WILL thank you!

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