The Mathematica GuideBook for Numerics: Mathematics and Physics Hardcover – 28 Nov 2005
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From the reviews:
"The numerics volume has two chapters. … offer a treasure trove of identities and formulas. … take the reader on a thrilling tour of the features of Mathematica. I am impressed with the breadth and depth of Trott’s coverage and his profound understanding of the strengths and limitations of Mathematica. … includes a multiplatform DVD-ROM which allows the reader to experiment with code and view graphics in color. … an invaluable resource and classic reference for scientists who use Mathematica in teaching or research." (Willy Hereman, SIAM Review, Vol. 49 (1), 2007)
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Several elucidating examples also further the knowledge about the mathematical or physical subject matter itself. Just to mention a few, I was amazed to discover that magnetic field lines of stationary flows are NOT closed in general, and some examples of modern mathematics and physics literature are used in contrast to classical physics, such as the continuous time random walk on a graph or quantum carpets in a solution, which provides an interesting classical vs. quantum variant. And next to textbook examples the reader will find many "real life" examples interspersed throughout the book as well, which makes the results quite "tangible".
Michael Trott includes a great comparison of Mathematica's superior compilable machine precision arithmetic (bytecode, instead of machine-code compilation), high-precision arithmetic, and interval arithmetic, all three of which are needed frequently depending on the numerics project at hand.
Needless to say, the book shows a profusion of carefully chosen graphics that help understanding the matter or result instantly. Here Michael Trott's expert ability as a graphics programmer is indispensible.
All references are hyperlinked through their respective DOI numbers. The DVD contains the texts and inputs of the other three GuideBooks as well.
Even if you use a "free" version of GNU-CAS or the home version of Mathematica (Wolfram Mathematica 9 Home Edition (Windows/Mac/Linux)), this series gives a peek into CAS levels that have long been unpatented trade secrets of TI and HP and very tough to decode. NO other good recent books cover the behind the curtain aspects of CAS as well as these volumes. If you're a math programmer, even if you love Maple, you'll find this series way helpful, at all kinds of levels.
Mathematica is a layered object language, and has multiple compile levels (with an overriding interpreter), which are essentially functions in what, as a programmer, you'd call arrays or hashes (or in Mathematica-speak, lists, functions, rules and patterns). You MUST read the programming book to "get" the rest of the series, otherwise you'll be stuck in what seems like a no-man's land of not being able to define your own classes yet not being able to use typical CAS-like functions in assembly-- when in fact both, and much more, are supported once you get it. We still use the graphics volume at our shader joes sub, and it is as relevant in 2013 as it was 5 years ago. I probably don't have to mention that a C++ compiler is a must beyond a certain level of coding. The difference in "getting" the unique symbolics is code that takes forever to run vs. code that executes as if directly compiled just like Fortran.
Now, for the Amazon deal. The simply AMAZING DVD that comes when you buy one of these volumes new (or used with a SPECIFIED unopened DVD) has an incredible reference with the texts for all the volumes. So, if you have an interest in one area over another (programming, symbolics, numerics, graphics), you can get that 1,000 to 1,300 page gem, and still read all the rest! I personally recommend getting the one you want new on Amazon to be sure you get the whole, functional DVD. Given that these are starting to skyrocket in price due to supply/demand, that could literally save you $300 or more. If you do find a good "deal" from a third party, follow this algorithm:
1. MAKE SURE the description on the LEFT (with the product condition) specifically says an unopened DVD is included. You can even comb the 4 volumes under Amazon's "see all offers" feature to find this statement.
2. Obviously, if the listing says "associated media MAY not be present" etc., do NOT buy there. Less obviously, if it says nothing but a condition like good or acceptable, remember that "good" itself CAN INCLUDE the UNSTATED spec that media "may" be missing. Pass there too!
3. In addition to the "includes unopened DVD," buy from a minimum 96% rated seller with at least 1,000 ratings, and be sure they SAY where they are shipping from. China and India do NOT have to tell you this, although they do have to meet Amazon's delivery time ranges. You just can't be sure what long transport times will do to media if it's not in the original, sealed shell. Joining Prime also gives the option of warehouse shipping free from Amazon itself if a third party tells you the above, yet Amazon warehouses and ships. That gives the added advantage of Amazon's professional packaging.
Regardless of which volume your prefer, you need to start by reading volume one, the programming text, or you'll be lost even in graphics. Programming not only gives you a crash course in CAS, but shows the many unique functions and methods of this software. It really is a hybrid between CAS functions and libraries and much more C++ looking calls and features if you get into it deeply. This allows you to come from many different points of view, and help learn the others if you wish. Truly an extraordinary series, in math/computer education as well as using the software to its full ability. The publisher advertises these as stand alone, which is true, but ONLY because the DVD allows you to read programming first! Without the DVD, that statement is misleading (which isn't the publisher's or Amazon's fault, let alone the author!).
One joke says that the definition of discrete math today is all the math/computer UI's they've removed from High School now! Don't get me started. Removing a calculus requirement means we're teaching 2,500 year old math to many of our HS grads. Adding just the basics of calc brings them up to 1666. If you teach, consider sneaking wisdom from these texts into your courses; we might even begin to bring our grandkids up to 1960 or so!!!
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To get the best use of the this long book, one should skim it and actually run the test code that might be of interest in Mathematica. Later, in the future, where one has a pertinent problem in writing Mathematica code, this book will be useful. In addition, one should go tot he Springer website and download the available materials. This will include the examples from the book which can be either copied into Mathematica or run native. Since some of the examples are quite complex, this is very useful to avoid mistakes although I tend to learn from the mistakes.
I would also recommend for the serious user the three other Guidebooks (Programming, Symbolics and Graphics) for a complete set.
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