This book is revolutionary. That is the only way to describe it. I have been a professional computer science researcher for almost 40 years, and only once or twice before have I seen a book that is destined to make such a profound change in the way we think about computation. It is hard to imagine that after 70 years or so of computer arithmetic that there is anything new to say about it, but this book reinvents the subject from the ground up, from the very notion of finite precision numbers to their bit-level representation, through the basic arithmetic operations, the calculation of elementary functions, all the way to the fundamental methods of numerical analysis, including completely new approaches to expression calculation, root finding, and the solution of differential equations. On every page from the beginning to the end of the book there are surprises that just astonished me, making me re-think material that I thought had been settled for decades.
The methods described in this book are profoundly different from all previous treatments of numerical methods. Unum arithmetic is an extension of floating point arithmetic, but mathematically much cleaner. It never does rounding, so there is no rounding error. It handles what in floating point arithmetic is called "overflow" and "underflow" in a far more natural and correct way that makes them normal rather than exceptional. It also handles exceptional values (NaN, +infinity, -infinity) cleanly and consistently. Those contributions alone would have been a profound contribution. But the book does much more.
One of the reasons I think the book is revolutionary is that unum-based numerical methods can effortlessly provide provable bounds on the error in numerical computation, something that is very rare for methods based on floating point calculations. And the bounds are generally as tight as possible (or as tight as you want them), rather than the useless or trivial bounds as often happens with floating point methods or even interval arithmetic methods.
Another reason I consider the book revolutionary is that many of the unum-based methods are cleanly parallelizable, even for problems that are normally considered to be unavoidably sequential. This was completely unexpected.
A third reason is that in most cases unum arithmetic uses fewer bits, and thus less power, storage, and bandwidth (the most precious resources in today’s computers) than the comparable floating point calculation. It hard to believe that we get this advantage in addition to all of the others, but it is amply demonstrated in the book. Doing efficient unum arithmetic takes more logic (e.g. transistors) than comparable floating point arithmetic does, but as the author points out, transistors are so cheap today that that hardly matters, especially when compared to the other benefits.
Some of the broader themes of the book are counterintuitive to people like me advanced conventional training, so that I have to re-think everything I “knew” before. For example, the discussion of just what it means to “solve” an equation numerically is extraordinarily thought provoking. Another example is the author’s extended discussion of how calculus is not the best inspiration for computational numerical methods, even for problems that would seem to absolutely require calculus-based thinking, such as the solution of ordinary differential equations.
Not only is the content of the book brilliant, but so is the presentation. The text is so well written, a mix of clarity, precision, and reader friendliness that it is a pure pleasure to read, rather then the dense struggle that mathematical textbooks usually require of the reader. But in addition, almost every page has full color graphics and diagrams that are completely compelling in their ability to clearly communicate the ideas. I cannot think of any technical book I have ever seen that is so beautifully illustrated all the way through.
I should add that I read the Kindle edition on an iPad, and for once Amazon did not screw up the presentation of a technical book, at least for this platform. It is beautifully produced, in full color and detail, and with all of the fonts and graphics reproduced perfectly.
Dr. Gustafson has also provided a Mathematica implementation of unums and of the many numerical methods discussed in the book. Let us hope that in the next few years there will be implementations in other languages, followed by hardware implementations. Over time there should be unum arithmetic units alongside of floating point arithmetic units on every CPU and GPU chip, and in the long run unums should replace floating point entirely. The case the author makes for this is overwhelming.
If you are at all interested in computer arithmetic or numerical methods, read this book. It is destined to be a classic.
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Top reviews from other countries
R S
I'm very glad to have read this and recommend it very highly
Reviewed in the United Kingdom 🇬🇧 on May 28, 2016David Wright
Very clear and fun to read
Reviewed in the United Kingdom 🇬🇧 on October 27, 2017
