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The Future of Numerical Computing
Written by one of the foremost experts in high-performance computing and the inventor of Gustafson’s Law, The End of Error: Unum Computing explains a new approach to computer arithmetic: the universal number (unum). The unum encompasses all IEEE floating-point formats as well as fixed-point and exact integer arithmetic. This new number type obtains more accurate answers than floating-point arithmetic yet uses fewer bits in many cases, saving memory, bandwidth, energy, and power.
A Complete Revamp of Computer Arithmetic from the Ground Up
Richly illustrated in color, this groundbreaking book represents a fundamental change in how to perform calculations automatically. It illustrates how this novel approach can solve problems that have vexed engineers and scientists for decades, including problems that have been historically limited to serial processing.
Suitable for Anyone Using Computers for Calculations
The book is accessible to anyone who uses computers for technical calculations, with much of the book only requiring high school math. The author makes the mathematics interesting through numerous analogies. He clearly defines jargon and uses color-coded boxes for mathematical formulas, computer code, important descriptions, and exercises.
- Sales Rank: #594275 in Books
- Published on: 2015-02-05
- Original language: English
- Number of items: 1
- Dimensions: 9.80" h x .80" w x 6.90" l, 2.02 pounds
- Binding: Paperback
- 416 pages
Review
"The author of the present book believes that it is time to supplement the century-old floating point arithmetic with something better: unum arithmetic. The book covers various operations with unum arithmetic and topics like polynomial evaluation, solving equations, two-body problem, etc. The appendices give a glossary of unum functions, ubox functions, and some algorithm listings."
―Zentralblatt MATH 1320
"This book is an extraordinary reinvention of computer arithmetic and elementary numerical methods from the ground up. Unum arithmetic is an extension of floating point in which it is also possible to represent the open intervals between two floating point numbers. This leads to arithmetic that is algebraically much cleaner, without rounding error, overflow underflow, or negative zero, and with clean and consistent treatment of positive and negative infinity and NaN. These changes are not just marginal technical improvements. As the book fully demonstrates, they lead to what can only be described as a radical re-foundation of elementary numerical analysis, with new methods that are free of rounding error, fully parallelizable, fully portable, easier for programmers to master, and often more economical of memory, bandwidth, and power than comparable floating point methods. The book is exceptionally well written and produced and is illustrated on every page with full-color diagrams that perfectly communicate the material. Anyone interested in computer arithmetic or numerical methods must read this book. It is surely destined to be a classic."
―David Jefferson, Center for Advanced Scientific Computing, Lawrence Livermore National Laboratory
"John Gustafson’s book The End of Error presents the ideas of computer arithmetic in a very easy-to-read and understandable form. While the title is provocative, the content provides an illuminating discussion of the issues. The examples are engaging, well thought out, and simple to follow."
―Jack Dongarra, University Distinguished Professor, University of Tennessee
"John Gustafson presents a bold and brilliant proposal for a revolutionary number representation system, unum, for scientific (and potentially all other) computers. Unum’s main advantage is that computing with these numbers gives scientists the correct answer all the time. Gustafson is able to show that the universal number, or unum, encompasses all standard floating-point formats as well as fixed-point and exact integer arithmetic. The book is a call to action for the next stage: implementation and testing that would lead to wide-scale adoption."
―Gordon Bell, Researcher Emeritus, Microsoft Research
"Reading more and more in [John Gustafson’s] book became a big surprise. I had not expected such an elaborate and sound piece of work. It is hard to believe that a single person could develop so many nice ideas and put them together into a sketch of what perhaps might be the future of computing. Reading [this] book is fascinating."
―Ulrich Kulisch, Karlsruhe Institute of Technology, Germany
About the Author
Dr. John L. Gustafson is an applied physicist and mathematician. He is a former Director at Intel Labs and former Chief Product Architect at AMD. A pioneer in high-performance computing, he introduced cluster computing in 1985 and first demonstrated scalable massively parallel performance on real applications in 1988. This became known as Gustafson’s Law, for which he won the inaugural ACM Gordon Bell Prize. He is also a recipient of the IEEE Computer Society’s Golden Core Award. Find more details on his website.
Most helpful customer reviews
18 of 18 people found the following review helpful.
This book is revolutionary
By Concerned small-D democrat
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.
18 of 18 people found the following review helpful.
Unum Computing Is A Breakthrough ... A Game Changer
By A. Shewmaker
I was a beta reader for "The End of Error: Unum Computing" by John Gustafson, and I wanted to highly recommend it to anyone who wants to get better numerical answers from a computer. I especially want to recommend it to anyone who is unaware of the pitfalls of floating point arithmetic. It provides a wonderfully readable overview of the history of computer arithmetic, gives many examples of troublesome computations, and describes in detail how our current number formats can be evolved to give us better answers. These benefits are made possible while making it so programmers don't have to worry nearly as much about how computer arithmetic differs from pure mathematics.
One of the key ideas of the new Universal Number (unum) format described in the book is the "inexact bit". It may not sound like much, but it enables computation on general mathematical intervals. While traditional interval arithmetic never lies by saying an answer lies within its bounds when it really is outside those bounds, it does lie by saying an answer may be one of the endpoints when we absolutely know that it cannot be.
It's not simply a matter of unum intervals being slightly more accurate than traditional interval arithmetic. It is of critical importance when dealing with singularities such as x = 1 / (0, 1] = [1, inf) where we know that the answer is absolutely not infinity. Unums can produce true and useful statements without underflowing or overflowing. If the answer needs to be tightened, then more bits can be used, but even if you've reached your system's limit you will never truly overflow and mathematical operations can continue to produce useful results.
This does mean that unums ask people to think more in terms of mathematical intervals (i.e. pairs of unums) instead of single numbers, but the payoff is that you don't have to worry about rounding errors, underflow, overflow, troublesome comparisons, and bounds that explode into uselessness. You may wonder if storing twice the amount of numbers is worth it (traditional arithmetic requires double the space for a given precision), but unums only store the number of bits necessary (plus a metadata tag) and that often ends up with an average of around half the typical floating point precision used. You'll have to try Unums out for your problem to see how space efficient it is.
Gustafson has created a Mathematica reference implementation, and has used it to show how he can get better answers with less programmer work and moving fewer bits. Among its benefits, unums also provide support for all your favorite mathematical laws: commutative, distributive, and associative. You no longer have to get by with only some of them. Among other things, that means serial and parallel computations will give the same answers. In other words, unums expose more parallelism than floats.
There's much more to the book, including discussions of how unums could be implemented in hardware. In particular, the book describes a well-defined and flexible scratchpad layer, an unpacked Unum format that is hardware friendly, and how unums can be clearly represented to a human. The first part of the book ends with a series of trial runs showing how unum arithmetic fares in situations that cause trouble for floating point, and the second part of the book continues with more difficult challenges and shows how Unums can be used in more interesting problems (e.g. solving equations, the n-body problem, and modeling gas).
The toolbox Gustafson builds around unums is extremely powerful, and I'm excited to see what people will be able to accomplish with unums in the next few years. The variable-size of unums will cause a lot of work for hardware designers, but I think the payoff will be well worth it.
15 of 15 people found the following review helpful.
Amazing! A brilliant technical work and a "page turner" in the same book
By Paul M Sweazey
Sure there will be naysayers, but 10 years from now we will look back to floating point as if it represented the dark ages. Structural engineering, Cosmology, and even high school calculus will be distinctly improved, startups will be born, and future generations will find that science and math are less baffling and more honest than today.
I bought this book because of my slight hope that it would show me an improved floating point variant. Instead I discovered a fundamental breakthrough. Gordon Bell says that the next step is implementation and testing that would lead to wide-scale adoption. I say, not would, but will. This book is easy to read, entertaining, and filled with surprises and revelations that keep you turning the pages.
I spent a career steering toward hardware architectures that could solve their problems using only integer data types, and away from problems without closed form solutions. This book helped me to understand what I feared and why. And now how do I feel? No project could be more satisfying than to build a unum processor. Congratulations to Mr. Gustavson for unums, which show how the real number line deserves the title "real".
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