Download PDF The End of Error: Unum Computing (Chapman & Hall/CRC Computational Science)
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The End of Error: Unum Computing (Chapman & Hall/CRC Computational Science)
Download PDF The End of Error: Unum Computing (Chapman & Hall/CRC Computational Science)
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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
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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.
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Product details
Series: Chapman & Hall/CRC Computational Science (Book 24)
Paperback: 416 pages
Publisher: Chapman and Hall/CRC; 1 edition (February 7, 2015)
Language: English
ISBN-10: 1482239868
ISBN-13: 978-1482239867
Product Dimensions:
7 x 0.5 x 10 inches
Shipping Weight: 1.6 pounds (View shipping rates and policies)
Average Customer Review:
4.3 out of 5 stars
10 customer reviews
Amazon Best Sellers Rank:
#1,195,189 in Books (See Top 100 in Books)
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.
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".
Review: The End of Error: Unum Computing by John L. GustafsonEmeritus Professor, Lenore M. Mullin PhDDepartment of Computer ScienceUniversity at Albany, SUNYScientist, and ConsultantI have spent my entire 35+ year career studying, programming (applications and systems) teaching, and researching scientific computing: algorithms, software, languages, hardware, design and verification). I have also worked and published with computer engineers, computer scientists and mathematicians. In addition, I have studied pedagogy and educational methods that work, e.g. “Guided Discovery†which contributed to my NSF Presidential Faculty Award ’92. It was from all these perspectives I read John’s book. Upon completion I felt this was one of the most profound books I have ever read from all perspectives and is a must read for anyone who seriously cares about how the computing industry is moving.I always knew there was imprecision in the way numbers were represented but John made it very clear how serious, and dangerous, this imprecision is. Through color photos, factual examples, and humor, he presents how costly this imprecision has, is and will continue to be, unless something is done, and soon, as we depend more and more on computers to run our lives using Cyber Physical Systems and an Internet of Everything. Until now one might say the errors discussed in his book are only a few examples but I believe anyone can extrapolate those examples to include what might happen if automated cars or surgical tools go awry, when many lives are lost. Sometimes a loss of money can be shoved under the table when large companies can write off such losses, but lives cannot.What was the most interesting thing about this book, besides the fact that we need a new look at how numbers are represented that guarantee accuracy and address the serious and dangerous issues mentioned above, is that it presented ideas using mathematics that most high school students can understand. Often scientific work is presented with such advanced mathematics that even when important the reader is turned off, and thus the impact is lost. Moreover, the use of color coding, and simple explanations are a pedagogically sound way of presenting even that. This is similarly true for software examples, which also can be a turn off to many readers.His justification of unum computing is convincing from all perspectives. I personally do not know why every computing vendor, every funding agency, everyone who is writing simulations, is not demanding that this be done. Perhaps our complacency to change is the reason for this but it is time to take what we have learned and see that new directions are essential for progress, precision, and correct answers. It is also important to take the burden away from users who spend an enormous amount of time, wasted time, on such issues, when they can be using their scientific minds to do science, or any applications that care about numeric precision, e.g. the Stock Market as mentioned in the book.Finally, he has built a simulator that is free to use and all the code is available in the book.I applaud his brilliance and courage to shake up the industry and encourage everyone to run out and buy this book. You WILL be blown away.
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