Below are a few reviews of John Gustafson's book "The End of Error" on Amazon books. I've ordered a copy of the book. I wanted a hard copy, but is back-ordered, and takes 3-4 weeks to ship. However, one can order a Kindle electronic copy, and get it immediately.
Also, this book answers Raul's question about worked examples. Gustafson provides a full Mathematica implementation of Unums, so one can explore all the ramifications of this new numerical representation. Skip <<<>>> "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 counter intuitive to people like me with 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. " David Jefferson <http://www.amazon.com/gp/pdp/profile/A3E5GY7K9GGYD0/ref=cm_cr_dp_pdp> on April 18, 2015 Other Review s (Amazon) "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 Skip Cave Cave Consulting LLC On Fri, Apr 29, 2016 at 12:17 PM, Skip Cave <[email protected]> wrote: > Interesting comment from John Gustafson on Google Groups: > <<<>>> > Incidentally, I've been challenged to a debate by William Kahan at the > ARITH23 conference, July 10-13 in San Jose, CA. (Kahn is the designer of > the IEEE floating point numerical format). > > Title: "The Great Debate: The End of Error?" > > Kahan has apparently prepared a 34-page response to my book (Gustafson's > "The End Of Error" book) though I have not seen it and he will probably > spring all kinds of surprises on me. It should be a good show! > <<<>> > > Skip > On Apr 29, 2016 10:27 AM, "Skip Cave" <[email protected]> wrote: > > Here's the Google Group on unum computing: > > https://groups.google.com/forum/#!forum/unum-computing > > > Here's Gustafson's home page: > > http://www.johngustafson.net/index.html > > > Skip > > > Skip Cave > Cave Consulting LLC > > On Fri, Apr 29, 2016 at 5:35 AM, Pierpaolo Bernardi <[email protected]> > wrote: > >> On Fri, Apr 29, 2016 at 7:41 AM, Skip Cave <[email protected]> >> wrote: >> > Here's a much newer presentation by Gustafson that goes into the >> > implementation of Unums in much more detail. >> > >> > http://www.johngustafson.net/presentations/Unums2.0slides-withNotes.pdf >> >> This is about Unums 2.0, which is a completely different idea. The >> choice of naming them Unums 2.0 is unfortunate IMO. >> >> Unums 2.0 is a way more exoteric idea then Unums, and as far as I >> understand it is still in an embryonal stage. >> >> For interested people there's a google group about unums where >> Gustafson participates, and up to now has always replied to questions. >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm >> > > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
