Hi Skip First thoughts on Gustafson's book: It is very impressive indeed: there is much of value and interest in it. I'm glad I bought it.
It predates Unums 2.0 and Unums 2.0 is certainly a major advance. Hence I'm pretty sure there will be a revised version of the book, maybe substantially different, but even so I'm glad to have this one. It is impressive, not just for the content but for the presentation, and even more so because Gustafson says that what we see is the output of a Mathematica notebook. Until recently I would have said that for us J programmers there was nothing comparable to Mathematica notebooks. However, there was a recent mention on one of our forums about Jupyter notebooks: I mean to check this out. Unums 2.0 I think is beautiful. It has a kind of compelling rightness. I have put on my website a J script for modelling arithmetic in Unums 2.0. Have just made a tiny beginning: can do addition and multiplication on 2-bit unums and addition on 3-bit unums. This may not seem much but anything more is just a matter of writing the lookup tables. The J code is very small and simple. The script is at: www.learningj.com/unums2.ijs Regards, Roger On Fri, May 13, 2016 at 6:57 PM, Skip Cave <[email protected]> wrote: > Great Roger! I'll be interested to see what you think about the book, and > Gustafson's proposed new binary number format. I haven't received my hard > copy of his book as yet. > On May 12, 2016 11:28 AM, "roger stokes" <[email protected]> wrote: > > > Many thanks Skip for posting the reviews. Having read your post, I > > couldn't hold out any longer and bought the Kindle version. > > > > Regards, > > Roger > > On May 12, 2016 8:22 AM, "Skip Cave" <[email protected]> wrote: > > > > > 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 > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
