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Art of Computer Programming, The: Seminumerical Algorithms, Volume 2
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The bible of all fundamental algorithms and the work that taught many of today's software developers most of what they know about computer programming.
–Byte, September 1995
I can't begin to tell you how many pleasurable hours of study and recreation they have afforded me! I have pored over them in cars, restaurants, at work, at home... and even at a Little League game when my son wasn't in the line-up.
–Charles Long
If you think you're a really good programmer... read [Knuth's] Art of Computer Programming... You should definitely send me a resume if you can read the whole thing.
–Bill Gates
It's always a pleasure when a problem is hard enough that you have to get the Knuths off the shelf. I find that merely opening one has a very useful terrorizing effect on computers.
–Jonathan Laventhol
The second volume offers a complete introduction to the field of seminumerical algorithms, with separate chapters on random numbers and arithmetic. The book summarizes the major paradigms and basic theory of such algorithms, thereby providing a comprehensive interface between computer programming and numerical analysis. Particularly noteworthy in this third edition is Knuth's new treatment of random number generators, and his discussion of calculations with formal power series.
- ISBN-100201896842
- ISBN-13978-0201896848
- Edition3rd
- PublisherAddison-Wesley Professional
- Publication dateNovember 4, 1997
- LanguageEnglish
- Dimensions9.5 x 6.6 x 1.64 inches
- Print length784 pages
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From the Publisher
Updated Box Set Now Available!
This new box set now includes Volume 4B - Combinatorial Algorithms, Part 2
"I've had loads of fun writing other parts of these volumes, but without doubt Section 7.2.2.1 [of Combinatorial Algorithms, Part 2] has been the funnest. And I know that my delight in good puzzles is shared by a significant number of leading computer scientists and mathematicians, who have told me that they chose their careers after having been inspired by such intellectual challenges."
— Donald Knuth
The Art of Computer Programming Volumes 1-4B
ISBN-10: 0137935102
Editorial Reviews
Amazon.com Review
The book begins with fundamental questions regarding random numbers and how to use algorithms to generate them. Subsequent chapters demonstrate efficient computation of single-precision and double-precision arithmetic calculations and modular arithmetic. The text then presents prime factorization (which can be used in cryptography, for instance) and algorithms for calculating fractions. This volume ends with algorithms for polynomial arithmetic and manipulation of power-series topics, which will benefit those with some knowledge of calculus.
Throughout this beautifully presented edition, Knuth incorporates hundreds of useful exercises for trying out the algorithms. These range from simple problems to larger research project topics. (The book provides answers, where appropriate, at the end of the book.) The result is a text that's suitable for college or graduate-level computer science courses or individual study by programmers. Volume 2 is an indispensable part of any working programmer's library.
From the Back Cover
The bible of all fundamental algorithms and the work that taught many of today's software developers most of what they know about computer programming.
―Byte, September 1995
I can't begin to tell you how many pleasurable hours of study and recreation they have afforded me! I have pored over them in cars, restaurants, at work, at home... and even at a Little League game when my son wasn't in the line-up.
―Charles Long
If you think you're a really good programmer... read [Knuth's] Art of Computer Programming... You should definitely send me a resume if you can read the whole thing.
―Bill Gates
It's always a pleasure when a problem is hard enough that you have to get the Knuths off the shelf. I find that merely opening one has a very useful terrorizing effect on computers.
―Jonathan Laventhol
The second volume offers a complete introduction to the field of seminumerical algorithms, with separate chapters on random numbers and arithmetic. The book summarizes the major paradigms and basic theory of such algorithms, thereby providing a comprehensive interface between computer programming and numerical analysis. Particularly noteworthy in this third edition is Knuth's new treatment of random number generators, and his discussion of calculations with formal power series.
About the Author
Donald E. Knuth is known throughout the world for his pioneering work on algorithms and programming techniques, for his invention of the Tex and Metafont systems for computer typesetting, and for his prolific and influential writing. Professor Emeritus of The Art of Computer Programming at Stanford University, he currently devotes full time to the completion of these fascicles and the seven volumes to which they belong.
Product details
- Publisher : Addison-Wesley Professional
- Publication date : November 4, 1997
- Edition : 3rd
- Language : English
- Print length : 784 pages
- ISBN-10 : 0201896842
- ISBN-13 : 978-0201896848
- Item Weight : 2.8 pounds
- Dimensions : 9.5 x 6.6 x 1.64 inches
- Part of series : The Art of Computer Programming
- Best Sellers Rank: #1,050,989 in Books (See Top 100 in Books)
- #1,770 in Computer Programming Languages
- #3,147 in Computer Software (Books)
- #6,433 in Computer Programming (Books)
- Customer Reviews:
About the author

Donald E. Knuth was born on January 10, 1938 in Milwaukee, Wisconsin. He studied mathematics as an undergraduate at Case Institute of Technology, where he also wrote software at the Computing Center. The Case faculty took the unprecedented step of awarding him a Master's degree together with the B.S. he received in 1960. After graduate studies at California Institute of Technology, he received a Ph.D. in Mathematics in 1963 and then remained on the mathematics faculty. Throughout this period he continued to be involved with software development, serving as consultant to Burroughs Corporation from 1960-1968 and as editor of Programming Languages for ACM publications from 1964-1967.
He joined Stanford University as Professor of Computer Science in 1968, and was appointed to Stanford's first endowed chair in computer science nine years later. As a university professor he introduced a variety of new courses into the curriculum, notably Data Structures and Concrete Mathematics. In 1993 he became Professor Emeritus of The Art of Computer Programming. He has supervised the dissertations of 28 students.
Knuth began in 1962 to prepare textbooks about programming techniques, and this work evolved into a projected seven-volume series entitled The Art of Computer Programming. Volumes 1-3 first appeared in 1968, 1969, and 1973. Having revised these three in 1997, he is now working full time on the remaining volumes. Volume 4A appeared at the beginning of 2011. More than one million copies have already been printed, including translations into ten languages.
He took ten years off from that project to work on digital typography, developing the TeX system for document preparation and the METAFONT system for alphabet design. Noteworthy by-products of those activities were the WEB and CWEB languages for structured documentation, and the accompanying methodology of Literate Programming. TeX is now used to produce most of the world's scientific literature in physics and mathematics.
His research papers have been instrumental in establishing several subareas of computer science and software engineering: LR(k) parsing; attribute grammars; the Knuth-Bendix algorithm for axiomatic reasoning; empirical studies of user programs and profiles; analysis of algorithms. In general, his works have been directed towards the search for a proper balance between theory and practice.
Professor Knuth received the ACM Turing Award in 1974 and became a Fellow of the British Computer Society in 1980, an Honorary Member of the IEEE in 1982. He is a member of the American Academy of Arts and Sciences, the National Academy of Sciences, and the National Academy of Engineering; he is also a foreign associate of l'Academie des Sciences (Paris), Det Norske Videnskaps-Akademi (Oslo), Bayerische Akademie der Wissenschaften (Munich), the Royal Society (London), and Rossiiskaya Akademia Nauk (Moscow). He holds five patents and has published approximately 160 papers in addition to his 28 books. He received the Medal of Science from President Carter in 1979, the American Mathematical Society's Steele Prize for expository writing in 1986, the New York Academy of Sciences Award in 1987, the J.D. Warnier Prize for software methodology in 1989, the Adelskøld Medal from the Swedish Academy of Sciences in 1994, the Harvey Prize from the Technion in 1995, and the Kyoto Prize for advanced technology in 1996. He was a charter recipient of the IEEE Computer Pioneer Award in 1982, after having received the IEEE Computer Society's W. Wallace McDowell Award in 1980; he received the IEEE's John von Neumann Medal in 1995. He holds honorary doctorates from Oxford University, the University of Paris, St. Petersburg University, and more than a dozen colleges and universities in America.
Professor Knuth lives on the Stanford campus with his wife, Jill. They have two children, John and Jennifer. Music is his main avocation.
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- 5 out of 5 stars
Good quality, classic
Reviewed in the United States on February 28, 2026Any serious student of computer science would be delighted to have this classical work, by one of the all-time greats, on their bookshelf. The binding looked solid, good paper quality, they will last for a couple of decades at least. It is a solid buy.
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it served its purpose
Reviewed in the United States on February 18, 2014this was a gift and the recipient loved it. It came in with excellent condition and arrived in time. BTW the recipient has a job related to the content of the book and hopefully this book had some positive effects.
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A must-have for your reference library!
Reviewed in the United States on September 18, 2016A leader in his field. Written so concepts are easy to understand. Good reference tool.
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The Classic
Reviewed in the United States on August 25, 2014A classic it remains. Worth the price and the time.
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Harder going than the first one, but still amazing reading
Reviewed in the United States on July 6, 2017I started reading this book immediately after I finished reading volume 1. I loved volume 1, and I feel the same way about this one, although this was actually a very different sort of book. For one thing, this book was way more mathematical than the first. The first half of volume 1 was all math and the last half was all programming. This book, instead, is almost all math and practically no programming. The author starts by covering random number generators - both implementing and testing them (bet you didn't know that you can actually test the randomness of a random number generator, did you? Knuth can spend a hundred pages showing you a dozen different ways!) This takes up about half the book (chapter 3), and then he moves on to discussing the implementation of arithmetic in computers. Again, you might think that this is worth maybe a dozen pages, but not for Knuth - he spends over two hundred pages discussing more tips and tricks for speeding up arithmetic than I would have guessed existed.
I attempted every single exercise in volume 1 and went into this one with the same intent. I got pretty close to at least trying each exercise; the last three sections, which covered exponents, polynomial evaluation and power series, went so deep that I ended up glossing over about half of the exercises in each section (the section on polynomial evaluation has over 70 exercises on its own). The exercises in this book were almost all more difficult, and more mathematically sophisticated, than volume 1 - I wouldn't have believed that was possible, but every time you think you've gone as deep as you can possibly go, Knuth shows you that you're just at the tip of the iceberg. I can't say I solved exercise, but I did at least try almost all of them - don't judge me until you try it yourself!
I did love this book, and enjoyed reading it, but I do think that this is one that a practicing programmer outside of very specific domains can probably safely skip - as fascinating as the topics are, and even after taking into consideration that this is not only the authoritative reference but in many cases the only printed material on some subjects - random number generation and lightning-fast (arbitrary precision) arithmetic aren't topics that most programmers have to deal with too often. Still, if you have the time, it is a really fun book to work through.
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the bible of programming;
Reviewed in the United States on February 22, 2023great college memories
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A classic
Reviewed in the United States on January 11, 2021A decent continuation of Knuth’s classic work, this goes into a sufficient level of depth that suits master and apprentice equally well.
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Numbers: random generations and arithmetic
Reviewed in the United States on August 10, 2006Volume 2 of "The Art of Computer Programming" is about random numbers and also about relearning one of the three Rs from grade school, viz. arithmetic. Each topic gets one chapter.
When you generate random numbers in Excel, or VBA, or Perl, or C using functions packaged with the software, you are really using a deterministic algorithm that is not random at all; the results do however look random and so we call them "pseudorandom".
Chapter 3 contains four main sections. First a section devoted to the linear congruence method (Xn+1=(aXn + c) mod m) of generating a pseudorandom sequence; with subsections on how to choose good values for a, c, and m. Second we get a section about how to test sequences to find if they are acceptably random or not. Third we find a section on other methods, expanding on linear congruence. Finally in a particularly fascinating section, DK provides a rigorous definition of randomness.
I haven't looked much at chapter 4 yet, on arithmetic. In it Knuth covers positional arithmetic, floating point arithmetic, multiplication and division at the machine level, prime numbers and efficient ways of investigating the primeness of very large numbers.
Again, DK is thorough and methodical. Again this is not a for dummies book. Again it is about theorems, algorithms, mechanical processes, and timeless truths. Again the exercises are a fascinating blend of the practical (investigate the random generating functions on the computers in your office) to the mathematical (he asks readers to formally prove many of the theorems he cites). And yes, again Knuth uses MIX, that wonderfully archaic fictional 60s machine language. But that should not stop readers; I use Perl.
Vincent Poirier, Tokyo
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sunflower5 out of 5 starsChristmas present
Reviewed in the United Kingdom on January 9, 2021Bought as Christmas gift.
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Dr K3 out of 5 starsThe Wikipedia of Algorithms - a start point for careful investigation..
Reviewed in the United Kingdom on August 3, 2022Mis-titled: all about algorithms and nothing on programming - any sensible programmer could do better than volume 1 page 2.
Sloppy pedantry. Long-windedness.
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