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The Art of Computer Programming, 3rd Edition, Volume 2: Seminumerical Algorithms


The Art of Computer Programming, 3rd Edition, Volume 2: Seminumerical Algorithms Cover




O dear Ophelia!

I am ill at these numbers:

I have not art to reckon my groans.

—Shakespeare, "Hamlet", Act II, Scene 2, Line 120

The algorithms discussed in this book deal directly with numbers; yet I believe they are properly called seminumerical, because they lie on theborderline between numeric and symbolic calculation. Each algorithm not onlycomputes the desired answers to a numerical problem, it also is intended toblend well with the internal operations of a digital computer. In many casespeople are not able to appreciate the full beauty of such an algorithm unlessthey also have some knowledge of a computer's machine language; the efficiencyof the corresponding machine program is a vital factor that cannot be divorcedfrom the algorithm itself. The problem is to find the best ways to make computers deal with numbers, and this involves tactical as well as numerical considerations. Therefore the subject matter of this book is unmistakably a part of computer science, as well as of numerical mathematics.

Some people working in "higher levels" of numerical analysis will regard thetopics treated here as the domain of system programmers. Other people working in"higher levels" of system programming will regard the topics treated here asthe domain of numerical analysts. But I hope that there are a few people left who will want to look carefully at these basic methods. Although the methods reside perhaps on a low level, they underlie all of the more grandiose applications of computers to numerical problems, so it is important to know them well. We are concerned here with the interface between numerical mathematics and computer programming, and it is the mating of both types of skills that makes the subject so interesting.

There is a noticeably higher percentage of mathematical material in this book than in other volumes of this series, because of the nature of the subjects treated. In most cases the necessary mathematical topics are developed here starting almost from scratch (or from results proved in Volume 1), but in several easily recognizable sections a knowledge of calculus has been assumed.

This volume comprises Chapters 3 and 4 of the complete series. Chapter 3 isconcerned with "random numbers": It is not only a study of various ways togenerate random sequences, it also investigates statistical tests forrandomness, as well as the transformation of uniform random numbers into othertypes of random quantities; the latter subject illustrates how random numbersare used in practice. I have also included a section about the nature ofrandomness itself. Chapter 4 is my attempt to tell the fascinating story ofwhat people have discovered about the processes of arithmetic, after centuriesof progress. It discusses various systems for representing numbers, and how toconvert between them; and it treats arithmetic on floating point numbers,high-precision integers, rational fractions, polynomials, and power series, including the questions of factoring and finding greatest common divisors.

Each of Chapters 3 and 4 can be used as the basis of a one-semester collegecourse at the junior to graduate level. Although courses on "Random Numbers"and on "Arithmetic" are not presently a part of many college curricula, Ibelieve the reader will find that the subject matter of these chapters lendsitself nicely to a unified treatment of material that has real educationalvalue. My own experience has been that these courses are a good means ofintroducing elementary probability theory and number theory to collegestudents. Nearly all of the topics usually treated in such introductorycourses arise naturally in connection with applications, and the presence ofthese applications can be an important motivation that helps the student tolearn and to appreciate the theory. Furthermore, each chapter gives a fewhints of more advanced topics that will whet the appetite of many students forfurther mathematical study.For the most part this book is self-contained, except for occasional discussions relating to the MIX computer explained in Volume 1. Appendix B contains a summary of the mathematical notations used, some of which are a little different from those found in traditional mathematics books.

Preface to the Third Edition

When the second edition of this book was completed in 1980, it represented thefirst major test case for prototype systems of electronic publishing calledTeX and METAFONT. I'am now pleased to celebrate the full development of those systems by returning to the book that inspired and shaped them. At last I am able to have all volumes of The Art of Computer Programming in a consistent format that will make them readily adaptable to future changes in printing and display technology. The new setup has allowed me to make many thousands of improvements that I have been wanting to incorporate for a long time.

In this new edition I have gone over every word of the text, trying to retainthe youthful exuberance of my original sentences while perhaps adding some moremature judgment. Dozens of new exercises have been added; dozens of old exercises have been given new and improved answers. Changes appear everywhere, but most significantly in Sections 3.5 (about theoretical guarantees of randomness), 3.6(about portable random-number generators), 4.5.2(about the binary gcd algorithm), and 4.7(about composition and iteration of powerseries).

The Art of Computer Programming is, however, still a work in progress. Research on seminumerical algorithms continues to grow at a phenomenal rate. Therefore some parts of this book are headed by an "under construction" icon, to apologize for the fact that the material is not up-to-date. My filesare bursting with important material that I plan to include in the final, glorious, fourth edition of Volume 2, perhaps 16 years from now; but I must finish Volumes 4 and 5 first, and I do not want to delay their publication any more than absolutely necessary.

I am enormously grateful to the many hundreds of people who have helped me togather and refine this material during the past 35 years. Most of the hard work of preparing the new edition was accomplished by Silvio Levy, who expertly edited the electronic text, and by Jeffrey Oldham, who converted nearly allof the original illustrations to METAPOST format. I have corrected every error that alert readers detected in the second edition (as well as some mistakes that, alas, nobody noticed); and I have tried to avoid introducing new errors in the new material. However, I suppose some defects still remain, and I want to fix them as soon as possible. Therefore I will cheerfully pay $2.56 to the first finder of each technical, typographical, or historical error. The Web page cited on pageiv contains a current listing of all corrections that have been reported to me.—D.E.K.

Stanford, California

July 1997

When a book has been eight years in the making,

there are too many colleagues, typists, students,

teachers, and friends to thank.

Besides, I have no intention of giving such people

the usual exoneration from responsibility for errors which remain.

They should have corrected me!

And sometimes they are even responsible for ideas

which may turn out in the long run to be wrong.

Anyway, to such fellow explorers, my thanks.

—Edward F. Campbell, Jr. (1975)

`Defendit numerus,' there is safety in numbers

is the maxim of the foolish;

`Deperdit numerus,' there is ruin in numbers

of the wise.

—C. C. Colton (1820)


Product Details

Knuth, Donald E
Addison-Wesley Professional
Preface by:
Knuth, Donald Ervin
Knuth, Donald Ervin
Knuth, Donald Ervin
Knuth, Donald E.
Reading, Mass.
Programming Languages - General
Programming - General
Electronic digital computers
Programming (electronic computers)
Computer programming
Computer algorithms
Software Engineering - Programming and Languages
Edition Number:
Edition Description:
Art of Computer Programming
Series Volume:
Publication Date:
November 1997
Grade Level:
Professional and scholarly
10 x 7 x 5 in 4241 gr

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The Art of Computer Programming, 3rd Edition, Volume 2: Seminumerical Algorithms Used Hardcover
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Product details 784 pages Addison-Wesley Professional - English 9780201896848 Reviews:
"Synopsis" by ,

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


Countless readers have spoken about the profound personal influence of Knuth’s work. Scientists have marveled at the beauty and elegance of his analysis, while ordinary programmers have successfully applied his “cookbook” solutions to their day-to-day problems. All have admired Knuth for the breadth, clarity, accuracy, and good humor found in his books.


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


Primarily written as a reference, some people have nevertheless found it possible and interesting to read each volume from beginning to end. A programmer in China even compared the experience to reading a poem.


If you think you’re a really good programmer… read [Knuth’s] Art of Computer Programming… You should definitely send me a résumé if you can read the whole thing.

Bill Gates


Whatever your background, if you need to do any serious computer programming, you will find your own good reason to make each volume in this series a readily accessible part of your scholarly or professional library.


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

In describing the new fourth volume, one reviewer listed the qualities that distinguish all of Knuth’s work.

[In sum:] detailed coverage of the basics, illustrated with well-chosen examples; occasional forays into more esoteric topics and problems at the frontiers of research; impeccable writing peppered with occasional bits of humor; extensive collections of exercises, all with solutions or helpful hints; a careful attention to history; implementations of many of the algorithms in his classic step-by-step form.

—Frank Ruskey

These four books comprise what easily could be the most important set of information on any serious programmer’s bookshelf.

"Synopsis" by ,

This boxed set consists of the following four volumes:   


0201896834 / 9780201896831 Art of Computer Programming, Volume 1: Fundamental Algorithms

0201896842 / 9780201896848 Art of Computer Programming, Volume 2: Seminumerical Algorithms

0201896850 / 9780201896855 Art of Computer Programming, Volume 3: Sorting and Searching

0201038048 / 9780201038040 Art of Computer Programming, Volume 4A: Combinatorial Algorithms

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