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We have been helped by Nancy Lee Snyder’s masteryof a complicated TEX manuscript. We express appreciation to our editors LaurenCowles and Alan Harvey at Cambridge University Press, and to our productioneditor Russell Hahn. We remain, of course, grateful to the individuals acknowledgedin the Preface to the First Edition.Special acknowledgment is due to programming consultant Seth Finkelstein,who wrote, rewrote, or influenced many of the routines in this book, as well as inits FORTRAN-language twin and the companion Example books.

Our project hasbenefited enormously from Seth’s talent for detecting, and following the trail of, evenvery slight anomalies (often compiler bugs, but occasionally our errors), and fromhis good programming sense. To the extent that this edition of Numerical Recipesin C has a more graceful and “C-like” programming style than its predecessor, mostof the credit goes to Seth. (Of course, we accept the blame for the FORTRANishlapses that still remain.)We prepared this book for publication on DEC and Sun workstations running the UNIX operating system, and on a 486/33 PC compatible runningMS-DOS 5.0/Windows 3.0.

(See §1.0 for a list of additional computers used inSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).••••••••••••••xiiiPreface to the Second EditionJune, 1992William H.

PressSaul A. TeukolskyWilliam T. VetterlingBrian P. FlannerySample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).program tests.) We enthusiastically recommend the principal software used: GNUEmacs, TEX, Perl, Adobe Illustrator, and PostScript.

Also used were a variety of Ccompilers – too numerous (and sometimes too buggy) for individual acknowledgment. It is a sobering fact that our standard test suite (exercising all the routinesin this book) has uncovered compiler bugs in many of the compilers tried. Whenpossible, we work with developers to see that such bugs get fixed; we encourageinterested compiler developers to contact us about such arrangements.WHP and SAT acknowledge the continued support of the U.S. National ScienceFoundation for their research on computational methods. D.A.R.P.A. support isacknowledged for §13.10 on wavelets.Preface to the First EditionxivSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.

Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).We call this book Numerical Recipes for several reasons. In one sense, this bookis indeed a “cookbook” on numerical computation. However there is an importantdistinction between a cookbook and a restaurant menu. The latter presents choicesamong complete dishes in each of which the individual flavors are blended anddisguised. The former — and this book — reveals the individual ingredients andexplains how they are prepared and combined.Another purpose of the title is to connote an eclectic mixture of presentationaltechniques. This book is unique, we think, in offering, for each topic considered,a certain amount of general discussion, a certain amount of analytical mathematics,a certain amount of discussion of algorithmics, and (most important) actual implementations of these ideas in the form of working computer routines.

Our task hasbeen to find the right balance among these ingredients for each topic. You willfind that for some topics we have tilted quite far to the analytic side; this where wehave felt there to be gaps in the “standard” mathematical training. For other topics,where the mathematical prerequisites are universally held, we have tilted towardsmore in-depth discussion of the nature of the computational algorithms, or towardspractical questions of implementation.We admit, therefore, to some unevenness in the “level” of this book.

About halfof it is suitable for an advanced undergraduate course on numerical computation forscience or engineering majors. The other half ranges from the level of a graduatecourse to that of a professional reference. Most cookbooks have, after all, recipes atvarying levels of complexity. An attractive feature of this approach, we think, is thatthe reader can use the book at increasing levels of sophistication as his/her experiencegrows. Even inexperienced readers should be able to use our most advanced routinesas black boxes. Having done so, we hope that these readers will subsequently goback and learn what secrets are inside.If there is a single dominant theme in this book, it is that practical methodsof numerical computation can be simultaneously efficient, clever, and — important— clear.

The alternative viewpoint, that efficient computational methods mustnecessarily be so arcane and complex as to be useful only in “black box” form,we firmly reject.Our purpose in this book is thus to open up a large number of computationalblack boxes to your scrutiny.

We want to teach you to take apart these black boxesand to put them back together again, modifying them to suit your specific needs.We assume that you are mathematically literate, i.e., that you have the normalmathematical preparation associated with an undergraduate degree in a physicalscience, or engineering, or economics, or a quantitative social science. We assumethat you know how to program a computer. We do not assume that you have anyprior formal knowledge of numerical analysis or numerical methods.The scope of Numerical Recipes is supposed to be “everything up to, butnot including, partial differential equations.” We honor this in the breach: First,we do have one introductory chapter on methods for partial differential equations(Chapter 19).

Second, we obviously cannot include everything else. All the so-called“standard” topics of a numerical analysis course have been included in this book:xvPreface to the First EditionAcknowledgmentsMany colleagues have been generous in giving us the benefit of their numericaland computational experience, in providing us with programs, in commenting onthe manuscript, or in general encouragement.

We particularly wish to thank GeorgeRybicki, Douglas Eardley, Philip Marcus, Stuart Shapiro, Paul Horowitz, BruceMusicus, Irwin Shapiro, Stephen Wolfram, Henry Abarbanel, Larry Smarr, RichardMuller, John Bahcall, and A.G.W. Cameron.We also wish to acknowledge two individuals whom we have never met:Forman Acton, whose 1970 textbook Numerical Methods that Work (New York:Harper and Row) has surely left its stylistic mark on us; and Donald Knuth, both forhis series of books on The Art of Computer Programming (Reading, MA: AddisonWesley), and for TEX, the computer typesetting language which immensely aidedproduction of this book.Research by the authors on computational methods was supported in part bythe U.S. National Science Foundation.October, 1985William H. PressBrian P.

FlannerySaul A. TeukolskyWilliam T. VetterlingSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.

Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).linear equations (Chapter 2), interpolation and extrapolation (Chaper 3), integration(Chaper 4), nonlinear root-finding (Chapter 9), eigensystems (Chapter 11), andordinary differential equations (Chapter 16). Most of these topics have been takenbeyond their standard treatments into some advanced material which we have feltto be particularly important or useful.Some other subjects that we cover in detail are not usually found in the standardnumerical analysis texts.

These include the evaluation of functions and of particularspecial functions of higher mathematics (Chapters 5 and 6); random numbers andMonte Carlo methods (Chapter 7); sorting (Chapter 8); optimization, includingmultidimensional methods (Chapter 10); Fourier transform methods, including FFTmethods and other spectral methods (Chapters 12 and 13); two chapters on thestatistical description and modeling of data (Chapters 14 and 15); and two-pointboundary value problems, both shooting and relaxation methods (Chapter 17).The programs in this book are included in ANSI-standard C. Versions of thebook in FORTRAN, Pascal, and BASIC are available separately.

We have moreto say about the C language, and the computational environment assumed by ourroutines, in §1.1 (Introduction)..