《Fortran数值算法宝典》：科学计算入门与参考

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《Numerical Recipes in Fortran》是一本经典的计算机数值算法指南，由William H. Press、Saul A. Teukolsky、William T. Vetterling、Brian P. Flannery和Exxon Research and Engineering Company的Flannery共同编著。该书专为科学计算领域的专业人士设计，特别是对于那些使用Fortran编程语言进行数值分析和解决问题的工程师和研究人员而言，它是一部不可或缺的参考书籍。该书的核心内容涵盖了广泛而深入的数值方法，包括但不限于求解微分方程、插值、数值积分、优化算法、统计分析、线性代数以及数值稳定性等内容。它将复杂的数学理论与实际编程技巧相结合，通过清晰易懂的语言和实用的示例，帮助读者理解和掌握数值计算的基本原理和技术。在第二版中，作者对原有的Fortran 77内容进行了更新，同时也增添了适应当时技术发展需求的部分，可能涉及了Fortran 90/95的新特性。书中强调实践性，提供了大量的程序代码，这些代码不仅能够帮助读者验证理论，而且可以直接用于他们的实际项目中，节省了大量的开发时间。版权方面，除了部分章节曾以不同形式发表于《Computers in Physics》杂志，享有美国物理学会的版权外，大部分内容受到Cambridge University Press的保护，除非特别标明，其余计算机程序和算法享有Numerical Recipes Software的版权。所有权利被保留，未经许可不得复制或再版。《Numerical Recipes in Fortran》不仅是初学者学习数值计算的入门教材，也是经验丰富的专业人士查阅和参考的重要资料库。无论是解决科研中的复杂问题，还是提升编程技能，这本书都是一个宝贵的资源，它将理论与实践完美结合，对于推动科学研究和工程应用的发展起到了关键作用。

Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X)

Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine-

readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs

visit 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).

Preface to the First Edition

We call thisbook Numerical Recipes for several reasons. In one sense, thisbook

is indeed a “cookbook” on numerical computation. However there is an important

distinction between a cookbook and a restaurant menu. The latter presents choices

among complete dishes in each of which the individual ﬂavors are blended and

disguised. The former — and this book — reveals the individual ingredients and

explains how they are prepared and combined.

Another purpose of the title is to connote an eclectic mixture of presentational

techniques. 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 imple-

mentations of these ideas in the form of working computer routines. Our task has

been to ﬁnd the right balance among these ingredients for each topic. You will

ﬁnd that for some topics we have tilted quite far to the analytic side; this where we

have felt there to be gaps in the “standard” mathematical training. For other topics,

where the mathematical prerequisites are universally held, we have tilted towards

more in-depth discussion of the nature of the computational algorithms, or towards

practical questions of implementation.

We admit, therefore, to some unevenness in the “level” of this book. About half

of it is suitable for an advanced undergraduate course on numerical computation for

science or engineering majors. The other half ranges from the level of a graduate

course to that of a professional reference. Most cookbooks have, after all, recipes at

varying levels of complexity. An attractive feature of thisapproach, we think, is that

the reader can use the book at increasinglevels of sophisticationas his/her experience

grows. Even inexperienced readers should be able to use our most advanced routines

as black boxes. Having done so, we hope that these readers will subsequently go

back and learn what secrets are inside.

If there is a single dominant theme in this book, it is that practical methods

of numerical computation can be simultaneously efﬁcient, clever, and — important

— clear. The alternative viewpoint, that efﬁcient computational methods must

necessarily 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 computational

black boxes to your scrutiny. We want to teach you to take apart these black boxes

and 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 normal

mathematical preparation associated with an undergraduate degree in a physical

science, or engineering, or economics, or a quantitative social science. We assume

that you know how to program a computer. We do not assume that you have any

prior formal knowledge of numerical analysis or numerical methods.

The scope of Numerical Recipes is supposed to be “everything up to, but

not 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 obviouslycannot includeeverything else. All the so-called

“standard” topics of a numerical analysis course have been included in this book:

xviii

Preface to the First Edition

xix

Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X)

Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine-

readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs

visit 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), and

ordinary differential equations (Chapter 16). Most of these topics have been taken

beyond their standard treatments into some advanced material which we have felt

to be particularly important or useful.

Some other subjects that we cover in detail are not usually found in the standard

numerical analysis texts. These include the evaluation of functions and of particular

special functions of higher mathematics (Chapters 5 and 6); random numbers and

Monte Carlo methods (Chapter 7); sorting (Chapter 8); optimization, including

multidimensional methods (Chapter 10); Fourier transform methods, including FFT

methods and other spectral methods (Chapters 12 and 13); two chapters on the

statistical description and modeling of data (Chapters 14 and 15); and two-point

boundary value problems, both shooting and relaxation methods (Chapter 17).

The programsin thisbookareincludedinANSI-standardFORTRAN-77. Versions

of the book in C, Pascal,andBASIC are available separately. We have more to

say about the FORTRAN language, and the computational environment assumed by

our routines, in §1.1 (Introduction).

Acknowledgments

Many colleagues have been generous in giving us the beneﬁt of their numerical

and computational experience, in providing us with programs, in commenting on

the manuscript, or in general encouragement. We particularly wish to thank George

Rybicki, Douglas Eardley, Philip Marcus, Stuart Shapiro, Paul Horowitz, Bruce

Musicus, Irwin Shapiro, Stephen Wolfram, Henry Abarbanel, Larry Smarr, Richard

Muller, 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 for

his series of books on The Art of Computer Programming (Reading, MA: Addison-

Wesley), and for T

X, the computer typesetting language which immensely aided

production of this book.

Research by the authors on computational methods was supported in part by

the U.S. National Science Foundation.

October, 1985 William H. Press

Brian P. Flannery

Saul A. Teukolsky

William T. Vetterling

Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X)

Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine-

readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs

visit 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).

License Information

Read this section if you want to use the programs in this book on a computer.

You’ll need to read the followingDisclaimer of Warranty, get the programsonto your

computer, and acquire a Numerical Recipes software license. (Without this license,

which can be the free “immediate license” under terms described below, the book is

intended as a text and reference book, for reading purposes only.)

Disclaimer of Warranty

We make no warranties, express or implied, that the programs contained

in this volume are free of error, or are consistent with any particular standard

of merchantability, or that they will meet your requirements for any particular

application. They should not be relied on for solving a problem whose incorrect

solution could result in injury to a person or loss of property. If you do use the

programs in such a manner, it is at your own risk. The authors and publisher

disclaim all liability for direct or consequential damages resulting from your

use of the programs.

How to Get the Code onto Your Computer

Pick one of the following methods:

• You can type the programs from this book directly into your computer. In

this case, the only kind of license available to you is the free “immediate

license” (see below). You are not authorized to transfer or distribute a

machine-readable copy to any other person, nor to have any other person

type the programs into a computer on your behalf. We do not want to hear

bug reports from you if you choose this option, because experience has

shown that virtually all reported bugs in such cases are typing errors!

• You can download the Numerical Recipes programs electronically from

the Numerical Recipes On-Line Software Store, located at our Web site

(http://www.nr.com). They are packaged as a password-protected

ﬁle, and you’ll need to purchase a license to unpack them. You can

get a single-screen license and password immediately, on-line, from the

On-Line Store, with fees ranging from $50 (PC, Macintosh, educational

institutions’ UNIX) to $140 (general UNIX). Downloading the packaged

software from the On-Line Store is also the way to start if you want to

acquire a more general (multiscreen, site, or corporate) license.

License Information

xxi

Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X)

Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine-

readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs

visit 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).

• You can purchase media containing the programs from Cambridge Uni-

versity Press. Diskette versions are available in IBM-compatible format

for machines running Windows 3.1, 95, or NT. CDROM versions in ISO-

9660 format for PC, Macintosh, and UNIX systems are also available;

these include both Fortran and C versions (as well as versions in Pascal

and BASIC from the ﬁrst edition) on a single CDROM. Diskettes pur-

chased from Cambridge University Press include a single-screen license

for PC or Macintosh only. The CDROM is available with a single-

screen license for PC or Macintosh (order ISBN 0 521 576083), or (at a

slightly higher price) with a single-screen license for UNIX workstations

(order ISBN 0 521 576075). Orders for media from Cambridge Univer-

sity Press can be placed at 800 872-7423 (North America only) or by

email to orders@cup.org (North America) or trade@cup.cam.ac.uk (rest

of world). Or, visit the Web sites http://www.cup.org (North America)

or http://www.cup.cam.ac.uk (rest of world).

Types of License Offered

Here are the types of licenses that we offer. Note that some types are

automatically acquired with the purchase of media from Cambridge University

Press, or of an unlocking password from the Numerical Recipes On-Line Software

Store, while other types of licenses require that you communicate speciﬁcally with

Numerical Recipes Software (email: orders@nr.com or fax: 781 863-1739). Our

Web site http://www.nr.com has additional information.

• [“Immediate License”] If you are the individual owner of a copy of this

book and you type one or more of its routines into your computer, we

authorize you to use them on that computer for your own personal and

noncommercial purposes. You are not authorized to transfer or distribute

machine-readable copies to any other person, or to use the routines on

more than one machine, or to distribute executable programs containing

our routines. This is the only free license.

• [“Single-Screen License”] This is the most common type of low-cost

license, with terms governed by our Single Screen (Shrinkwrap) License

document (complete terms available through our Web site). Basically, this

license lets you use Numerical Recipes routines on any one screen (PC,

workstation, X-terminal, etc.). You may also, under this license, transfer

pre-compiled, executable programs incorporating our routines to other,

unlicensed, screens or computers, providing that (i) your application is

noncommercial (i.e., does not involve the selling of your program for a

fee), (ii) the programs were ﬁrst developed, compiled, and successfully

run on a licensed screen, and (iii)our routines are bound into the programs

in such a manner that they cannot be accessed as individual routines and

cannot practicably be unbound and used in other programs. That is, under

this license, your program user must not be able to use our programs as

part of a program library or “mix-and-match” workbench. Conditions for

other types of commercial or noncommercial distribution may be found

on our Web site (http://www.nr.com).

xxii

License Information

Sample page from NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X)

Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine-

readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs

visit 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).

• [“Multi-Screen, Server, Site, and Corporate Licenses”] The terms of

the Single Screen License can be extended to designated groups of

machines, deﬁned by number of screens, number of machines, locations,

or ownership. Signiﬁcant discounts from the corresponding single-screen

prices are available when the estimated number of screens exceeds 40.

Contact Numerical Recipes Software (email: orders@nr.com or fax: 781

863-1739) for details.

• [“Course Right-to-Copy License”] Instructors at accredited educational

institutions who have adopted this book for a course, and who have

already purchased a Single Screen License (either acquired with the

purchase of media, or from the Numerical Recipes On-Line Software

Store), may license the programs for use in that course as follows: Mail

your name, title, and address; the course name, number, dates, and

estimated enrollment; and advance payment of $5 per (estimated) student

to Numerical Recipes Software, at this address: P.O. Box 243, Cambridge,

MA 02238 (USA). You will receive by return mail a license authorizing

you to make copies of the programs for use by your students, and/or to

transfer the programs to a machine accessible to your students (but only

for the duration of the course).

About Copyrights on Computer Programs

Like artistic or literary compositions, computer programs are protected by

program from a copyrighted source. (It is also not a friendly thing to do, since it

deprives the program’s author of compensation for his or her creative effort.) Under

computer language) also come under the same copyright as the original work.

a particular form. In the case of a computer program, the ideas consist of the

program’s methodology and algorithm, including the necessary sequence of steps

adopted by the programmer. The expression of those ideas is the program source

code (particularly any arbitrary or stylistic choices embodied in it), its derived object

code, and any other derivative works.

If you analyze the ideas contained in a program, and then express those

ideas in your own completely different implementation, then that new program

implementation belongs to you. That is what we have done for those programs in

this book that are not entirely of our own devising. When programs in this book are

said to be “based” on programs published in copyright sources, we mean that the

ideas are the same. The expression of these ideas as source code is our own. We

believe that no material in this book infringes on an existing copyright.

Trademarks

Several registered trademarks appear within the text of this book: Sun is a

trademark of Sun Microsystems, Inc. SPARC and SPARCstation are trademarks of

SPARC International, Inc. Microsoft, Windows 95, Windows NT, PowerStation,

and MS are trademarks of Microsoft Corporation. DEC, VMS, Alpha AXP, and

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