《数值算法》C语言实现：科学计算的艺术

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"《数值算法》(Numerical Recipes in C) 是一本由William H. Press、Brian P. Flannery、Saul A. Teukolsky 和 William T. Vetterling合著的书籍，提供了300多个用C语言实现的数值算法。这本书覆盖了广泛的科学计算主题，包括线性代数、插值与外推、积分、特殊函数、随机数生成、非线性方程求解、傅里叶变换、谱分析、统计学、微分方程求解和编码技术等。书中每个章节包含数学理论、算法讨论、程序实现及技巧，程序可在多种C语言环境下运行。此书适合科研人员、软件开发者以及大学师生作为参考和学习材料。" 《数值算法》一书深入介绍了科学计算中的各种重要算法，以下是其中一些关键知识点的详细说明： 1. **线性方程组的求解**：涵盖了如高斯消元法、LU分解、Cholesky分解等方法，用于解决大型线性系统的求解问题。 2. **逆矩阵和行列式计算**：讲解如何计算矩阵的逆和行列式，这对于理解和解决线性系统至关重要。 3. **多项式和有理函数的内插与外推**：通过拉格朗日插值、牛顿插值、样条插值等方法，对离散数据进行平滑处理和预测。 4. **函数的积分和估值**：介绍不同的数值积分方法，如梯形法则、辛普森法则、高斯积分等，用于近似求解不定积分和定积分。 5. **特殊函数的数值计算**：如伽马函数、贝塞尔函数、勒让德多项式等，这些在物理、工程等领域广泛应用的函数的计算方法。 6. **随机数的产生**：讨论各种随机数生成器，如线性同余法，用于模拟和统计实验。 7. **非线性方程求解**：包括牛顿迭代法、二分法等，解决单变量或多变量的非线性方程。 8. **傅里叶变换和FFT**：快速傅里叶变换(FFT)是信号处理和图像处理中的关键算法，用于频率域分析。 9. **谱分析和小波变换**：提供频谱分析工具，用于识别信号的频率成分；小波变换则允许在时间和频率上局部化分析。 10. **统计描述和数据建模**：涵盖基本统计量计算、假设检验、回归分析等，用于数据分析和预测模型构建。 11. **常微分方程和偏微分方程求解**：介绍欧拉方法、龙格-库塔方法等数值解法，解决动态系统问题。 12. **线性预测和线性预测编码**：在通信和信号处理领域，线性预测模型用于预测序列和压缩数据。 13. **数字滤波**：探讨数字滤波器设计，如IIR和FIR滤波器，用于信号的噪声消除和特征提取。 14. **格雷码和算术码**：在信息编码和通信中，这些编码技术用于提高数据传输的效率和可靠性。本书不仅提供了详尽的理论介绍，还附带了可以直接使用的C语言程序，这使得读者能够立即实践所学，进一步理解和应用这些数值算法。无论是专业科研工作者还是初学者，都能从中受益匪浅。

xviii License Information

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)

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 or CDROMs, visit website

http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).

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

ULTRIX are trademarks of Digital Equipment Corporation. IBM is a trademark of

International Business Machines Corporation. Apple and Macintosh are trademarks

of Apple Computer, Inc. UNIX is a trademark licensed exclusively through X/Open

Co. Ltd. IMSL is a trademark of Visual Numerics, Inc. NAG refers to proprietary

computer software of Numerical Algorithms Group (USA) Inc. PostScript and

Adobe Illustrator are trademarks of Adobe Systems Incorporated. Last, and no doubt

least, Numerical Recipes (when identifying products) is a trademark of Numerical

Recipes Software.

Attributions

The fact that ideas are legally “free as air” in no way supersedes the ethical

requirement that ideas be credited to their known originators. When programs in

this book are based on known sources, whether copyrighted or in the public domain,

published or “handed-down,” we have attempted to give proper attribution. Unfor-

tunately, the lineage of many programs in common circulation is often unclear. We

would be grateful to readers for new or corrected information regarding attributions,

which we will attempt to incorporate in subsequent printings.

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)

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 or CDROMs, visit website

http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).

Computer Programs

by Chapter and Section

1.0 flmoon calculate phases of the moon by date

1.1 julday Julian Day number from calendar date

1.1 badluk Friday the 13th when the moon is full

1.1 caldat calendar date from Julian day number

2.1 gaussj Gauss-Jordan matrix inversion and linear equation

solution

2.3 ludcmp linear equation solution, LU decomposition

2.3 lubksb linear equation solution, backsubstitution

2.4 tridag solution of tridiagonal systems

2.4 banmul multiply vector by band diagonal matrix

2.4 bandec band diagonal systems, decomposition

2.4 banbks band diagonal systems, backsubstitution

2.5 mprove linear equation solution, iterative improvement

2.6 svbksb singular value backsubstitution

2.6 svdcmp singular value decomposition of a matrix

2.6 pythag calculate (a

+ b

)

1/2

without overﬂow

2.7 cyclic solution of cyclic tridiagonal systems

2.7 sprsin convert matrix to sparse format

2.7 sprsax product of sparse matrix and vector

2.7 sprstx product of transpose sparse matrix and vector

2.7 sprstp transpose of sparse matrix

2.7 sprspm pattern multiply two sparse matrices

2.7 sprstm threshold multiply two sparse matrices

2.7 linbcg biconjugate gradient solution of sparse systems

2.7 snrm used by linbcg for vector norm

2.7 atimes used by linbcg for sparse multiplication

2.7 asolve used by linbcg for preconditioner

2.8 vander solve Vandermonde systems

2.8 toeplz solve Toeplitz systems

2.9 choldc Cholesky decomposition

2.9 cholsl Cholesky backsubstitution

2.10 qrdcmp QR decomposition

2.10 qrsolv QR backsubstitution

2.10 rsolv right triangular backsubstitution

2.10 qrupdt update a QR decomposition

2.10 rotate Jacobi rotation used by qrupdt

3.1 polint polynomial interpolation

3.2 ratint rational function interpolation

3.3 spline construct a cubic spline

3.3 splint cubic spline interpolation

3.4 locate search an ordered table by bisection

xix

xx Computer Programs by Chapter and Section

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)

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 or CDROMs, visit website

http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).

3.4 hunt search a table when calls are correlated

3.5 polcoe polynomial coefﬁcients from table of values

3.5 polcof polynomial coefﬁcients from table of values

3.6 polin2 two-dimensional polynomial interpolation

3.6 bcucof construct two-dimensional bicubic

3.6 bcuint two-dimensional bicubic interpolation

3.6 splie2 construct two-dimensional spline

3.6 splin2 two-dimensional spline interpolation

4.2 trapzd trapezoidal rule

4.2 qtrap integrate using trapezoidal rule

4.2 qsimp integrate using Simpson’s rule

4.3 qromb integrate using Romberg adaptive method

4.4 midpnt extended midpoint rule

4.4 qromo integrate using open Romberg adaptive method

4.4 midinf integrate a function on a semi-inﬁnite interval

4.4 midsql integrate a function with lower square-root singularity

4.4 midsqu integrate a function with upper square-root singularity

4.4 midexp integrate a function that decreases exponentially

4.5 qgaus integrate a function by Gaussian quadratures

4.5 gauleg Gauss-Legendre weights and abscissas

4.5 gaulag Gauss-Laguerre weights and abscissas

4.5 gauher Gauss-Hermite weights and abscissas

4.5 gaujac Gauss-Jacobi weights and abscissas

4.5 gaucof quadrature weights from orthogonal polynomials

4.5 orthog construct nonclassical orthogonal polynomials

4.6 quad3d integrate a function over a three-dimensional space

5.1 eulsum sum a series by Euler–van Wijngaarden algorithm

5.3 ddpoly evaluate a polynomial and its derivatives

5.3 poldiv divide one polynomial by another

5.3 ratval evaluate a rational function

5.7 dfridr numerical derivative by Ridders’ method

5.8 chebft ﬁt a Chebyshev polynomial to a function

5.8 chebev Chebyshev polynomial evaluation

5.9 chder derivative of a function already Chebyshev ﬁtted

5.9 chint integrate a function already Chebyshev ﬁtted

5.10 chebpc polynomial coefﬁcients from a Chebyshev ﬁt

5.10 pcshft polynomial coefﬁcients of a shifted polynomial

5.11 pccheb inverse of chebpc; use to economize power series

5.12 pade Pad

e approximant from power series coefﬁcients

5.13 ratlsq rational ﬁt by least-squares method

6.1 gammln logarithm of gamma function

6.1 factrl factorial function

6.1 bico binomial coefﬁcients function

6.1 factln logarithm of factorial function

Computer Programs by Chapter and Section xxi

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)

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 or CDROMs, visit website

http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).

6.1 beta beta function

6.2 gammp incomplete gamma function

6.2 gammq complement of incomplete gamma function

6.2 gser series used by gammp and gammq

6.2 gcf continued fraction used by gammp and gammq

6.2 erff error function

6.2 erffc complementary error function

6.2 erfcc complementary error function, concise routine

6.3 expint exponential integral E

6.3 ei exponential integral Ei

6.4 betai incomplete beta function

6.4 betacf continued fraction used by betai

6.5 bessj0 Bessel function J

6.5 bessy0 Bessel function Y

6.5 bessj1 Bessel function J

6.5 bessy1 Bessel function Y

6.5 bessy Bessel function Y of general integer order

6.5 bessj Bessel function J of general integer order

6.6 bessi0 modiﬁed Bessel function I

6.6 bessk0 modiﬁed Bessel function K

6.6 bessi1 modiﬁed Bessel function I

6.6 bessk1 modiﬁed Bessel function K

6.6 bessk modiﬁed Bessel function K of integer order

6.6 bessi modiﬁed Bessel function I of integer order

6.7 bessjy Bessel functions of fractional order

6.7 beschb Chebyshev expansion used by bessjy

6.7 bessik modiﬁed Bessel functions of fractional order

6.7 airy Airy functions

6.7 sphbes spherical Bessel functions j

and y

6.8 plgndr Legendre polynomials, associated (spherical harmonics)

6.9 frenel Fresnel integrals S(x) and C(x)

6.9 cisi cosine and sine integrals Ci and Si

6.10 dawson Dawson’s integral

6.11 rf Carlson’s elliptic integral of the ﬁrst kind

6.11 rd Carlson’s elliptic integral of the second kind

6.11 rj Carlson’s elliptic integral of the third kind

6.11 rc Carlson’s degenerate elliptic integral

6.11 ellf Legendre elliptic integral of the ﬁrst kind

6.11 elle Legendre elliptic integral of the second kind

6.11 ellpi Legendre elliptic integral of the third kind

6.11 sncndn Jacobian elliptic functions

6.12 hypgeo complex hypergeometric function

6.12 hypser complex hypergeometric function, series evaluation

6.12 hypdrv complex hypergeometric function, derivative of

7.1 ran0 random deviate by Park and Miller minimal standard

7.1 ran1 random deviate, minimal standard plus shufﬂe

xxii Computer Programs by Chapter and Section

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)

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 or CDROMs, visit website

http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).

7.1 ran2 random deviate by L’Ecuyer long period plus shufﬂe

7.1 ran3 random deviate by Knuth subtractive method

7.2 expdev exponential random deviates

7.2 gasdev normally distributed random deviates

7.3 gamdev gamma-law distribution random deviates

7.3 poidev Poisson distributed random deviates

7.3 bnldev binomial distributed random deviates

7.4 irbit1 random bit sequence

7.4 irbit2 random bit sequence

7.5 psdes “pseudo-DES” hashing of 64 bits

7.5 ran4 random deviates from DES-like hashing

7.7 sobseq Sobol’s quasi-random sequence

7.8 vegas adaptive multidimensional Monte Carlo integration

7.8 rebin sample rebinning used by vegas

7.8 miser recursive multidimensional Monte Carlo integration

7.8 ranpt get random point, used by miser

8.1 piksrt sort an array by straight insertion

8.1 piksr2 sort two arrays by straight insertion

8.1 shell sort an array by Shell’s method

8.2 sort sort an array by quicksort method

8.2 sort2 sort two arrays by quicksort method

8.3 hpsort sort an array by heapsort method

8.4 indexx construct an index for an array

8.4 sort3 sort, use an index to sort 3 or more arrays

8.4 rank construct a rank table for an array

8.5 select ﬁnd the N th largest in an array

8.5 selip ﬁnd the N th largest, without altering an array

8.5 hpsel ﬁnd M largest values, without altering an array

8.6 eclass determine equivalence classes from list

8.6 eclazz determine equivalence classes from procedure

9.0 scrsho graph a function to search for roots

9.1 zbrac outward search for brackets on roots

9.1 zbrak inward search for brackets on roots

9.1 rtbis ﬁnd root of a function by bisection

9.2 rtflsp ﬁnd root of a function by false-position

9.2 rtsec ﬁnd root of a function by secant method

9.2 zriddr ﬁnd root of a function by Ridders’ method

9.3 zbrent ﬁnd root of a function by Brent’s method

9.4 rtnewt ﬁnd root of a function by Newton-Raphson

9.4 rtsafe ﬁnd root of a function by Newton-Raphson and bisection

9.5 laguer ﬁnd a root of a polynomial by Laguerre’s method

9.5 zroots roots of a polynomial by Laguerre’s method with

deﬂation

9.5 zrhqr roots of a polynomial by eigenvalue methods

9.5 qroot complex or double root of a polynomial, Bairstow

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