Numerical Recipes in C: 科学计算的艺术

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"Numerical Recipes in C - The Art of Scientific Computing" 本书《Numerical Recipes in C》是科学计算领域的经典著作，由William H. Press、Saul A. Teukolsky、William T. Vetterling和Brian P. Flannery四位作者共同撰写。他们分别来自哈佛-史密森天体物理中心、康奈尔大学物理系、宝丽来公司和埃克森研究与工程公司，具有深厚的学术背景和实践经验。这本书第二版主要关注用C语言进行数值计算的方法和技术。数值计算是计算机科学与数学交叉的一个重要领域，它涉及到在实际问题中利用计算机对数学模型进行求解。C语言因其高效和灵活性，成为编写数值算法的理想选择。书中的内容可能涵盖了广泛的科学计算主题，包括但不限于： 1. 数值线性代数：如矩阵运算、求解线性方程组、特征值和特征向量的计算等。 2. 插值和拟合：如多项式插值、样条插值、最小二乘法拟合等方法，用于数据处理和预测。 3. 微积分和数值积分：包括高斯积分、辛方法等，用于求解连续函数的积分。 4. 常微分方程（ODE）求解：如欧拉方法、龙格-库塔方法等，常用于模拟动态系统。 5. 最优化问题：如梯度下降、牛顿法、遗传算法等，用于寻找函数的极值。 6. 随机数生成和统计分析：在模拟实验或概率模型中必不可少。 7. 图像处理和信号处理：可能包含傅立叶变换、小波分析等相关内容。书中提供的程序代码是C语言实现的，旨在帮助读者理解并应用这些算法。值得注意的是，尽管本书允许互联网用户为个人学习目的打印一份纸质副本，但复制机器可读文件或在任何服务器计算机上复制是被严格禁止的。欲购买《Numerical Recipes》系列图书、磁盘或CD-ROM，可以通过访问官方网址或联系出版社进行订购。《Numerical Recipes in C》是一部深入浅出的科学计算指南，适合于科研工作者、工程师以及对数值计算感兴趣的学者和学生使用。通过本书，读者不仅可以学习到C语言编程技能，还能掌握如何利用计算机解决各种科学和工程问题的数值方法。

xviii

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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.

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

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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).

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

Computer Programs by Chapter and Section

World Wide Web 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,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).

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

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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

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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

World Wide Web 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,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).

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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