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762 Chapter 17. Two Point Boundary Value Problems } for (i=1;i
17.3 Relaxation Methods 763 dependent variables y1,y2,...,yN at point xk. At an arbitrary point k in the middle of the mesh, we approximate the set of N first-order ODEs by algebraic relations of the form 0=Ek ≡ y k − y k−1 − (xk − xk−1)g k (xk,xk−1, y k , y k−1 ), k =2, 3,...,M (17.3.3) The notation signifies that g k can be evaluated using information from both points k, k − 1. The FDEs labeled by Ek provide N equations coupling 2N variables at points k, k − 1. There are M − 1 points, k =2, 3,...,M, at which difference equations of the form (17.3.3) apply. Thus the FDEs provide a total of (M − 1)N equations for the MN unknowns. The remaining N equations come from the boundary conditions. At the first boundary we have while at the second boundary 0=E1 ≡ B(x1, y 1 ) (17.3.4) 0=EM+1 ≡ C(xM, y M ) (17.3.5) The vectors E1 and B have only n1 nonzero components, corresponding to the n1 boundary conditions at x1. It will turn out to be useful to take these nonzero components to be the last n1 components. In other words, Ej,1 �= 0only for j = n2 +1,n2 +2,...,N. At the other boundary, only the first n2 components of EM+1 and C are nonzero: Ej,M+1 �= 0 only for j =1, 2,...,n2. The “solution” of the FDE problem in (17.3.3)–(17.3.5) consists of a set of variables yj,k, the values of the N variables yj at the M points xk. The algorithm we describe below requires an initial guess for the yj,k. We then determine increments ∆yj,k such that yj,k +∆yj,k is an improved approximation to the solution. Equations for the increments are developed by expanding the FDEs in first-order Taylor series with respect to small changes ∆yk . At an interior point, k =2, 3,...,M this gives: Ek(yk +∆yk , yk−1 +∆yk−1 ) ≈ Ek(yk , yk−1 ) + N� ∂Ek N� ∂Ek ∆yn,k−1 + ∆yn,k ∂yn,k−1 ∂yn,k n=1 n=1 (17.3.6) For a solution we want the updated value E(y +∆y) to be zero, so the general set of equations at an interior point can be written in matrix form as N� 2N� where Sj,n∆yn,k−1 + Sj,n∆yn−N,k = −Ej,k, j =1, 2,...,N (17.3.7) n=1 n=N+1 Sj,n = ∂Ej,k , Sj,n+N = ∂yn,k−1 ∂Ej,k , n =1, 2,...,N (17.3.8) ∂yn,k The quantity Sj,n is an N × 2N matrix at each point k. Each interior point thus supplies a block of N equations coupling 2N corrections to the solution variables at the points k, k − 1. Similarly, the algebraic relations at the boundaries can be expanded in a first-order Taylor series for increments that improve the solution. Since E1 depends only on y1 ,we find at the first boundary: where N� At the second boundary, N� Sj,n∆yn,1 = −Ej,1, j = n2 +1,n2 +2,...,N (17.3.9) n=1 Sj,n = ∂Ej,1 ∂yn,1 , n =1, 2,...,N (17.3.10) Sj,n∆yn,M = −Ej,M+1, j =1, 2,...,n2 (17.3.11) n=1 Sample 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 <strong>Numerical</strong> 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 server computer, is strictly prohibited. To order <strong>Numerical</strong> 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).
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Numerical Recipes in C The Art of S
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Contents Preface to the Second Edit
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Contents vii 7.2 Transformation Met
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Contents ix 15.6 Confidence Limits
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Preface to the Second Edition Our a
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Preface to the Second Edition xiii
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Preface to the First Edition xv lin
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Types of License Offered License In
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Computer Programs by Chapter and Se
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Chapter 1. Preliminaries 1.0 Introd
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1.0 Introduction 3 Tested Machines
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1.1 Program Organization and Contro
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true 1.1 Program Organization and C
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} 1.2 Some C Conventions for Scient
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1.2 Some C Conventions for Scientif
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(a) **m (b) **m 1.2 Some C Conventi
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= 0 3 = 0 = 0 = 0 = 0 = 0 1 ⁄ 2 1
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1.3 Error, Accuracy, and Stability
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2.0 Introduction 33 Accumulated rou
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2.0 Introduction 35 between singula
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2.1 Gauss-Jordan Elimination 37 Eli
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2.1 Gauss-Jordan Elimination 39 pre
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2.2 Gaussian Elimination with Backs
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2.3 LU Decomposition and Its Applic
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a b c d 2.3 LU Decomposition and It
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} 2.3 LU Decomposition and Its Appl
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2.3 LU Decomposition and Its Applic
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2.4 Tridiagonal and Band Diagonal S
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2.4 Tridiagonal and Band Diagonal S
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δx 2.5 Iterative Improvement of a
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2.5 Iterative Improvement of a Solu
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2.6 Singular Value Decomposition 59
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SVD of a Square Matrix 2.6 Singular
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solutions of A ⋅ x = d null space
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} 2.6 Singular Value Decomposition
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2.7 Sparse Linear Systems 2.7 Spars
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2.7 Sparse Linear Systems 73 prescr
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2.7 Sparse Linear Systems 75 Here
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2.7 Sparse Linear Systems 77 Finall
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2.7 Sparse Linear Systems 79 In row
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} 2.7 Sparse Linear Systems 81 jp=i
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2.7 Sparse Linear Systems 83 Matrix
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2.7 Sparse Linear Systems 85 while
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p=dvector(1,n); pp=dvector(1,n); r=
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} 2.7 Sparse Linear Systems 89 void
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2.8 Vandermonde Matrices and Toepli
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2.9 Cholesky Decomposition 97 If yo
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2.10 QR Decomposition 99 The standa
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#include #include "nrutil.h" 2.10
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in terms of which 2.11 Is Matrix In
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Chapter 3. Interpolation and Extrap
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(a) (b) 3.0 Introduction 107 Figure
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3.1 Polynomial Interpolation and Ex
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3.2 Rational Function Interpolation
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3.3 Cubic Spline Interpolation 113
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3.3 Cubic Spline Interpolation 115
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3.4 How to Search an Ordered Table
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} 3.4 How to Search an Ordered Tabl
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#include "nrutil.h" 3.5 Coefficient
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3.6 Interpolation in Two or More Di
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} int j; float *ymtmp; 3.6 Interpol
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3.6 Interpolation in Two or More Di
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Chapter 4. Integration of Functions
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4.1 Classical Formulas for Equally
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#define FUNC(x) ((*func)(x)) 4.2 El
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4.2 Elementary Algorithms 139 trapz
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4.4 Improper Integrals 141 which co
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4.4 Improper Integrals 143 The rout
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4.4 Improper Integrals 145 If you n
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4.5 Gaussian Quadratures and Orthog
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Gauss-Legendre: Gauss-Chebyshev: Ga
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} 4.5 Gaussian Quadratures and Orth
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4.6 Multidimensional Integrals 161
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y 4.6 Multidimensional Integrals 16
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Chapter 5. Evaluation of Functions
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5.1 Series and Their Convergence 16
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5.2 Evaluation of Continued Fractio
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5.2 Evaluation of Continued Fractio
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5.3 Polynomials and Rational Functi
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5.3 Polynomials and Rational Functi
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5.4 Complex Arithmetic 177 Actually
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5.5 Recurrence Relations and Clensh
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5.5 Recurrence Relations and Clensh
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5.6 Quadratic and Cubic Equations 1
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5.6 Quadratic and Cubic Equations 1
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5.7 Numerical Derivatives 187 With
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} 5.7 Numerical Derivatives 189 The
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Chebyshev polynomials 1 .5 0 −.5
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} 5.8 Chebyshev Approximation 193 f
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5.9 Derivatives or Integrals of a C
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5.10 Polynomial Approximation from
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5.11 Economization of Power Series
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5.12 Padé Approximants 201 then R(
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5.12 Padé Approximants 203 cof[2*n
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5.13 Rational Chebyshev Approximati
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5.13 Rational Chebyshev Approximati
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5.14 Evaluation of Functions by Pat
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5.14 Evaluation of Functions by Pat
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6.1 Gamma, Beta, and Related Functi
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6.1 Gamma, Beta, and Related Functi
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incomplete gamma function P(a,x) 1.
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} void nrerror(char error_text[]);
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#include 6.2 Incomplete Gamma Func
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6.3 Exponential Integrals 223 where
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6.3 Exponential Integrals 225 #incl
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6.4 Incomplete Beta Function 227 Th
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6.4 Incomplete Beta Function 229 So
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Bessel functions 1 .5 0 −.5 −1
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} 6.5 Bessel Functions of Integer O
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6.5 Bessel Functions of Integer Ord
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modified Bessel functions 4 3 2 1 6
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6.6 Modified Bessel Functions of In
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6.7 Bessel Functions of Fractional
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} 6.7 Bessel Functions of Fractiona
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6.8 Spherical Harmonics 253 The fir
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6.9 Fresnel Integrals, Cosine and S
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} 6.9 Fresnel Integrals, Cosine and
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} 6.10 Dawson’s Integral 259 } if
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6.11 Elliptic Integrals and Jacobia
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where 6.11 Elliptic Integrals and J
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#define TINY 1.0e-25 #define BIG 4.
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6.11 Elliptic Integrals and Jacobia
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6.12 Hypergeometric Functions 271 C
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#include "complex.h" #define ONE Co
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7.1 Uniform Deviates 275 As for ref
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7.1 Uniform Deviates 277 This corre
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#define IA 16807 #define IM 2147483
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RAN 3 1 7.1 Uniform Deviates 281 iy
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7.1 Uniform Deviates 283 from the w
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7.1 Uniform Deviates 285 Constants
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7.2 Transformation Method: Exponent
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7.2 Transformation Method: Exponent
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first random deviate in 7.3 Rejecti
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} 7.3 Rejection Method: Gamma, Pois
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} 7.3 Rejection Method: Gamma, Pois
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7.4 Generation of Random Bits 297 s
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} } *iseed
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7.5 Random Sequences Based on Data
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} } 7.5 Random Sequences Based on D
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7.6 Simple Monte Carlo Integration
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7.6 Simple Monte Carlo Integration
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7.7 Quasi- (that is, Sub-) Random S
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} 7.7 Quasi- (that is, Sub-) Random
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7.8 Adaptive and Recursive Monte Ca
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} 7.8 Adaptive and Recursive Monte
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Chapter 8. Sorting 8.0 Introduction
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8.1 Straight Insertion and Shell’
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8.2 Quicksort 333 question of the b
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8.2 Quicksort 335 istack=lvector(1,
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a 4 a 2 8.3 Heapsort 337 a 5 a 1 a
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original array 1 2 3 4 5 6 14 8 32
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8.5 Selecting the Mth Largest 341 w
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8.5 Selecting the Mth Largest 343 r
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} } } } 8.6 Determination of Equiva
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Chapter 9. Root Finding and Nonline
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9.0 Introduction 349 good first gue
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(a) (b) (c) (d) 9.1 Bracketing and
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Bisection Method 9.1 Bracketing and
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9.2 Secant Method, False Position M
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} 9.2 Secant Method, False Position
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} 9.3 Van Wijngaarden-Dekker-Brent
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9.3 Van Wijngaarden-Dekker-Brent Me
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9.4 Newton-Raphson Method Using Der
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9.4 Newton-Raphson Method Using Der
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} 9.4 Newton-Raphson Method Using D
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9.5 Roots of Polynomials 9.5 Roots
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9.5 Roots of Polynomials 371 tentat
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9.5 Roots of Polynomials 373 The me
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Eigenvalue Methods 9.5 Roots of Pol
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9.5 Roots of Polynomials 377 of goi
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9.6 Newton-Raphson Method for Nonli
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9.6 Newton-Raphson Method for Nonli
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9.7 Globally Convergent Methods for
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X1 A B C 10.0 Introduction 395 D Fi
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10.1 Golden Section Search in One D
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10.1 Golden Section Search in One D
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} } 10.1 Golden Section Search in O
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10.2 Parabolic Interpolation and Br
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} 10.3 One-Dimensional Search with
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10.3 One-Dimensional Search with Fi
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(a) (b) (c) (d) 10.4 Downhill Simpl
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10.4 Downhill Simplex Method in Mul
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10.5 Direction Set (Powell’s) Met
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(a) 10.6 Conjugate Gradient Methods
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10.6 Conjugate Gradient Methods in
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} 10.7 Variable Metric Methods in M
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10.7 Variable Metric Methods in Mul
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} 10.7 Variable Metric Methods in M
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10.8 Linear Programming and the Sim
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} 10.8 Linear Programming and the S
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10.9 Simulated Annealing Methods 44
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(a) (b) (c) 1 .5 0 1 .5 0 1 .5 0 10
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} } 10.9 Simulated Annealing Method
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} free_ivector(jorder,1,ncity); #in
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Definitions and Basic Facts 11.0 In
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11.0 Introduction 459 In both the d
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11.0 Introduction 461 “Eigenpacka
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11.1 Jacobi Transformations of a Sy
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} } 11.2 Reduction of a Symmetric M
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11.2 Reduction of a Symmetric Matri
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11.2 Reduction of a Symmetric Matri
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11.3 Eigenvalues and Eigenvectors o
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} } 11.4 Hermitian Matrices 481 for
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11.5 Reduction of a General Matrix
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11.5 Reduction of a General Matrix
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11.6 The QR Algorithm for Real Hess
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11.7 Eigenvalues or Eigenvectors by
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11.7 Eigenvalues or Eigenvectors by
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and equation (12.0.1) looks like th
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12.1 Fourier Transform of Discretel
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12.1 Fourier Transform of Discretel
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12.2 Fast Fourier Transform (FFT) 5
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12.2 Fast Fourier Transform (FFT) 5
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Other FFT Algorithms 12.2 Fast Four
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12.3 FFT of Real Functions, Sine an
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12.3 FFT of Real Functions, Sine an
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#include 12.3 FFT of Real Function
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} 12.3 FFT of Real Functions, Sine
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} } } 12.4 FFT in Two or More Dimen
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data [1] Re Im 12.4 FFT in Two or M
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12.5 Fourier Transforms of Real Dat
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} } 12.6 External Storage or Memory
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Chapter 13. Fourier and Spectral Ap
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(t) r*s(t) 13.1 Convolution and Dec
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13.1 Convolution and Deconvolution
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13.1 Convolution and Deconvolution
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13.2 Correlation and Autocorrelatio
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13.3 Optimal (Wiener) Filtering wit
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log scale 13.4 Power Spectrum Estim
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13.4 Power Spectrum Estimation Usin
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amplitude 1 .8 .6 .4 .2 0 13.4 Powe
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13.5 Digital Filtering in the Time
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IIR (Recursive) Filters 13.5 Digita
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(a) (b) 13.5 Digital Filtering in t
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13.6 Linear Prediction and Linear P
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} 13.6 Linear Prediction and Linear
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13.7 Maximum Entropy (All Poles) Me
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13.8 Spectral Analysis of Unevenly
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} 13.8 Spectral Analysis of Unevenl
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13.9 Computing Fourier Integrals Us
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Cubic order: 13.9 Computing Fourier
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13.9 Computing Fourier Integrals Us
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13.10 Wavelet Transforms 591 The sp
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13.10 Wavelet Transforms 593 p.”
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13.10 Wavelet Transforms 595 comple
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} 13.10 Wavelet Transforms 597 wfil
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.2 0 −.2 .2 0 −.2 DAUB4 e10 + e
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wavelet amplitude 1.5 1 .5 .0001 10
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13.10 Wavelet Transforms 605 Figure
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13.11 Numerical Use of the Sampling
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Chapter 14. Statistical Description
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14.1 Moments of a Distribution: Mea
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14.1 Moments of a Distribution: Mea
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14.2 Do Two Distributions Have the
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14.3 Are Two Distributions Differen
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1. male 2. female . 14.4 Contingenc
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14.4 Contingency Table Analysis of
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14.5 Linear Correlation 637 two dis
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14.6 Nonparametric or Rank Correlat
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} } 14.7 Do Two-Dimensional Distrib
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14.7 Do Two-Dimensional Distributio
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14.8 Savitzky-Golay Smoothing Filte
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8 6 4 2 0 8 6 4 2 0 8 6 4 2 0 befor
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14.8 Savitzky-Golay Smoothing Filte
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15.1 Least Squares as a Maximum Lik
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15.1 Least Squares as a Maximum Lik
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15.2 Fitting Data to a Straight Lin
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15.2 Fitting Data to a Straight Lin
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15.3 Straight-Line Data with Errors
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15.3 Straight-Line Data with Errors
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15.4 General Linear Least Squares 6
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} 15.4 General Linear Least Squares
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15.5 Nonlinear Models 681 Lawson, C
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15.5 Nonlinear Models 683 Note that
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15.5 Nonlinear Models 685 pivots, b
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#include "nrutil.h" 15.5 Nonlinear
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15.6 Confidence Limits on Estimated
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actual data set 15.6 Confidence Lim
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15.7 Robust Estimation 699 M� 1 C
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15.7 Robust Estimation 701 where th
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15.7 Robust Estimation 703 algorith
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} *a=aa; *b=bb; *abdev=abdevt/ndata
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Chapter 16. Integration of Ordinary
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16.0 Introduction 709 problem where
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Page 815 and 816: 18.1 Fredholm Equations of the Seco
Page 817 and 818: #include "nrutil.h" 18.1 Fredholm E
Page 819 and 820: 18.2 Volterra Equations 795 in §18
Page 821 and 822: 18.3 Integral Equations with Singul
Page 827 and 828: f (x) 1 .5 0 −.5 0 18.3 Integral
Page 829 and 830: 18.4 Inverse Problems and the Use o
Page 831 and 832: 18.4 Inverse Problems and the Use o
Page 833 and 834: 18.5 Linear Regularization Methods
Page 835 and 836: 18.5 Linear Regularization Methods
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18.5 Linear Regularization Methods
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18.6 Backus-Gilbert Method 815 nece
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18.6 Backus-Gilbert Method 817 Subs
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18.7 Maximum Entropy Image Restorat
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Page 849 and 850:
Page 851 and 852:
Chapter 19. Partial Differential Eq
Page 853 and 854:
19.0 Introduction 829 possible stab
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y L ∆ y1 19.0 Introduction 831 A
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19.0 Introduction 833 So-called rap
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19.1 Flux-Conservative Initial Valu
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Page 863 and 864:
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Page 867 and 868:
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t or n 19.1 Flux-Conservative Initi
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19.2 Diffusive Initial Value Proble
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19.2 Diffusive Initial Value Proble
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and the heuristic stability criteri
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19.3 Initial Value Problems in Mult
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19.3 Initial Value Problems in Mult
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19.4 Fourier and Cyclic Reduction M
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FACR Method 19.5 Relaxation Methods
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19.5 Relaxation Methods for Boundar
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19.6 Multigrid Methods for Boundary
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S S S E γ = 1 19.6 Multigrid Metho
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} 19.6 Multigrid Methods for Bounda
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#include 19.6 Multigrid Methods fo
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Chapter 20. Less-Numerical Algorith
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20.1 Diagnosing Machine Parameters
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20.1 Diagnosing Machine Parameters
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(a) G(i) (b) MSB 4 3 2 1 0 LSB i MS
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20.3 Cyclic Redundancy and Other Ch
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Page 925 and 926:
Page 927 and 928:
} 20.4 Huffman Coding and Compressi
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20.4 Huffman Coding and Compression
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} 20.4 Huffman Coding and Compressi
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} } } 20.4 Huffman Coding and Compr
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1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
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20.5 Arithmetic Coding 913 Individu
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20.6 Arithmetic at Arbitrary Precis
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} for (j=n;j>=1;j--) { ireg=u[j]+HI
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#include "nrutil.h" #define RX 256.
Page 945 and 946:
} 20.6 Arithmetic at Arbitrary Prec
Page 947 and 948:
Page 949:
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