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860 Chapter 19. Partial Differential Equations If f(y = l∆) ≡ fl, then we get An from the inverse formula An = The complete solution to the problem is L−1 1 � fl sin sinh (πnJ/L) πnl L l=1 u = ujl + u H jl (19.4.14) (19.4.15) By adding appropriate terms of the form (19.4.12), we can handle inhomogeneous terms on any boundary surface. A much simpler procedure for handling inhomogeneous terms is to note that whenever boundary terms appear on the left-hand side of (19.0.6), they can be taken over to the right-hand side since they are known. The effective source term is therefore ρjl plus a contribution from the boundary terms. To implement this idea formally, write the solution as u = u ′ + u B (19.4.16) where u ′ = 0 on the boundary, while u B vanishes everywhere except on the boundary. There it takes on the given boundary value. In the above example, the only nonzero values of u B would be The model equation (19.0.3) becomes or, in finite-difference form, u B J,l = fl u ′ j+1,l + u ′ j−1,l + u ′ j,l+1 + u ′ j,l−1 − 4u ′ j,l = (19.4.17) ∇ 2 u ′ = −∇ 2 u B + ρ (19.4.18) − (u B j+1,l + uB j−1,l + uB j,l+1 + uB j,l−1 − 4uB j,l )+∆2 ρj,l (19.4.19) All the u B terms in equation (19.4.19) vanish except when the equation is evaluated at j = J − 1, where u ′ J,l + u′ J−2,l + u′ J−1,l+1 + u′ J−1,l−1 − 4u′ J−1,l = −fl +∆ 2 ρJ−1,l (19.4.20) Thus the problem is now equivalent to the case of zero boundary conditions, except that one row of the source term is modified by the replacement ∆ 2 ρJ−1,l → ∆ 2 ρJ−1,l − fl (19.4.21) The case of Neumann boundary conditions ∇u =0is handled by the cosine expansion (12.3.17): ujl = 2 2 J L J�′′ m=0 L�′′ n=0 �umn cos πjm J cos πln L (19.4.22) 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 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 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).
19.4 Fourier and Cyclic Reduction Methods 861 Here the double prime notation means that the terms for m =0and m = J should be multiplied by 1 2 , and similarly for n =0and n = L. Inhomogeneous terms ∇u = g can be again included by adding a suitable solution of the homogeneous equation, or more simply by taking boundary terms over to the right-hand side. For example, the condition becomes ∂u = g(y) at x =0 (19.4.23) ∂x u1,l − u−1,l 2∆ = gl (19.4.24) where gl ≡ g(y = l∆). Once again we write the solution in the form (19.4.16), where now ∇u ′ =0on the boundary. This time ∇u B takes on the prescribed value on the boundary, but u B vanishes everywhere except just outside the boundary. Thus equation (19.4.24) gives u B −1,l = −2∆gl (19.4.25) All the u B terms in equation (19.4.19) vanish except when j =0: u ′ 1,l + u ′ −1,l + u ′ 0,l+1 + u ′ 0,l−1 − 4u ′ 0,l =2∆gl +∆ 2 ρ0,l (19.4.26) Thus u ′ is the solution of a zero-gradient problem, with the source term modified by the replacement ∆ 2 ρ0,l → ∆ 2 ρ0,l +2∆gl (19.4.27) Sometimes Neumann boundary conditions are handled by using a staggered grid, with the u’s defined midway between zone boundaries so that first derivatives are centered on the mesh points. You can solve such problems using similar techniques to those described above if you use the alternative form of the cosine transform, equation (12.3.23). Cyclic Reduction Evidently the FFT method works only when the original PDE has constant coefficients, and boundaries that coincide with the coordinate lines. An alternative algorithm, which can be used on somewhat more general equations, is called cyclic reduction (CR). We illustrate cyclic reduction on the equation ∂2u ∂x2 + ∂2u ∂y + b(y)∂u + c(y)u = g(x, y) (19.4.28) 2 ∂y This form arises very often in practice from the Helmholtz or Poisson equations in polar, cylindrical, or spherical coordinate systems. More general separable equations are treated in [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 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 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).
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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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10.9 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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and equation (12.0.1) looks like th
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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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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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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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13.6 Linear Prediction and Linear P
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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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.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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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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14.7 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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y(x) 16.1 Runge-Kutta Method 711 1
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} 16.1 Runge-Kutta Method 713 yt=ve
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16.2 Adaptive Stepsize Control for
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16.3 Modified Midpoint Method 723 H
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16.4 Richardson Extrapolation and t
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16.5 Second-Order Conservative Equa
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y 16.6 Stiff Sets of Equations 735
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16.6 Stiff Sets of Equations 737 If
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16.6 Stiff Sets of Equations 739 od
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} 16.6 Stiff Sets of Equations 741
Page 767 and 768:
16.6 Stiff Sets of Equations 743 Co
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16.6 Stiff Sets of Equations 745 Se
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} 16.7 Multistep, Multivalue, and P
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16.7 Multistep, Multivalue, and Pre
Page 775 and 776:
16.7 Multistep, Multivalue, and Pre
Page 777 and 778:
Chapter 17. Two Point Boundary Valu
Page 779 and 780:
equired boundary value 17.0 Introdu
Page 781 and 782:
17.1 The Shooting Method 17.1 The S
Page 783 and 784:
} 17.1 The Shooting Method 759 floa
Page 785 and 786:
17.2 Shooting to a Fitting Point 76
Page 787 and 788:
17.3 Relaxation Methods 763 depende
Page 789 and 790:
X X X X X X X X X X X X X X X X X X
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17.3 Relaxation Methods 767 Next co
Page 793 and 794:
17.3 Relaxation Methods 769 calcula
Page 795 and 796:
} 17.3 Relaxation Methods 771 for (
Page 797 and 798:
17.4 A Worked Example: Spheroidal H
Page 799 and 800:
Page 801 and 802:
Page 803 and 804:
} 17.4 A Worked Example: Spheroidal
Page 805 and 806:
Page 807 and 808:
17.5 Automated Allocation of Mesh P
Page 809 and 810:
17.6 Handling Internal Boundary Con
Page 811 and 812:
17.6 Handling Internal Boundary Con
Page 813 and 814:
18.0 Introduction 789 is called a F
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 823 and 824:
Page 825 and 826:
Page 827 and 828:
f (x) 1 .5 0 −.5 0 18.3 Integral
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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 839 and 840: 18.6 Backus-Gilbert Method 815 nece
Page 841 and 842: 18.6 Backus-Gilbert Method 817 Subs
Page 843 and 844: 18.7 Maximum Entropy Image Restorat
Page 851 and 852: Chapter 19. Partial Differential Eq
Page 853 and 854: 19.0 Introduction 829 possible stab
Page 855 and 856: y L ∆ y1 19.0 Introduction 831 A
Page 857 and 858: 19.0 Introduction 833 So-called rap
Page 859 and 860: 19.1 Flux-Conservative Initial Valu
Page 869 and 870: t or n 19.1 Flux-Conservative Initi
Page 871 and 872: 19.2 Diffusive Initial Value Proble
Page 873 and 874: 19.2 Diffusive Initial Value Proble
Page 875 and 876: and the heuristic stability criteri
Page 877 and 878: 19.3 Initial Value Problems in Mult
Page 879 and 880: 19.3 Initial Value Problems in Mult
Page 881 and 882: 19.4 Fourier and Cyclic Reduction M
Page 883: 19.4 Fourier and Cyclic Reduction M
Page 887 and 888: FACR Method 19.5 Relaxation Methods
Page 889 and 890: 19.5 Relaxation Methods for Boundar
Page 895 and 896: 19.6 Multigrid Methods for Boundary
Page 899 and 900: S S S E γ = 1 19.6 Multigrid Metho
Page 905 and 906: } 19.6 Multigrid Methods for Bounda
Page 911 and 912: #include 19.6 Multigrid Methods fo
Page 913 and 914: Chapter 20. Less-Numerical Algorith
Page 915 and 916: 20.1 Diagnosing Machine Parameters
Page 917 and 918: 20.1 Diagnosing Machine Parameters
Page 919 and 920: (a) G(i) (b) MSB 4 3 2 1 0 LSB i MS
Page 921 and 922: 20.3 Cyclic Redundancy and Other Ch
Page 927 and 928: } 20.4 Huffman Coding and Compressi
Page 929 and 930: 20.4 Huffman Coding and Compression
Page 931 and 932: } 20.4 Huffman Coding and Compressi
Page 933 and 934: } } } 20.4 Huffman Coding and Compr
Page 935 and 936:
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
Page 937 and 938:
20.5 Arithmetic Coding 913 Individu
Page 939 and 940:
20.6 Arithmetic at Arbitrary Precis
Page 941 and 942:
} for (j=n;j>=1;j--) { ireg=u[j]+HI
Page 943 and 944:
#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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