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756 Chapter 17. Two Point Boundary Value Problems Problems Reducible to the Standard Boundary Problem There are two important problems that can be reduced to the standard boundary value problem described by equations (17.0.1) – (17.0.3). The first is the eigenvalue problem for differential equations. Here the right-hand side of the system of differential equations depends on a parameter λ, dyi(x) dx = gi(x, y1,...,yN ,λ) (17.0.4) and one has to satisfy N +1boundary conditions instead of just N. The problem is overdetermined and in general there is no solution for arbitrary values of λ. For certain special values of λ, the eigenvalues, equation (17.0.4) does have a solution. We reduce this problem to the standard case by introducing a new dependent variable and another differential equation yN+1 ≡ λ (17.0.5) dyN+1 dx =0 (17.0.6) An example of this trick is given in §17.4. The other case that can be put in the standard form is a free boundary problem. Here only one boundary abscissa x1 is specified, while the other boundary x2 is to be determined so that the system (17.0.1) has a solution satisfying a total of N +1 boundary conditions. Here we again add an extra constant dependent variable: yN+1 ≡ x2 − x1 dyN+1 dx We also defineanewindependent variable t by setting (17.0.7) =0 (17.0.8) x − x1 ≡ tyN+1, 0 ≤ t ≤ 1 (17.0.9) The system of N +1differential equations for dy i/dt is now in the standard form, with t varying between the known limits 0 and 1. CITED REFERENCES AND FURTHER READING: Keller, H.B. 1968, Numerical Methods for Two-Point Boundary-Value Problems (Waltham, MA: Blaisdell). Kippenhan, R., Weigert, A., and Hofmeister, E. 1968, in Methods in Computational Physics, vol. 7 (New York: Academic Press), pp. 129ff. Eggleton, P.P. 1971, Monthly Notices of the Royal Astronomical Society, vol. 151, pp. 351–364. London, R.A., and Flannery, B.P. 1982, Astrophysical Journal, vol. 258, pp. 260–269. Stoer, J., and Bulirsch, R. 1980, Introduction to Numerical Analysis (New York: Springer-Verlag), §§7.3–7.4. 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).
17.1 The Shooting Method 17.1 The Shooting Method 757 In this section we discuss “pure” shooting, where the integration proceeds from x1 to x2, and we try to match boundary conditions at the end of the integration. In the next section, we describe shooting to an intermediate fitting point, where the solution to the equations and boundary conditions is found by launching “shots” from both sides of the interval and trying to match continuity conditions at some intermediate point. Our implementation of the shooting method exactly implements multidimensional, globally convergent Newton-Raphson (§9.7). It seeks to zero n 2 functions of n2 variables. The functions are obtained by integrating N differential equations from x1 to x2. Let us see how this works: At the starting point x1 there are N starting values yi to be specified, but subject to n1 conditions. Therefore there are n2 = N − n1 freely specifiable starting values. Let us imagine that these freely specifiable values are the components of a vector V that lives in a vector space of dimension n 2. Then you, the user, knowing the functional form of the boundary conditions (17.0.2), can write a function that generates a complete set of N starting values y, satisfying the boundary conditions at x1, from an arbitrary vector value of V in which there are no restrictions on the n 2 component values. In other words, (17.0.2) converts to a prescription yi(x1) =yi(x1; V1,...,Vn2) i =1,...,N (17.1.1) Below, the function that implements (17.1.1) will be called load. Notice that the components of V might be exactly the values of certain “free” components of y, with the other components of y determined by the boundary conditions. Alternatively, the components of V might parametrize the solutions that satisfy the starting boundary conditions in some other convenient way. Boundary conditions often impose algebraic relations among the y i, rather than specific values for each of them. Using some auxiliary set of parameters often makes it easier to “solve” the boundary relations for a consistent set of y i’s. It makes no difference which way you go, as long as your vector space of V’s generates (through 17.1.1) all allowed starting vectors y. Given a particular V, a particular y(x1) is thus generated. It can then be turned into a y(x2) by integrating the ODEs to x2 as an initial value problem (e.g., using Chapter 16’s odeint). Now, at x2, let us define a discrepancy vector F, also of dimension n2, whose components measure how far we are from satisfying the n 2 boundary conditions at x2 (17.0.3). Simplest of all is just to use the right-hand sides of (17.0.3), Fk = B2k(x2, y) k =1,...,n2 (17.1.2) As in the case of V, however, you can use any other convenient parametrization, as long as your space of F’s spans the space of possible discrepancies from the desired boundary conditions, with all components of F equal to zero if and only if the boundary conditions at x2 are satisfied. Below, you will be asked to supply a user-written function score which uses (17.0.3) to convert an N-vector of ending values y(x2) into an n2-vector of discrepancies F. 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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5.14 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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6.8 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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7.5 Random Sequences Based on Data
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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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7.8 Adaptive and Recursive Monte Ca
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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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} } } } 8.6 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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10.2 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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10.7 Variable Metric Methods in Mul
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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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} } 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 Matri
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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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and equation (12.0.1) looks like th
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12.2 Fast Fourier Transform (FFT) 5
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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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13.9 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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.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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13.11 Numerical Use of the Sampling
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Chapter 14. Statistical Description
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14.4 Contingency Table Analysis of
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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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15.5 Nonlinear Models 681 Lawson, C
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15.5 Nonlinear Models 683 Note that
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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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Page 759 and 760: y 16.6 Stiff Sets of Equations 735
Page 761 and 762: 16.6 Stiff Sets of Equations 737 If
Page 763 and 764: 16.6 Stiff Sets of Equations 739 od
Page 765 and 766: } 16.6 Stiff Sets of Equations 741
Page 767 and 768: 16.6 Stiff Sets of Equations 743 Co
Page 769 and 770: 16.6 Stiff Sets of Equations 745 Se
Page 771 and 772: } 16.7 Multistep, Multivalue, and P
Page 773 and 774: 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: equired boundary value 17.0 Introdu
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
Page 791 and 792: 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 803 and 804: } 17.4 A Worked Example: Spheroidal
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 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:
Page 837 and 838:
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 845 and 846:
Page 847 and 848:
Page 849 and 850:
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 861 and 862:
Page 863 and 864:
Page 865 and 866:
Page 867 and 868:
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 and 884:
Page 885 and 886:
Page 887 and 888:
FACR Method 19.5 Relaxation Methods
Page 889 and 890:
19.5 Relaxation Methods for Boundar
Page 891 and 892:
Page 893 and 894:
Page 895 and 896:
19.6 Multigrid Methods for Boundary
Page 897 and 898:
Page 899 and 900:
S S S E γ = 1 19.6 Multigrid Metho
Page 901 and 902:
Page 903 and 904:
Page 905 and 906:
} 19.6 Multigrid Methods for Bounda
Page 907 and 908:
Page 909 and 910:
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 923 and 924:
Page 925 and 926:
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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