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726 Chapter 16. Integration of Ordinary Differential Equations — not at all infinitesimal — distance. That single step is a grand leap consisting of many (e.g., dozens to hundreds) substeps of modified midpoint method, which are then extrapolated to zero stepsize. The sequence of separate attempts to cross the interval H is made with increasing values of n, the number of substeps. Bulirsch and Stoer originally proposed the sequence n =2, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96,...,[nj =2nj−2],... (16.4.1) More recent work by Deuflhard [2,3] suggests that the sequence n =2, 4, 6, 8, 10, 12, 14,...,[nj =2j],... (16.4.2) is usually more efficient. For each step, we do not know in advance how far up this sequence we will go. After each successive n is tried, a polynomial extrapolation is attempted. That extrapolation gives both extrapolated values and error estimates. If the errors are not satisfactory, we go higher in n. If they are satisfactory, we go on to the next step and begin anew with n =2. Of course there must be some upper limit, beyond which we conclude that there is some obstacle in our path in the interval H, so that we must reduce H rather than just subdivide it more finely. In the implementations below, the maximum number of n’s to be tried is called KMAXX. For reasons described below we usually take this equal to 8; the 8th value of the sequence (16.4.2) is 16, so this is the maximum number of subdivisions of H that we allow. We enforce error control, as in the Runge-Kutta method, by monitoring internal consistency, and adapting stepsize to match a prescribed bound on the local truncation error. Each new result from the sequence of modified midpoint integrations allows a tableau like that in §3.1 to be extended by one additional set of diagonals. The size of the new correction added at each stage is taken as the (conservative) error estimate. How should we use this error estimate to adjust the stepsize? The best strategy now known is due to Deuflhard [2,3]. For completeness we describe it here: Suppose the absolute value of the error estimate returned from the kth column (and hence the k +1st row) of the extrapolation tableau is ɛk+1,k. Error control is enforced by requiring ɛk+1,k
16.4 Richardson Extrapolation and the Bulirsch-Stoer Method 727 The work per unit step to get column k is Ak+1/Hk, which we nondimensionalize with a factor of H and write as Wk = Ak+1 Hk H (16.4.7) � � ɛk+1,k 1/(2k+1) = Ak+1 (16.4.8) ɛ The quantities Wk can be calculated during the integration. The optimal column index q is then defined by Wq = min Wk k=1,...,kf (16.4.9) where kf is the final column, in which the error criterion (16.4.3) was satisfied. The q determined from (16.4.9) defines the stepsize Hq to be used as the next basic stepsize, so that we can expect to get convergence in the optimal column q. Two important refinements have to be made to the strategy outlined so far: • If the current H is “too small,” then kf will be “too small,” and so q remains “too small.” It may be desirable to increase H and aim for convergence in a column q > kf. • If the current H is “too big,” we may not converge at all on the current step and we will have to decrease H. We would like to detect this by monitoring the quantities ɛk+1,k for each k so we can stop the current step as soon as possible. Deuflhard’s prescription for dealing with these two problems uses ideas from communication theory to determine the “average expected convergence behavior” of the extrapolation. His model produces certain correction factors α(k, q) by which Hk is to be multiplied to try to get convergence in column q. The factors α(k, q) depend only on ɛ and the sequence {ni} and so can be computed once during initialization: α(k, q) =ɛ Ak+1−Aq+1 (2k+1)(Aq+1 −A1 +1) for kAq+2 (16.4.13) During initialization, this inequality can be checked for q =1, 2,...to determine kmax, the largest allowed column. Then when (16.4.12) is satisfied it will always be efficient to use Hq+1. (In practice we limit kmax to 8 even when ɛ is very small as there is very little further gain in efficiency whereas roundoff can become a problem.) The problem of stepsize reduction is handled by computing stepsize estimates > Aq+2 Hq+1 ¯Hk ≡ Hkα(k, q), k =1,...,q− 1 (16.4.14) during the current step. The ¯ H’s are estimates of the stepsize to get convergence in the optimal column q. Ifany ¯ Hk is “too small,” we abandon the current step and restart using ¯ Hk. The criterion of being “too small” is taken to be The α’s satisfy α(k, q +1) >α(k, q). Hkα(k, q +1)
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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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6.8 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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9.4 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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10.4 Downhill Simplex Method 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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and equation (12.0.1) looks like th
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13.1 Convolution and Deconvolution
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log scale 13.4 Power Spectrum Estim
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13.9 Computing Fourier Integrals Us
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Cubic order: 13.9 Computing Fourier
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13.10 Wavelet Transforms 591 The sp
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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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15.1 Least Squares as a Maximum Lik
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15.2 Fitting Data to a Straight Lin
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Page 795 and 796: } 17.3 Relaxation Methods 771 for (
Page 797 and 798: 17.4 A Worked Example: Spheroidal H
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17.4 A Worked Example: Spheroidal H
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} 17.4 A Worked Example: Spheroidal
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17.4 A Worked Example: Spheroidal H
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17.5 Automated Allocation of Mesh P
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17.6 Handling Internal Boundary Con
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17.6 Handling Internal Boundary Con
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18.0 Introduction 789 is called a F
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18.1 Fredholm Equations of the Seco
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#include "nrutil.h" 18.1 Fredholm E
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18.2 Volterra Equations 795 in §18
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18.3 Integral Equations with Singul
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f (x) 1 .5 0 −.5 0 18.3 Integral
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18.4 Inverse Problems and the Use o
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18.4 Inverse Problems and the Use o
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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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Chapter 19. Partial Differential Eq
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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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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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} 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.
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} 20.6 Arithmetic at Arbitrary Prec
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