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718 Chapter 16. Integration of Ordinary Differential Equations for the ith equation will be taken to be ∆0 = eps × yscal[i] (16.2.8) If you desire constant fractional errors, plug a pointer to y into the pointer to yscal calling slot (no need to copy the values into a different array). If you desire constant absolute errors relative to some maximum values, set the elements of yscal equal to those maximum values. A useful “trick” for getting constant fractional errors except “very” near zero crossings is to set yscal[i] equal to |y[i]| + |h × dydx[i]|. (The routine odeint, below, does this.) Here is a more technical point. We have to consider one additional possibility for yscal. The error criteria mentioned thus far are “local,” in that they bound the error of each step individually. In some applications you may be unusually sensitive about a “global” accumulation of errors, from beginning to end of the integration and in the worst possible case where the errors all are presumed to add with the same sign. Then, the smaller the stepsize h, the smaller the value ∆ 0 that you will need to impose. Why? Because there will be more steps between your starting and ending values of x. In such cases you will want to set yscal proportional to h, typically to something like ∆0 = ɛh × dydx[i] (16.2.9) This enforces fractional accuracy ɛ not on the values of y but (much more stringently) on the increments to those values at each step. But now look back at (16.2.7). If ∆ 0 has an implicit scaling with h, then the exponent 0.20 is no longer correct: When the stepsize is reduced from a too-large value, the new predicted value h 1 will fail to meet the desired accuracy when yscal is also altered to this new h 1 value. Instead of 0.20 = 1/5, we must scale by the exponent 0.25 = 1/4 for things to work out. The exponents 0.20 and 0.25 are not really very different. This motivates us to adopt the following pragmatic approach, one that frees us from having to know in advance whether or not you, the user, plan to scale your yscal’s with stepsize. Whenever we decrease a stepsize, let us use the larger value of the exponent (whether we need it or not!), and whenever we increase a stepsize, let us use the smaller exponent. Furthermore, because our estimates of error are not exact, but only accurate to the leading order in h, we are advised to put in a safety factor S which is a few percent smaller than unity. Equation (16.2.7) is thus replaced by h0 = ⎧ � � ⎪⎨ Sh1 � � � � ⎪⎩ Sh1 � � ∆0 ∆1 ∆0 ∆1 � � � � � � � � 0.20 0.25 ∆0 ≥ ∆1 ∆0 < ∆1 (16.2.10) We have found this prescription to be a reliable one in practice. Here, then, is a stepper program that takes one “quality-controlled” Runge- Kutta step. 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).
16.2 Adaptive Stepsize Control for Runge-Kutta 719 #include #include "nrutil.h" #define SAFETY 0.9 #define PGROW -0.2 #define PSHRNK -0.25 #define ERRCON 1.89e-4 The value ERRCON equals (5/SAFETY) raised to the power (1/PGROW), see use below. void rkqs(float y[], float dydx[], int n, float *x, float htry, float eps, float yscal[], float *hdid, float *hnext, void (*derivs)(float, float [], float [])) Fifth-order Runge-Kutta step with monitoring of local truncation error to ensure accuracyand adjust stepsize. Input are the dependent variable vector y[1..n] and its derivative dydx[1..n] at the starting value of the independent variable x. Also input are the stepsize to be attempted htry, the required accuracy eps, and the vector yscal[1..n] against which the error is scaled. On output, y and x are replaced bytheir new values, hdid is the stepsize that was actuallyaccomplished, and hnext is the estimated next stepsize. derivs is the user-supplied routine that computes the right-hand side derivatives. { void rkck(float y[], float dydx[], int n, float x, float h, float yout[], float yerr[], void (*derivs)(float, float [], float [])); int i; float errmax,h,htemp,xnew,*yerr,*ytemp; } yerr=vector(1,n); ytemp=vector(1,n); h=htry; Set stepsize to the initial trial value. for (;;) { rkck(y,dydx,n,*x,h,ytemp,yerr,derivs); Take a step. errmax=0.0; Evaluate accuracy. for (i=1;i ERRCON) *hnext=SAFETY*h*pow(errmax,PGROW); else *hnext=5.0*h; No more than a factor of 5 increase. *x += (*hdid=h); for (i=1;i
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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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4.6 Multidimensional Integrals 161
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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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5.3 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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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 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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6.8 Spherical Harmonics 253 The fir
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6.9 Fresnel Integrals, Cosine and S
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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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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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7.7 Quasi- (that is, Sub-) Random S
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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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a 4 a 2 8.3 Heapsort 337 a 5 a 1 a
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8.5 Selecting the Mth Largest 341 w
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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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9.5 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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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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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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15.2 Fitting Data to a Straight Lin
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Page 787 and 788: 17.3 Relaxation Methods 763 depende
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Page 791 and 792: 17.3 Relaxation Methods 767 Next co
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17.3 Relaxation Methods 769 calcula
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} 17.3 Relaxation Methods 771 for (
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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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17.5 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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