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248 Chapter 6. Special Functions #include #define EPS 1.0e-10 #define FPMIN 1.0e-30 #define MAXIT 10000 #define XMIN 2.0 #define PI 3.141592653589793 void bessik(float x, float xnu, float *ri, float *rk, float *rip, float *rkp) Returns the modified Bessel functions ri = Iν, rk = Kν and their derivatives rip = I ′ ν, rkp = K ′ ν ,forpositivexand for xnu = ν ≥ 0. The relative accuracy is within one or two significant digits of EPS. FPMIN is a number close to the machine’s smallest floating-point number. All internal arithmetic is in double precision. To convert the entire routine to double precision, change the float declarations above to double and decrease EPS to 10−16 . Also convert the function beschb. { void beschb(double x, double *gam1, double *gam2, double *gampl, double *gammi); void nrerror(char error_text[]); int i,l,nl; double a,a1,b,c,d,del,del1,delh,dels,e,f,fact,fact2,ff,gam1,gam2, gammi,gampl,h,p,pimu,q,q1,q2,qnew,ril,ril1,rimu,rip1,ripl, ritemp,rk1,rkmu,rkmup,rktemp,s,sum,sum1,x2,xi,xi2,xmu,xmu2; if (x =1;l--) { ritemp=fact*ril+ripl; fact -= xi; ripl=fact*ritemp+ril; ril=ritemp; } f=ripl/ril; Now have unnormalized Iµ and I ′ µ. if (x < XMIN) { Use series. x2=0.5*x; pimu=PI*xmu; fact = (fabs(pimu) < EPS ? 1.0 : pimu/sin(pimu)); d = -log(x2); e=xmu*d; fact2 = (fabs(e) < EPS ? 1.0 : sinh(e)/e); beschb(xmu,&gam1,&gam2,&gampl,&gammi); Chebyshev evaluation of Γ1 and Γ2. ff=fact*(gam1*cosh(e)+gam2*fact2*d); f0. 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).
} 6.7 Bessel Functions of Fractional Order 249 sum=ff; e=exp(e); p=0.5*e/gampl; p0. q=0.5/(e*gammi); q0. c=1.0; d=x2*x2; sum1=p; for (i=1;i MAXIT) nrerror("bessk series failed to converge"); rkmu=sum; rk1=sum1*xi2; } else { Evaluate CF2 by Steed’s algorithm b=2.0*(1.0+x); d=1.0/b; h=delh=d; (§5.2), which is OK because there can be no zero denominators. q1=0.0; Initializations for recurrence (6.7.35). q2=1.0; a1=0.25-xmu2; q=c=a1; First term in equation (6.7.34). a = -a1; s=1.0+q*delh; for (i=2;i MAXIT) nrerror("bessik: failure to converge in cf2"); h=a1*h; rkmu=sqrt(PI/(2.0*x))*exp(-x)/s; Omit the factor exp(−x) to scale rk1=rkmu*(xmu+x+0.5-h)*xi; all the returned functions by exp(x) } for x ≥ XMIN. rkmup=xmu*xi*rkmu-rk1; rimu=xi/(f*rkmu-rkmup); Get Iµ from Wronskian. *ri=(rimu*ril1)/ril; Scale original Iν and I ′ ν. *rip=(rimu*rip1)/ril; 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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} 4.5 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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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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} 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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} } } } 8.6 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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} 9.3 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 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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9.7 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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10.7 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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11.5 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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12.1 Fourier Transform of Discretel
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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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} 12.3 FFT of Real Functions, Sine
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Chapter 13. Fourier and Spectral Ap
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(t) r*s(t) 13.1 Convolution and Dec
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13.1 Convolution and Deconvolution
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13.1 Convolution and Deconvolution
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13.2 Correlation and Autocorrelatio
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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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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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} 13.10 Wavelet Transforms 597 wfil
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.2 0 −.2 .2 0 −.2 DAUB4 e10 + e
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wavelet amplitude 1.5 1 .5 .0001 10
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} nprev=nnew; } free_vector(wksp,1,
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13.10 Wavelet Transforms 605 Figure
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13.11 Numerical Use of the Sampling
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Chapter 14. Statistical Description
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14.1 Moments of a Distribution: Mea
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14.5 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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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
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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
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16.7 Multistep, Multivalue, and Pre
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Chapter 17. Two Point Boundary Valu
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equired boundary value 17.0 Introdu
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17.1 The Shooting Method 17.1 The S
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} 17.1 The Shooting Method 759 floa
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17.2 Shooting to a Fitting Point 76
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17.3 Relaxation Methods 763 depende
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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
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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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