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244 Chapter 6. Special Functions rjpl=fact*rjtemp-rjl; rjl=rjtemp; } if (rjl == 0.0) rjl=EPS; f=rjpl/rjl; Now have unnormalized Jµ and J ′ 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=2.0/PI*fact*(gam1*cosh(e)+gam2*fact2*d); f0. e=exp(e); p=e/(gampl*PI); p0. q=1.0/(e*PI*gammi); q0. pimu2=0.5*pimu; fact3 = (fabs(pimu2) < EPS ? 1.0 : sin(pimu2)/pimu2); r=PI*pimu2*fact3*fact3; c=1.0; d = -x2*x2; sum=ff+r*q; sum1=p; for (i=1;i MAXIT) nrerror("bessy series failed to converge"); rymu = -sum; ry1 = -sum1*xi2; rymup=xmu*xi*rymu-ry1; rjmu=w/(rymup-f*rymu); Equation (6.7.13). } else { Evaluate CF2 by modified Lentz’s method (§5.2). a=0.25-xmu2; p = -0.5*xi; q=1.0; br=2.0*x; bi=2.0; fact=a*xi/(p*p+q*q); cr=br+q*fact; ci=bi+p*fact; den=br*br+bi*bi; dr=br/den; di = -bi/den; dlr=cr*dr-ci*di; dli=cr*di+ci*dr; temp=p*dlr-q*dli; q=p*dli+q*dlr; p=temp; for (i=2;i
} 6.7 Bessel Functions of Fractional Order 245 cr=br+cr*fact; ci=bi-ci*fact; if (fabs(cr)+fabs(ci) < FPMIN) cr=FPMIN; den=dr*dr+di*di; dr /= den; di /= -den; dlr=cr*dr-ci*di; dli=cr*di+ci*dr; temp=p*dlr-q*dli; q=p*dli+q*dlr; p=temp; if (fabs(dlr-1.0)+fabs(dli) < EPS) break; } if (i > MAXIT) nrerror("cf2 failed in bessjy"); gam=(p-f)/q; Equations (6.7.6) – (6.7.10). rjmu=sqrt(w/((p-f)*gam+q)); rjmu=SIGN(rjmu,rjl); rymu=rjmu*gam; rymup=rymu*(p+q/gam); ry1=xmu*xi*rymu-rymup; } fact=rjmu/rjl; *rj=rjl1*fact; Scale original Jν and J ′ ν. *rjp=rjp1*fact; 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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Page 299 and 300: 7.1 Uniform Deviates 275 As for ref
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} 7.3 Rejection Method: Gamma, Pois
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7.4 Generation of Random Bits 297 s
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} } *iseed
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7.5 Random Sequences Based on Data
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} } 7.5 Random Sequences Based on D
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7.6 Simple Monte Carlo Integration
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7.6 Simple Monte Carlo Integration
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7.7 Quasi- (that is, Sub-) Random S
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} 7.7 Quasi- (that is, Sub-) Random
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7.8 Adaptive and Recursive Monte Ca
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} 7.8 Adaptive and Recursive Monte
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Chapter 8. Sorting 8.0 Introduction
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8.1 Straight Insertion and Shell’
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8.2 Quicksort 333 question of the b
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8.2 Quicksort 335 istack=lvector(1,
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a 4 a 2 8.3 Heapsort 337 a 5 a 1 a
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original array 1 2 3 4 5 6 14 8 32
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8.5 Selecting the Mth Largest 341 w
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} } } } 8.6 Determination of Equiva
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Chapter 9. Root Finding and Nonline
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9.0 Introduction 349 good first gue
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(a) (b) (c) (d) 9.1 Bracketing and
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Bisection Method 9.1 Bracketing and
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9.2 Secant Method, False Position M
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} 9.2 Secant Method, False Position
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} 9.3 Van Wijngaarden-Dekker-Brent
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9.3 Van Wijngaarden-Dekker-Brent Me
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9.4 Newton-Raphson Method Using Der
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9.4 Newton-Raphson Method Using Der
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} 9.4 Newton-Raphson Method Using D
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9.5 Roots of Polynomials 9.5 Roots
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9.5 Roots of Polynomials 371 tentat
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9.5 Roots of Polynomials 373 The me
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Eigenvalue Methods 9.5 Roots of Pol
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9.5 Roots of Polynomials 377 of goi
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9.6 Newton-Raphson Method for Nonli
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9.6 Newton-Raphson Method for Nonli
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9.7 Globally Convergent Methods for
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X1 A B C 10.0 Introduction 395 D Fi
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10.1 Golden Section Search in One D
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10.1 Golden Section Search in One D
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} } 10.1 Golden Section Search in O
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10.2 Parabolic Interpolation and Br
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} 10.3 One-Dimensional Search with
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10.3 One-Dimensional Search with Fi
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(a) (b) (c) (d) 10.4 Downhill Simpl
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10.4 Downhill Simplex Method in Mul
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10.5 Direction Set (Powell’s) Met
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(a) 10.6 Conjugate Gradient Methods
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10.7 Variable Metric Methods in Mul
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10.8 Linear Programming and the Sim
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} 10.8 Linear Programming and the S
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10.9 Simulated Annealing Methods 44
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(a) (b) (c) 1 .5 0 1 .5 0 1 .5 0 10
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} } 10.9 Simulated Annealing Method
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} free_ivector(jorder,1,ncity); #in
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Definitions and Basic Facts 11.0 In
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11.0 Introduction 459 In both the d
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11.0 Introduction 461 “Eigenpacka
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11.1 Jacobi Transformations of a Sy
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} } 11.4 Hermitian Matrices 481 for
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11.5 Reduction of a General Matrix
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11.6 The QR Algorithm for Real Hess
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11.7 Eigenvalues or Eigenvectors by
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11.7 Eigenvalues or Eigenvectors by
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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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} } } 12.4 FFT in Two or More Dimen
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data [1] Re Im 12.4 FFT in Two or M
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Chapter 13. Fourier and Spectral Ap
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(t) r*s(t) 13.1 Convolution and Dec
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13.1 Convolution and Deconvolution
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13.1 Convolution and Deconvolution
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13.3 Optimal (Wiener) Filtering wit
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log scale 13.4 Power Spectrum Estim
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13.4 Power Spectrum Estimation Usin
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amplitude 1 .8 .6 .4 .2 0 13.4 Powe
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13.5 Digital Filtering in the Time
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IIR (Recursive) Filters 13.5 Digita
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13.6 Linear Prediction and Linear P
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} 13.6 Linear Prediction and Linear
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13.7 Maximum Entropy (All Poles) Me
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13.8 Spectral Analysis of Unevenly
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} 13.8 Spectral Analysis of Unevenl
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} 13.8 Spectral Analysis of Unevenl
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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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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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1. male 2. female . 14.4 Contingenc
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14.4 Contingency Table Analysis of
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14.5 Linear Correlation 637 two dis
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14.6 Nonparametric or Rank Correlat
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} } 14.7 Do Two-Dimensional Distrib
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14.7 Do Two-Dimensional Distributio
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14.8 Savitzky-Golay Smoothing Filte
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8 6 4 2 0 8 6 4 2 0 8 6 4 2 0 befor
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14.8 Savitzky-Golay Smoothing Filte
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15.1 Least Squares as a Maximum Lik
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15.1 Least Squares as a Maximum Lik
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15.2 Fitting Data to a Straight Lin
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15.2 Fitting Data to a Straight Lin
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15.3 Straight-Line Data with Errors
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15.4 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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