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664 Chapter 15. Modeling of Data Here gammq is our routine for the incomplete gamma function Q(a, x), §6.2. If Q is larger than, say, 0.1, then the goodness-of-fit is believable. If it is larger than, say, 0.001, then the fit may be acceptable if the errors are nonnormal or have been moderately underestimated. If Q is less than 0.001 then the model and/or estimation procedure can rightly be called into question. In this latter case, turn to §15.7 to proceed further. If you do not know the individual measurement errors of the points σ i, and are proceeding (dangerously) to use equation (15.1.6) for estimating these errors, then here is the procedure for estimating the probable uncertainties of the parameters a and b: Set σi ≡ 1 in all equations through (15.2.6), and multiply σ a and σb, as obtained from equation (15.2.9), by the additional factor � χ 2 /(N − 2), where χ 2 is computed by (15.2.2) using the fitted parameters a and b. As discussed above, this procedure is equivalent to assuming a good fit, so you get no independent goodness-of-fit probability Q. In §14.5 we promised a relation between the linear correlation coefficient r (equation 14.5.1) and a goodness-of-fit measure, χ 2 (equation 15.2.2). For unweighted data (all σi =1), that relation is where χ 2 =(1− r 2 )NVar (y1 ...yN ) (15.2.13) NVar (y1 ...yN) ≡ N� (yi − y) 2 i=1 (15.2.14) For data with varying weights σi, the above equations remain valid if the sums in equation (14.5.1) are weighted by 1/σ 2 i . The following function, fit, carries out exactly the operations that we have discussed. When the weights σ are known in advance, the calculations exactly correspond to the formulas above. However, when weights σ are unavailable, the routine assumes equal values of σ for each point and assumes a good fit, as discussed in §15.1. The formulas (15.2.6) are susceptible to roundoff error. rewrite them as follows: Define Accordingly, we ti = 1 � xi − σi Sx � , S i =1, 2,...,N (15.2.15) and N� Stt = (15.2.16) t i=1 2 i Then, as you can verify by direct substitution, b = 1 N� Stt i=1 a = Sy − Sxb S tiyi σi (15.2.17) (15.2.18) 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).
#include #include "nrutil.h" 15.2 Fitting Data to a Straight Line 665 σ 2 � 1 a = 1+ S S2 � x SStt σ 2 b = 1 Stt Cov(a, b) =− Sx rab = SStt Cov(a, b) σaσb (15.2.19) (15.2.20) (15.2.21) (15.2.22) void fit(float x[], float y[], int ndata, float sig[], int mwt, float *a, float *b, float *siga, float *sigb, float *chi2, float *q) Given a set of data points x[1..ndata],y[1..ndata] with individual standard deviations sig[1..ndata], fit them to a straight line y = a + bx by minimizing χ 2 . Returned are a,b and their respective probable uncertainties siga and sigb, the chi-square chi2, andthe goodness-of-fit probability q (that the fit would have χ 2 this large or larger). If mwt=0 on input, then the standard deviations are assumed to be unavailable: q is returned as 1.0 and the normalization of chi2 is to unit standard deviation on all points. { float gammq(float a, float x); int i; float wt,t,sxoss,sx=0.0,sy=0.0,st2=0.0,ss,sigdat; *b=0.0; if (mwt) { Accumulate sums ... ss=0.0; 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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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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} 6.7 Bessel Functions of Fractiona
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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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#include #include "nrutil.h" 6.11
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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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} } 7.5 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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8.5 Selecting the Mth Largest 343 r
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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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} 10.7 Variable Metric Methods in M
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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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11.2 Reduction of a Symmetric Matri
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11.3 Eigenvalues and Eigenvectors o
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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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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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#include 12.3 FFT of Real Function
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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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12.5 Fourier Transforms of Real Dat
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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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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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(a) (b) 13.5 Digital Filtering in t
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13.6 Linear Prediction and Linear P
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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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13.9 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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.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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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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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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