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412 Chapter 10. Minimization or Maximization of Functions<br />
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412 Chapter 10. Minimization or Maximization of Functions } if (i != ilo) { for (j=1;j
10.5 Direction Set (Powell’s) Methods in Multidimensions 413 direction n, then any function of N variables f(P) can be minimized along the line n by our one-dimensional methods. One can dream up various multidimensional minimization methods that consist of sequences of such line minimizations. Different methods will differ only by how, at each stage, they choose the next direction n to try. All such methods presume the existence of a “black-box” sub-algorithm, which we might call linmin (given as an explicit routine at the end of this section), whose definition can be taken for now as linmin: Given as input the vectors P and n, and the function f, find the scalar λ that minimizes f(P + λn). Replace P by P + λn. Replace n by λn. Done. All the minimization methods in this section and in the two sections following fall under this general schema of successive line minimizations. (The algorithm in §10.7 does not need very accurate line minimizations. Accordingly, it has its own approximate line minimization routine, lnsrch.) In this section we consider a class of methods whose choice of successive directions does not involve explicit computation of the function’s gradient; the next two sections do require such gradient calculations. You will note that we need not specify whether linmin uses gradient information or not. That choice is up to you, and its optimization depends on your particular function. You would be crazy, however, to use gradients in linmin and not use them in the choice of directions, since in this latter role they can drastically reduce the total computational burden. But what if, in your application, calculation of the gradient is out of the question. You might first think of this simple method: Take the unit vectors e 1, e2,...eN as a set of directions. Using linmin, move along the first direction to its minimum, then from there along the second direction to its minimum, and so on, cycling through the whole set of directions as many times as necessary, until the function stops decreasing. This simple method is actually not too bad for many functions. Even more interesting is why it is bad, i.e. very inefficient, for some other functions. Consider a function of two dimensions whose contour map (level lines) happens to define a long, narrow valley at some angle to the coordinate basis vectors (see Figure 10.5.1). Then the only way “down the length of the valley” going along the basis vectors at each stage is by a series of many tiny steps. More generally, in N dimensions, if the function’s second derivatives are much larger in magnitude in some directions than in others, then many cycles through all N basis vectors will be required in order to get anywhere. This condition is not all that unusual; according to Murphy’s Law, you should count on it. Obviously what we need is a better set of directions than the e i’s. All direction set methods consist of prescriptions for updating the set of directions as the method proceeds, attempting to come up with a set which either (i) includes some very good directions that will take us far along narrow valleys, or else (more subtly) (ii) includes some number of “non-interfering” directions with the special property that minimization along one is not “spoiled” by subsequent minimization along another, so that interminable cycling through the set of directions can be avoided. 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).
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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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Computer Programs by Chapter and Se
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Computer Programs by Chapter and Se
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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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1.1 Program Organization and Contro
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true 1.1 Program Organization and C
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1.1 Program Organization and Contro
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1.1 Program Organization and Contro
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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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1.2 Some C Conventions for Scientif
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(a) **m (b) **m 1.2 Some C Conventi
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1.2 Some C Conventions for Scientif
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1.2 Some C Conventions for Scientif
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1.2 Some C Conventions for Scientif
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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 65
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2.6 Singular Value Decomposition 67
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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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2.8 Vandermonde Matrices and Toepli
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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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4.1 Classical Formulas for Equally
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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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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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4.5 Gaussian Quadratures and Orthog
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4.5 Gaussian Quadratures and Orthog
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4.5 Gaussian Quadratures and Orthog
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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 Fractional
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} 6.7 Bessel Functions of Fractiona
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6.7 Bessel Functions of Fractional
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} 6.7 Bessel Functions of Fractiona
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} 6.7 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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} } *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 S
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} 7.7 Quasi- (that is, Sub-) Random
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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 Ca
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7.8 Adaptive and Recursive Monte Ca
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} } 7.8 Adaptive and Recursive Mont
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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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11.1 Jacobi Transformations of a Sy
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11.1 Jacobi Transformations of a Sy
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11.1 Jacobi Transformations of a Sy
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} } 11.2 Reduction of a Symmetric M
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11.2 Reduction of a Symmetric Matri
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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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11.3 Eigenvalues and Eigenvectors o
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11.3 Eigenvalues and Eigenvectors o
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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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11.5 Reduction of a General Matrix
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11.6 The QR Algorithm for Real Hess
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11.6 The QR Algorithm for Real Hess
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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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⏐h(t)⏐ 2 (a) Ph( f ) (one-sided
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12.1 Fourier Transform of Discretel
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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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(a) (b) (c) +1 0 −1 +1 0 −1 +1
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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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12.5 Fourier Transforms of Real Dat
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12.5 Fourier Transforms of Real Dat
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12.5 Fourier Transforms of Real Dat
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12.6 External Storage or Memory-Loc
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} } 12.6 External Storage or Memory
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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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13.4 Power Spectrum Estimation Usin
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amplitude 1 .8 .6 .4 .2 0 13.4 Powe
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13.4 Power Spectrum Estimation Usin
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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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13.6 Linear Prediction and Linear P
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} 13.6 Linear Prediction and Linear
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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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13.8 Spectral Analysis of Unevenly
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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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} 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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14.1 Moments of a Distribution: Mea
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14.2 Do Two Distributions Have the
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14.2 Do Two Distributions Have the
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14.2 Do Two Distributions Have the
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14.3 Are Two Distributions Differen
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14.3 Are Two Distributions Differen
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14.3 Are Two Distributions Differen
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14.3 Are Two Distributions Differen
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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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14.4 Contingency Table Analysis of
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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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14.6 Nonparametric or Rank Correlat
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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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#include #include "nrutil.h" 14.7
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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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#include #include "nrutil.h" 15.2
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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 6
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} 15.4 General Linear Least Squares
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15.4 General Linear Least Squares 6
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15.4 General Linear Least Squares 6
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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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15.6 Confidence Limits on Estimated
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15.6 Confidence Limits on Estimated
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15.6 Confidence Limits on Estimated
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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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16.2 Adaptive Stepsize Control for
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16.2 Adaptive Stepsize Control for
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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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16.4 Richardson Extrapolation and t
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16.4 Richardson Extrapolation and t
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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
- Page 763 and 764:
16.6 Stiff Sets of Equations 739 od
- Page 765 and 766:
} 16.6 Stiff Sets of Equations 741
- Page 767 and 768:
16.6 Stiff Sets of Equations 743 Co
- Page 769 and 770:
16.6 Stiff Sets of Equations 745 Se
- Page 771 and 772:
} 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
- Page 779 and 780:
equired boundary value 17.0 Introdu
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17.1 The Shooting Method 17.1 The S
- Page 783 and 784:
} 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 H
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17.4 A Worked Example: Spheroidal H
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} 17.4 A Worked Example: Spheroidal
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17.4 A Worked Example: Spheroidal H
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17.5 Automated Allocation of Mesh P
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17.6 Handling Internal Boundary Con
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17.6 Handling Internal Boundary Con
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18.0 Introduction 789 is called a F
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18.1 Fredholm Equations of the Seco
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#include "nrutil.h" 18.1 Fredholm E
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18.2 Volterra Equations 795 in §18
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18.3 Integral Equations with Singul
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18.3 Integral Equations with Singul
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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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18.5 Linear Regularization Methods
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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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18.7 Maximum Entropy Image Restorat
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18.7 Maximum Entropy Image Restorat
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18.7 Maximum Entropy Image Restorat
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Chapter 19. Partial Differential Eq
- Page 853 and 854:
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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19.1 Flux-Conservative Initial Valu
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19.1 Flux-Conservative Initial Valu
- Page 865 and 866:
19.1 Flux-Conservative Initial Valu
- Page 867 and 868:
19.1 Flux-Conservative Initial Valu
- Page 869 and 870:
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
- Page 875 and 876:
and the heuristic stability criteri
- Page 877 and 878:
19.3 Initial Value Problems in Mult
- Page 879 and 880:
19.3 Initial Value Problems in Mult
- Page 881 and 882:
19.4 Fourier and Cyclic Reduction M
- Page 883 and 884:
19.4 Fourier and Cyclic Reduction M
- Page 885 and 886:
19.4 Fourier and Cyclic Reduction M
- Page 887 and 888:
FACR Method 19.5 Relaxation Methods
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19.5 Relaxation Methods for Boundar
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19.5 Relaxation Methods for Boundar
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19.5 Relaxation Methods for Boundar
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19.6 Multigrid Methods for Boundary
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19.6 Multigrid Methods for Boundary
- Page 899 and 900:
S S S E γ = 1 19.6 Multigrid Metho
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19.6 Multigrid Methods for Boundary
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19.6 Multigrid Methods for Boundary
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} 19.6 Multigrid Methods for Bounda
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19.6 Multigrid Methods for Boundary
- Page 909 and 910:
19.6 Multigrid Methods for Boundary
- Page 911 and 912:
#include 19.6 Multigrid Methods fo
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Chapter 20. Less-Numerical Algorith
- Page 915 and 916:
20.1 Diagnosing Machine Parameters
- Page 917 and 918:
20.1 Diagnosing Machine Parameters
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(a) G(i) (b) MSB 4 3 2 1 0 LSB i MS
- Page 921 and 922:
20.3 Cyclic Redundancy and Other Ch
- Page 923 and 924:
20.3 Cyclic Redundancy and Other Ch
- Page 925 and 926:
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
- Page 931 and 932:
} 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
- Page 937 and 938:
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
- Page 943 and 944:
#include "nrutil.h" #define RX 256.
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} 20.6 Arithmetic at Arbitrary Prec
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20.6 Arithmetic at Arbitrary Precis
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20.6 Arithmetic at Arbitrary Precis
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