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Numerical Recipes in C
Numerical Recipes in C
Acrobat ® Edition
l Front Matter, Contents, and Prefaces xi
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Legal Matters xvi
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Computer Programs by Chapter and Section xix
1 Preliminaries
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1.0 Introduction 1
l 1.1 Program Organization and Control Structures 5
l 1.2 Some C Conventions for Scientific Computing 15
l 1.3 Error, Accuracy, and Stability 15
2 Solution of Linear Algebraic Equations
l 2.0 Introduction 32
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2.1 Gauss-Jordan Elimination 36
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2.2 Gaussian Elimination with Backsubstitution 41
l 2.3 LU Decomposition and Its Applications 43
l 2.4 Tridiagonal and Band Diagonal Systems of Equations 50
l 2.5 Iterative Improvement of a Solution to Linear Equations 55
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2.6 Singular Value Decomposition 59
l 2.7 Sparse Linear Systems 71
l 2.8 Vandermonde Matrices and Toeplitz Matrices 90
l 2.9 Cholesky Decomposition 96
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2.10 QR Decomposition 98
l 2.11 Is Matrix Inversion an $N^3$ Process? 102
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3 Interpolation and Extrapolation
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3.0 Introduction 105
l 3.1 Polynomial Interpolation and Extrapolation 108
l 3.2 Rational Function Interpolation and Extrapolation 111
l 3.3 Cubic Spline Interpolation 113
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3.4 How to Search an Ordered Table 117
l 3.5 Coefficients of the Interpolating Polynomial 120
l 3.6 Interpolation in Two or More Dimensions 123
4 Integration of Functions
l 4.0 Introduction 129
l 4.1 Classical Formulas for Equally Spaced Abscissas 130
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4.2 Elementary Algorithms 136
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4.3 Romberg Integration 140
l 4.4 Improper Integrals 141
l 4.5 Gaussian Quadratures and Orthogonal Polynomials 147
l 4.6 Multidimensional Integrals 161
5 Evaluation of Functions
l 5.0 Introduction 165
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5.1 Series and Their Convergence 165
l 5.2 Evaluation of Continued Fractions 169
l 5.3 Polynomials and Rational Functions 173
l 5.4 Complex Arithmetic 176
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5.5 Recurrence Relations and Clenshaw's Recurrence Formula 178
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5.6 Quadratic and Cubic Equations 183
l 5.7 Numerical Derivatives 186
l 5.8 Chebyshev Approximation 190
l 5.9 Derivatives or Integrals of a Chebyshev-approximated Function 195
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l 5.10 Polynomial Approximation from Chebyshev Coefficients 197
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5.11 Economization of Power Series 198
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5.12 Pad\'e Approximants 200
l 5.13 Rational Chebyshev Approximation 204
l 5.14 Evaluation of Functions by Path Integration 208
6 Special Functions
l 6.0 Introduction 212
l 6.1 Gamma Function, Beta Function, Factorials, Binomial Coefficients 213
l
6.2 Incomplete Gamma Function, Error Function, Chi-Square Probability Function,
Cumulative Poisson Function 216
l 6.3 Exponential Integrals 222
l 6.4 Incomplete Beta Function, Student's Distribution, F-Distribution,Cumulative Binomial
Distribution 226
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6.5 Bessel Functions of Integer Order 230
l 6.6 Modified Bessel Functions of Integer Order 236
l 6.7 Bessel Functions of Fractional Order, Airy Functions, SphericalBessel Functions 240
l 6.8 Spherical Harmonics 252
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6.9 Fresnel Integrals, Cosine and Sine Integrals 255
l 6.10 Dawson's Integral 259
l 6.11 Elliptic Integrals and Jacobian Elliptic Functions 261
l 6.12 Hypergeometric Functions 271
7 Random Numbers
l 7.0 Introduction 274
l 7.1 Uniform Deviates 275
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7.2 Transformation Method: Exponential and Normal Deviates 287
l 7.3 Rejection Method: Gamma, Poisson, Binomial Deviates 290
l 7.4 Generation of Random Bits 296
l 7.5 Random Sequences Based on Data Encryption 300
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7.6 Simple Monte Carlo Integration 304
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l 7.7 Quasi- (that is, Sub-) Random Sequences 309
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7.8 Adaptive and Recursive Monte Carlo Methods 316
8 Sorting
l 8.0 Introduction 329
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8.1 Straight Insertion and Shell's Method 330
l 8.2 Quicksort 332
l 8.3 Heapsort 336
l 8.4 Indexing and Ranking 338
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8.5 Selecting the $M$th Largest 341
l 8.6 Determination of Equivalence Classes 345
9 Root Finding and Nonlinear Sets of Equations
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9.0 Introduction 347
l 9.1 Bracketing and Bisection 350
l 9.2 Secant Method, False Position Method, and Ridders' Method 354
l 9.3 Van Wijngaarden--Dekker--Brent Method 359
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9.4 Newton-Raphson Method Using Derivative 362
l 9.5 Roots of Polynomials 369
l 9.6 Newton-Raphson Method for Nonlinear Systems of Equations 379
l 9.7 Globally Convergent Methods for Nonlinear Systems of Equations 383
10 Minimization or Maximization of Functions
l 10.0 Introduction 394
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10.1 Golden Section Search in One Dimension 397
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10.2 Parabolic Interpolation and Brent's Method in One Dimension 402
l 10.3 One-Dimensional Search with First Derivatives 305
l 10.4 Downhill Simplex Method in Multidimensions 408
l 10.5 Direction Set (Powell's) Methods in Multidimensions 412
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10.6 Conjugate Gradient Methods in Multidimensions 420
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l 10.7 Variable Metric Methods in Multidimensions 425
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10.8 Linear Programming and the Simplex Method 430
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10.9 Simulated Annealing Methods 444
11 Eigensystems
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11.0 Introduction 456
l 11.1 Jacobi Transformations of a Symmetric Matrix 463
l 11.2 Reduction of a Symmetric Matrix to Tridiagonal Form: Givens and Householder
Reductions 469
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11.3 Eigenvalues and Eigenvectors of a Tridiagonal Matrix 475
l 11.4 Hermitian Matrices 481
l 11.5 Reduction of a General Matrix to Hessenberg Form 482
l 11.6 The QR Algorithm for Real Hessenberg Matrices 486
l 11.7 Improving Eigenvalues and/or Finding Eigenvectors by Inverse Iteration 493
12 Fast Fourier Transform
l 12.0 Introduction 496
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12.1 Fourier Transform of Discretely Sampled Data 500
l 12.2 Fast Fourier Transform (FFT) 504
l 12.3 FFT of Real Functions, Sine and Cosine Transforms 510
l 12.4 FFT in Two or More Dimensions 521
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12.5 Fourier Transforms of Real Data in Two and Three Dimensions 525
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12.6 External Storage or Memory-Local FFTs 532
13 Fourier and Spectral Applications
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13.0 Introduction 537
l 13.1 Convolution and Deconvolution Using the FFT 538
l 13.2 Correlation and Autocorrelation Using the FFT 545
l 13.3 Optimal (Wiener) Filtering with the FFT 547
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13.4 Power Spectrum Estimation Using the FFT 549
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