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<html><head><title>Numerical Recipes Table of Contents</title></head>
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<h1> Contents of Numerical Recipes</h1>
<h2>(Second Edition in C++, C, or Fortran) </h2>
<I> (Page numbers may vary slightly among the language versions.) </I>
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<LI> Preface to the Second Edition xi
<LI> Preface to the First Edition xiv
<LI> Legal Matters xvi
<LI> Computer Programs by Chapter and Section xix
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<h2><a name="C1"></A><A HREF="progs.htm#C1">1 Preliminaries </a></h2>
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<LI> 1.0 Introduction 1
<LI> 1.1 Program Organization and Control Structures 5
<LI> 1.2 Error, Accuracy, and Stability 18
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<h2><a name="C2"></A><A HREF="progs.htm#C2">2 Solution of Linear Algebraic Equations </a></h2>
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<LI> 2.0 Introduction 22
<LI> 2.1 Gauss-Jordan Elimination 27
<LI> 2.2 Gaussian Elimination with Backsubstitution 33
<LI> 2.3 LU Decomposition and Its Applications 34
<LI> 2.4 Tridiagonal and Band Diagonal Systems of Equations 42
<LI> 2.5 Iterative Improvement of a Solution to Linear Equations 47
<LI> 2.6 Singular Value Decomposition 51
<LI> 2.7 Sparse Linear Systems 63
<LI> 2.8 Vandermonde Matrices and Toeplitz Matrices 82
<LI> 2.9 Cholesky Decomposition 89
<LI> 2.10 QR Decomposition 91
<LI> 2.11 Is Matrix Inversion an $N^3$ Process? 95
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<h2><a name="C3"></A><A HREF="progs.htm#C3">3 Interpolation and Extrapolation </a></h2>
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<LI> 3.0 Introduction 99
<LI> 3.1 Polynomial Interpolation and Extrapolation 102
<LI> 3.2 Rational Function Interpolation and Extrapolation 104
<LI> 3.3 Cubic Spline Interpolation 107
<LI> 3.4 How to Search an Ordered Table 110
<LI> 3.5 Coefficients of the Interpolating Polynomial 113
<LI> 3.6 Interpolation in Two or More Dimensions 116
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<h2><a name="C4"></A><A HREF="progs.htm#C4">4 Integration of Functions </a></h2>
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<LI> 4.0 Introduction 123
<LI> 4.1 Classical Formulas for Equally Spaced Abscissas 124
<LI> 4.2 Elementary Algorithms 130
<LI> 4.3 Romberg Integration 134
<LI> 4.4 Improper Integrals 135
<LI> 4.5 Gaussian Quadratures and Orthogonal Polynomials 140
<LI> 4.6 Multidimensional Integrals 155
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<h2><a name="C5"></A><A HREF="progs.htm#C5">5 Evaluation of Functions </a></h2>
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<LI> 5.0 Introduction 159
<LI> 5.1 Series and Their Convergence 159
<LI> 5.2 Evaluation of Continued Fractions 163
<LI> 5.3 Polynomials and Rational Functions 167
<LI> 5.4 Complex Arithmetic 171
<LI> 5.5 Recurrence Relations and Clenshaw's Recurrence Formula 172
<LI> 5.6 Quadratic and Cubic Equations 178
<LI> 5.7 Numerical Derivatives 180
<LI> 5.8 Chebyshev Approximation 184
<LI> 5.9 Derivatives or Integrals of a Chebyshev-approximated
Function 189
<LI> 5.10 Polynomial Approximation from Chebyshev Coefficients 191
<LI> 5.11 Economization of Power Series 192
<LI> 5.12 Pad\'e Approximants 194
<LI> 5.13 Rational Chebyshev Approximation 197
<LI> 5.14 Evaluation of Functions by Path Integration 201
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<h2><a name="C6"></A><A HREF="progs.htm#C6">6 Special Functions </a></h2>
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<LI> 6.0 Introduction 205
<LI> 6.1 Gamma Function, Beta Function, Factorials, Binomial
Coefficients 206
<LI> 6.2 Incomplete Gamma Function, Error Function, Chi-Square
Probability
Function, Cumulative Poisson Function 209
<LI> 6.3 Exponential Integrals 215
<LI> 6.4 Incomplete Beta Function, Student's Distribution,
F-Distribution,Cumulative Binomial Distribution 219
<LI> 6.5 Bessel Functions of Integer Order 223
<LI> 6.6 Modified Bessel Functions of Integer Order 229
<LI> 6.7 Bessel Functions of Fractional Order, Airy Functions,
SphericalBessel Functions 234
<LI> 6.8 Spherical Harmonics 246
<LI> 6.9 Fresnel Integrals, Cosine and Sine Integrals 248
<LI> 6.10 Dawson's Integral 252
<LI> 6.11 Elliptic Integrals and Jacobian Elliptic Functions 254
<LI> 6.12 Hypergeometric Functions 263
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<h2><a name="C7"></A><A HREF="progs.htm#C7">7 Random Numbers </a></h2>
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<LI> 7.0 Introduction 266
<LI> 7.1 Uniform Deviates 267
<LI> 7.2 Transformation Method: Exponential and Normal Deviates 277
<LI> 7.3 Rejection Method: Gamma, Poisson, Binomial Deviates 281
<LI> 7.4 Generation of Random Bits 287
<LI> 7.5 Random Sequences Based on Data Encryption 290
<LI> 7.6 Simple Monte Carlo Integration 295
<LI> 7.7 Quasi- (that is, Sub-) Random Sequences 299
<LI> 7.8 Adaptive and Recursive Monte Carlo Methods 306
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<h2><a name="C8"></A><A HREF="progs.htm#C8">8 Sorting </a></h2>
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<LI> 8.0 Introduction 320
<LI> 8.1 Straight Insertion and Shell's Method 321
<LI> 8.2 Quicksort 323
<LI> 8.3 Heapsort 327
<LI> 8.4 Indexing and Ranking 329
<LI> 8.5 Selecting the $M$th Largest 333
<LI> 8.6 Determination of Equivalence Classes 337
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<h2><a name="C9"></A><A HREF="progs.htm#C9">9 Root Finding and Nonlinear Sets of Equations </a></h2>
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<LI> 9.0 Introduction 340
<LI> 9.1 Bracketing and Bisection 343
<LI> 9.2 Secant Method, False Position Method, and Ridders' Method 347
<LI> 9.3 Van Wijngaarden--Dekker--Brent Method 352
<LI> 9.4 Newton-Raphson Method Using Derivative 355
<LI> 9.5 Roots of Polynomials 362
<LI> 9.6 Newton-Raphson Method for Nonlinear Systems of Equations 372
<LI> 9.7 Globally Convergent Methods for Nonlinear Systems of
Equations 376
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<h2><a name="C10"></A><A HREF="progs.htm#C10">10 Minimization or Maximization of Functions </a></h2>
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<LI> 10.0 Introduction 387
<LI> 10.1 Golden Section Search in One Dimension 390
<LI> 10.2 Parabolic Interpolation and Brent's Method in One Dimension 395
<LI> 10.3 One-Dimensional Search with First Derivatives 399
<LI> 10.4 Downhill Simplex Method in Multidimensions 402
<LI> 10.5 Direction Set (Powell's) Methods in Multidimensions 406
<LI> 10.6 Conjugate Gradient Methods in Multidimensions 413
<LI> 10.7 Variable Metric Methods in Multidimensions 418
<LI> 10.8 Linear Programming and the Simplex Method 423
<LI> 10.9 Simulated Annealing Methods 436
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<h2><a name="C11"></A><A HREF="progs.htm#C11">11 Eigensystems </a></h2>
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<LI> 11.0 Introduction 449
<LI> 11.1 Jacobi Transformations of a Symmetric Matrix 456
<LI> 11.2 Reduction of a Symmetric Matrix to Tridiagonal Form:
Givens and Householder Reductions 462
<LI> 11.3 Eigenvalues and Eigenvectors of a Tridiagonal Matrix 469
<LI> 11.4 Hermitian Matrices 475
<LI> 11.5 Reduction of a General Matrix to Hessenberg Form 476
<LI> 11.6 The QR Algorithm for Real Hessenberg Matrices 480
<LI> 11.7 Improving Eigenvalues and/or Finding Eigenvectors by
Inverse Iteration 487
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<h2><a name="C12"></A><A HREF="progs.htm#C12">12 Fast Fourier Transform </a></h2>
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<LI> 12.0 Introduction 490
<LI> 12.1 Fourier Transform of Discretely Sampled Data 494
<LI> 12.2 Fast Fourier Transform (FFT) 498
<LI> 12.3 FFT of Real Functions, Sine and Cosine Transforms 504
<LI> 12.4 FFT in Two or More Dimensions 515
<LI> 12.5 Fourier Transforms of Real Data in Two and Three Dimensions 519
<LI> 12.6 External Storage or Memory-Local FFTs 525
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<h2><a name="C13"></A><A HREF="progs.htm#C13">13 Fourier and Spectral Applications </a></h2>
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<LI> 13.0 Introduction 530
<LI> 13.1 Convolution and Deconvolution Using the FFT 531
<LI> 13.2 Correlation and Autocorrelation Using the FFT 538
<LI> 13.3 Optimal (Wiener) Filtering with the FFT 539
<LI> 13.4 Power Spectrum Estimation Using the FFT 542
<LI> 13.5 Digital Filtering in the Time Domain 551
<LI> 13.6 Linear Prediction and Linear Predictive Coding 557
<LI> 13.7 Power Spectrum Estimation by the Maximum Entropy
(All Poles) Method 565
<LI> 13.8 Spectral Analysis of Unevenly Sampled Data 569
<LI> 13.9 Computing Fourier Integrals Using the FFT 577
<LI> 13.10 Wavelet Transforms 584
<LI> 13.11 Numerical Use of the Sampling Theorem 600
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<h2><a name="C14"></A><A HREF="progs.htm#C14">14 Statistical Description of Data </a></h2>
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<LI> 14.0 Introduction 603
<LI> 14.1 Moments of a Distribution: Mean, Variance, Skewness,
and So Forth 604
<LI> 14.2 Do Two Distributions Have the Same Means or Variances? 609
<LI> 14.3 Are Two Distributions Different? 614
<LI> 14.4 Contingency Table Analysis of Two Distributions 622
<LI> 14.5 Linear Correlation 630
<LI> 14.6 Nonparametric or Rank Correlation 633
<LI> 14.7 Do Two-Dimensional Distributions Differ? 640
<LI> 14.8 Savitzky-Golay Smoothing Filters 644
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<h2><a name="C15"></A><A HREF="progs.htm#C15">15 Modeling of Data </a></h2>
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<LI> 15.0 Introduction 650
<LI> 15.1 Least Squares as a Maximum Likelihood Estimator 651
<LI> 15.2 Fitting Data to a Straight Line 655
<LI> 15.3 Straight-Line Data with Errors in Both Coordinates 660
<LI> 15.4 General Linear Least Squares 665
<LI> 15.5 Nonlinear Models 675
<LI> 15.6 Confidence Limits on Estimated Model Parameters 684
<LI> 15.7 Robust Estimation 694
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<h2><a name="C16"></A><A HREF="progs.htm#C16">16 Integration of Ordinary Differential Equations </a></h2>
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<LI> 16.0 Introduction 701
<LI> 16.1 Runge-Kutta Method 704
<LI> 16.2 Adaptive Stepsize Control for Runge-Kutta 708
<LI> 16.3 Modified Midpoint Method 716
<LI> 16.4 Richardson Extrapolation and the Bulirsch-Stoer Method 718
<LI> 16.5 Second-Order Conservative Equations 726
<LI> 16.6 Stiff Sets of Equations 727
<LI> 16.7 Multistep, Multivalue, and Predictor-Corrector Methods 740
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<h2><a name="C17"></A><A HREF="progs.htm#C17">17 Two Point Boundary Value Problems </a></h2>
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<LI> 17.0 Introduction 745
<LI> 17.1 The Shooting Method 749
<LI> 17.2 Shooting to a Fitting Point 751
<LI> 17.3 Relaxation Methods 753
<LI> 17.4 A Worked Example: Spheroidal Harmonics 764
<LI> 17.5 Automated Allocation of Mesh Points 774
<LI> 17.6 Handling Internal Boundary Conditions or Singular Points 775
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<h2><a name="C18"></A><A HREF="progs.htm#C18">18 Integral Equations and Inverse Theory </a></h2>
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<LI> 18.0 Introduction 779
<LI> 18.1 Fredholm Equations of the Second Kind 782
<LI> 18.2 Volterra Equations 786
<LI> 18.3 Integral Equations with Singular Kernels 788
<LI> 18.4 Inverse Problems and the Use of A Priori Information 795
<LI> 18.5 Linear Regularization Methods 799
<LI> 18.6 Backus-Gilbert Method 806
<LI> 18.7 Maximum Entropy Image Restoration 809
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<h2><a name="C19"></A><A HREF="progs.htm#C19">19 Partial Differential Equations </a></h2>
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<LI> 19.0 Introduction 818
<LI> 19.1 Flux-Conservative Initial Value Problems 825
<LI> 19.2 Diffusive Initial Value Problems 838
<LI> 19.3 Initial Value Problems in Multidimensions 844
<LI> 19.4 Fourier and Cyclic Reduction Methods for Boundary
Value Problems 848
<LI> 19.5 Relaxation Methods for Boundary Value Problems 854
<LI> 19.6 Multigrid Methods for Boundary Value Problems 862
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<h2><a name="C20"></A><A HREF="progs.htm#C20">20 Less-Numerical Algorithms </a></h2>
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<LI> 20.0 Introduction 881
<LI> 20.1 Diagnosing Machine Parameters 881
<LI> 20.2 Gray Codes 886
<LI> 20.3 Cyclic Redundancy and Other Checksums 888
<LI> 20.4 Huffman Coding and Compression of Data 896
<LI> 20.5 Arithmetic Coding 902
<LI> 20.6 Arithmetic at Arbitrary Precision 906
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<LI> References 916
<LI> Index of Programs and Dependencies 921
<LI> General Index 935
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