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Numerical Recipes in C++ by Press e.a.

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  1. Jan 22, 2013 #1

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    Last edited by a moderator: May 6, 2017
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  3. Jan 22, 2013 #2

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    The extensive reference to learn the most important numerical algorithms and get your hands on their implementations.
     
  4. Jan 26, 2013 #3

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    1 Preliminaries 1
    1.0 Introduction 1
    1.1 Program Organization and Control Structures 5
    1.2 Some C++ Conventions for Scientific Computing 16
    1.3 Implementation of the Vector and Matrix Classes 25
    1.4 Error, Accuracy, and Stability 31

    2 Solution of Linear Algebraic Equations 35
    2.0 Introduction 35
    2.1 Gauss-Jordan Elimination 39
    2.2 Gaussian Elimination with Backsubstitution 44
    2.3 LU Decomposition and Its Applications 46
    2.4 Tridiagonal and Band Diagonal Systems of Equations 53
    2.5 Iterative Improvement of a Solution to Linear Equations 58
    2.6 Singular Value Decomposition 62
    2.7 Sparse Linear Systems 74
    2.8 Vandermonde Matrices and Toeplitz Matrices 93
    2.9 Cholesky Decomposition 99
    2.10 QR Decomposition 101
    2.11 Is Matrix Inversion an N3 Process? 105

    3 Interpolation and Extrapolation 108
    3.0 Introduction 108
    3.1 Polynomial Interpolation and Extrapolation 111
    3.2 Rational Function Interpolation and Extrapolation 114
    3.3 Cubic Spline Interpolation 116
    3.4 How to Search an Ordered Table 120
    3.5 Coef?cients of the Interpolating Polynomial 123
    3.6 Interpolation in Two or More Dimensions 126

    4 Integration of Functions 133
    4.0 Introduction 133
    4.1 Classical Formulas for Equally Spaced Abscissas 134
    4.2 Elementary Algorithms 141
    4.3 Romberg Integration 144
    4.4 Improper Integrals 146
    4.5 Gaussian Quadratures and Orthogonal Polynomials 152
    4.6 Multidimensional Integrals 166

    5 Evaluation of Functions 171
    5.0 Introduction 171
    5.1 Series and Their Convergence 171
    5.2 Evaluation of Continued Fractions 175
    5.3 Polynomials and Rational Functions 179
    5.4 Complex Arithmetic 182
    5.5 Recurrence Relations and Clenshaw’s Recurrence Formula 184
    5.6 Quadratic and Cubic Equations 189
    5.7 Numerical Derivatives 192
    5.8 Chebyshev Approximation 196
    5.9 Derivatives or Integrals of a Chebyshev-approximated Function 201
    5.10 Polynomial Approximation from Chebyshev Coef?cients 203
    5.11 Economization of Power Series 204
    5.12 Pade Approximants 206 ´
    5.13 Rational Chebyshev Approximation 209
    5.14 Evaluation of Functions by Path Integration 213

    6 Special Functions 217
    6.0 Introduction 217
    6.1 Gamma Function, Beta Function, Factorials, Binomial Coef?cients 218
    6.2 Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function 221
    6.3 Exponential Integrals 227
    6.4 Incomplete Beta Function, Student’s Distribution, F-Distribution, Cumulative Binomial Distribution 231
    6.5 Bessel Functions of Integer Order 235
    6.6 Modi?ed Bessel Functions of Integer Order 241
    6.7 Bessel Functions of Fractional Order, Airy Functions, Spherical
    Bessel Functions 245
    6.8 Spherical Harmonics 257
    6.9 Fresnel Integrals, Cosine and Sine Integrals 259
    6.10 Dawson’s Integral 264
    6.11 Elliptic Integrals and Jacobian Elliptic Functions 265
    6.12 Hypergeometric Functions 275

    7 Random Numbers 278
    7.0 Introduction 278
    7.1 Uniform Deviates 279Contents vii
    7.2 Transformation Method: Exponential and Normal Deviates 291
    7.3 Rejection Method: Gamma, Poisson, Binomial Deviates 294
    7.4 Generation of Random Bits 300
    7.5 Random Sequences Based on Data Encryption 304
    7.6 Simple Monte Carlo Integration 308
    7.7 Quasi- (that is, Sub-) Random Sequences 313
    7.8 Adaptive and Recursive Monte Carlo Methods 320

    8 Sorting 332
    8.0 Introduction 332
    8.1 Straight Insertion and Shell’s Method 333
    8.2 Quicksort 336
    8.3 Heapsort 339
    8.4 Indexing and Ranking 341
    8.5 Selecting the Mth Largest 344
    8.6 Determination of Equivalence Classes 348

    9 Root Finding and Nonlinear Sets of Equations 351
    9.0 Introduction 351
    9.1 Bracketing and Bisection 354
    9.2 Secant Method, False Position Method, and Ridders’ Method 358
    9.3 Van Wijngaarden–Dekker–Brent Method 363
    9.4 Newton-Raphson Method Using Derivative 366
    9.5 Roots of Polynomials 373
    9.6 Newton-Raphson Method for Nonlinear Systems of Equations 383
    9.7 Globally Convergent Methods for Nonlinear Systems of Equations 387

    10 Minimization or Maximization of Functions 398
    10.0 Introduction 398
    10.1 Golden Section Search in One Dimension 401
    10.2 Parabolic Interpolation and Brent’s Method in One Dimension 406
    10.3 One-Dimensional Search with First Derivatives 410
    10.4 Downhill Simplex Method in Multidimensions 413
    10.5 Direction Set (Powell’s) Methods in Multidimensions 417
    10.6 Conjugate Gradient Methods in Multidimensions 424
    10.7 Variable Metric Methods in Multidimensions 430
    10.8 Linear Programming and the Simplex Method 434
    10.9 Simulated Annealing Methods 448

    11 Eigensystems 461
    11.0 Introduction 461
    11.1 Jacobi Transformations of a Symmetric Matrix 468
    11.2 Reduction of a Symmetric Matrix to Tridiagonal Form: Givens and Householder Reductions 474
    11.3 Eigenvalues and Eigenvectors of a Tridiagonal Matrix 481
    11.4 Hermitian Matrices 486
    11.5 Reduction of a General Matrix to Hessenberg Form 487viii Contents
    11.6 The QR Algorithm for Real Hessenberg Matrices 491
    11.7 Improving Eigenvalues and/or Finding Eigenvectors by Inverse Iteration 498

    12 Fast Fourier Transform 501
    12.0 Introduction 501
    12.1 Fourier Transform of Discretely Sampled Data 505
    12.2 Fast Fourier Transform (FFT) 509
    12.3 FFT of Real Functions, Sine and Cosine Transforms 515
    12.4 FFT in Two or More Dimensions 526
    12.5 Fourier Transforms of Real Data in Two and Three Dimensions 530
    12.6 External Storage or Memory-Local FFTs 536

    13 Fourier and Spectral Applications 542
    13.0 Introduction 542
    13.1 Convolution and Deconvolution Using the FFT 543
    13.2 Correlation and Autocorrelation Using the FFT 550
    13.3 Optimal (Wiener) Filtering with the FFT 552
    13.4 Power Spectrum Estimation Using the FFT 555
    13.5 Digital Filtering in the Time Domain 563
    13.6 Linear Prediction and Linear Predictive Coding 569
    13.7 Power Spectrum Estimation by the Maximum Entropy (All Poles) Method 577
    13.8 Spectral Analysis of Unevenly Sampled Data 580
    13.9 Computing Fourier Integrals Using the FFT 589
    13.10 Wavelet Transforms 596
    13.11 Numerical Use of the Sampling Theorem 611

    14 Statistical Description of Data 614
    14.0 Introduction 614
    14.1 Moments of a Distribution: Mean, Variance, Skewness, and So Forth 615
    14.2 Do Two Distributions Have the Same Means or Variances? 620
    14.3 Are Two Distributions Different? 625
    14.4 Contingency Table Analysis of Two Distributions 633
    14.5 Linear Correlation 641
    14.6 Nonparametric or Rank Correlation 644
    14.7 Do Two-Dimensional Distributions Differ? 650
    14.8 Savitzky-Golay Smoothing Filters 655

    15 Modeling of Data 661
    15.0 Introduction 661
    15.1 Least Squares as a Maximum Likelihood Estimator 662
    15.2 Fitting Data to a Straight Line 666
    15.3 Straight-Line Data with Errors in Both Coordinates 671
    15.4 General Linear Least Squares 676
    15.5 Nonlinear Models 686Contents ix
    15.6 Con?dence Limits on Estimated Model Parameters 694
    15.7 Robust Estimation 704

    16 Integration of Ordinary Differential Equations 712
    16.0 Introduction 712
    16.1 Runge-Kutta Method 715
    16.2 Adaptive Stepsize Control for Runge-Kutta 719
    16.3 Modi?ed Midpoint Method 727
    16.4 Richardson Extrapolation and the Bulirsch-Stoer Method 729
    16.5 Second-Order Conservative Equations 737
    16.6 Stiff Sets of Equations 739
    16.7 Multistep, Multivalue, and Predictor-Corrector Methods 751

    17 Two Point Boundary Value Problems 756
    17.0 Introduction 756
    17.1 The Shooting Method 760
    17.2 Shooting to a Fitting Point 762
    17.3 Relaxation Methods 765
    17.4 A Worked Example: Spheroidal Harmonics 775
    17.5 Automated Allocation of Mesh Points 785
    17.6 Handling Internal Boundary Conditions or Singular Points 787

    18 Integral Equations and Inverse Theory 790
    18.0 Introduction 790
    18.1 Fredholm Equations of the Second Kind 793
    18.2 Volterra Equations 796
    18.3 Integral Equations with Singular Kernels 799
    18.4 Inverse Problems and the Use of A Priori Information 806
    18.5 Linear Regularization Methods 811
    18.6 Backus-Gilbert Method 818
    18.7 Maximum Entropy Image Restoration 821

    19 Partial Differential Equations 829
    19.0 Introduction 829
    19.1 Flux-Conservative Initial Value Problems 836
    19.2 Diffusive Initial Value Problems 849
    19.3 Initial Value Problems in Multidimensions 855
    19.4 Fourier and Cyclic Reduction Methods for Boundary Value Problems 859
    19.5 Relaxation Methods for Boundary Value Problems 865
    19.6 Multigrid Methods for Boundary Value Problems 873

    20 Less-Numerical Algorithms 891
    20.0 Introduction 891
    20.1 Diagnosing Machine Parameters 891
    20.2 Gray Codes 896x Contents
    20.3 Cyclic Redundancy and Other Checksums 898
    20.4 Huffman Coding and Compression of Data 906
    20.5 Arithmetic Coding 912
    20.6 Arithmetic at Arbitrary Precision 916

    References 927

    Appendix A: Table of Function Declarations 931

    Appendix B: Utility Routines and Classes 939

    Appendix C: Converting to Single Precision 957

    Index of Programs and Dependencies 959

    General Index 972
     
    Last edited: Jan 26, 2013
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