Numerical Recipes 3rd Edition: The Art of Scientific ComputingPDF|Epub|txt|kindle电子书版本网盘下载

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1 Preliminaries1

1.0 Introduction1

1.1 Error，Accuracy，and Stability8

1.2 C Family Syntax12

1.3 Objects，Classes，and Inheritance17

1.4 Vector and Matrix Objects24

1.5 Some Further Conventions and Capabilities30

2 Solution of Linear Algebraic Equations37

2.0 Introduction37

2.1 Gauss-Jordan Elimination41

2.2 Gaussian Elimination with Backsubstitution46

2.3 LU Decomposition and Its Applications48

2.4 Tridiagonal and Band-Diagonal Systems of Equations56

2.5 Iterative Improvement of a Solution to Linear Equations61

2.6 Singular Value Decomposition65

2.7 Sparse Linear Systems75

2.8 Vandermonde Matrices and Toeplitz Matrices93

2.9 Cholesky Decomposition100

2.10 QR Decomposition102

2.11 Is Matrix Inversion an N3 Process？106

3 Interpolation and Extrapolation110

3.0 Introduction110

3.1 Preliminaries：Searching an Ordered Table114

3.2 Polynomial Interpolation and Extrapolation118

3.3 Cubic Spline Interpolation120

3.4 Rational Function Interpolation and Extrapolation124

3.5 Coefficients of the Interpolating Polynomial129

3.6 Interpolation on a Grid in Multidimensions132

3.7 Interpolation on Scattered Data in Multidimensions139

3.8 Laplace Interpolation150

4 Integration of Functions155

4.0 Introduction155

4.1 Classical Formulas for Equally Spaced Abscissas156

4.2 Elementary Algorithms162

4.3 Romberg Integration166

4.4 Improper Integrals167

4.5 Quadrature by Variable Transformation172

4.6 Gaussian Quadratures and Orthogonal Polynomials179

4.7 Adaptive Quadrature194

4.8 Multidimensional Integrals196

5 Evaluation of Functions201

5.0 Introduction201

5.1 Polynomials and Rational Functions201

5.2 Evaluation of Continued Fractions206

5.3 Series and Their Convergence209

5.4 Recurrence Relations and Clenshaw’s Recurrence Formula219

5.5 Complex Arithmetic225

5.6 Quadratic and Cubic Equations227

5.7 Numerical Derivatives229

5.8 Chebyshev Approximation233

5.9 Derivatives or Integrals of a Chebyshev-Approximated Function240

5.10 Polynomial Approximation from Chebyshev Coefficients241

5.11 Economization of Power Series243

5.12 Pade Approximants245

5.13 Rational Chebyshev Approximation247

5.14 Evaluation of Functions by Path Integration251

6 Special Functions255

6.0 Introduction255

6.1 Gamma Function，Beta Function，Factorials，Binomial Coefficients256

6.2 Incomplete Gamma Function and Error Function259

6.3 Exponential Integrals266

6.4 Incomplete Beta Function270

6.5 Bessel Functions of Integer Order274

6.6 Bessel Functions of Fractional Order，Airy Functions，Spherical Bessel Functions283

6.7 Spherical Harmonics292

6.8 Fresnel Integrals，Cosine and Sine Integrals297

6.9 Dawson’s Integral302

6.10 Generalized Fermi-Dirac Integrals304

6.11 Inverse of the Function x log（x）307

6.12 Elliptic Integrals and Jacobian Elliptic Functions309

6.13 Hypergeometric Functions318

6.14 Statistical Functions320

7 Random Numbers340

7.0 Introduction340

7.1 Uniform Deviates341

7.2 Completely Hashing a Large Array358

7.3 Deviates from Other Distributions361

7.4 Multivariate Normal Deviates378

7.5 Linear Feedback Shift Registers380

7.6 Hash Tables and Hash Memories386

7.7 Simple Monte Carlo Integration397

7.8 Quasi- （that is，Sub-） Random Sequences403

7.9 Adaptive and Recursive Monte Carlo Methods410

8 Sorting and Selection419

8.0 Introduction419

8.1 Straight Insertion and Shell’s Method420

8.2 Quicksort423

8.3 Heapsort426

8.4 Indexing and Ranking428

8.5 Selecting the Mth Largest431

8.6 Determination of Equivalence Classes439

9 Root Finding and Nonlinear Sets of Equations442

9.0 Introduction442

9.1 Bracketing and Bisection445

9.2 Secant Method，False Position Method，and Ridders’ Method449

9.3 Van Wijngaarden-Dekker-Brent Method454

9.4 Newton-Raphson Method Using Derivative456

9.5 Roots of Polynomials463

9.6 Newton-Raphson Method for Nonlinear Systems of Equations473

9.7 Globally Convergent Methods for Nonlinear Systems of Equations477

10 Minimization or Maximization of Functions487

10.0 Introduction487

10.1 Initially Bracketing a Minimum490

10.2 Golden Section Search in One Dimension492

10.3 Parabolic Interpolation and Brent’s Method in One Dimension496

10.4 One-Dimensional Search with First Derivatives499

10.5 Downhill Simplex Method in Multidimensions502

10.6 Line Methods in Multidimensions507

10.7 Direction Set （Powell’s） Methods in Multidimensions509

10.8 Conjugate Gradient Methods in Multidimensions515

10.9 Quasi-Newton or Variable Metric Methods in Multidimensions521

10.10 Linear Programming：The Simplex Method526

10.11 Linear Programming：Interior-Point Methods537

10.12 Simulated Annealing Methods549

10.13 Dynamic Programming555

11 Eigensystems563

11.0 Introduction563

11.1 Jacobi Transformations of a Symmetric Matrix570

11.2 Real Symmetric Matrices576

11.3 Reduction of a Symmetric Matrix to Tridiagonal Form：Givens and Householder Reductions578

11.4 Eigenvalues and Eigenvectors of a Tridiagonal Matrix583

11.5 Hermitian Matrices590

11.6 Real Nonsymmetric Matrices590

11.7 The QR Algorithm for Real Hessenberg Matrices596

11.8 Improving Eigenvalues and/or Finding Eigenvectors by Inverse Iteration597

12 Fast Fourier Transform600

12.0 Introduction600

12.1 Fourier Transform of Discretely Sampled Data605

12.2 Fast Fourier Transform （FFT）608

12.3 FFT of Real Functions617

12.4 Fast Sine and Cosine Transforms620

12.5 FFT in Two or More Dimensions627

12.6 Fourier Transforms of Real Data in Two and Three Dimensions631

12.7 External Storage or Memory-Local FFTs637

13 Fourier and Spectral Applications640

13.0 Introduction640

13.1 Convolution and Deconvolution Using the FFT641

13.2 Correlation and Autocorrelation Using the FFT648

13.3 Optimal （Wiener） Filtering with the FFT649

13.4 Power Spectrum Estimation Using the FFT652

13.5 Digital Filtering in the Time Domain667

13.6 Linear Prediction and Linear Predictive Coding673

13.7 Power Spectrum Estimation by the Maximum Entropy （All-Poles） Method681

13.8 Spectral Analysis of Unevenly Sampled Data685

13.9 Computing Fourier Integrals Using the FFT692

13.10 Wavelet Transforms699

13.11 Numerical Use of the Sampling Theorem717

14 Statistical Description of Data720

14.0 Introduction720

14.1 Moments of a Distribution：Mean，Variance，Skewness，and So Forth721

14.2 Do Two Distributions Have the Same Means or Variances？726

14.3 Are Two Distributions Different？730

14.4 Contingency Table Analysis of Two Distributions741

14.5 Linear Correlation745

14.6 Nonparametric or Rank Correlation748

14.7 Information-Theoretic Properties of Distributions754

14.8 Do Two-Dimensional Distributions Differ？762

14.9 Savitzky-Golay Smoothing Filters766

15 Modeling of Data773

15.0 Introduction773

15.1 Least Squares as a Maximum Likelihood Estimator776

15.2 Fitting Data to a Straight Line780

15.3 Straight-Line Data with Errors in Both Coordinates785

15.4 General Linear Least Squares788

15.5 Nonlinear Models799

15.6 Confidence Limits on Estimated Model Parameters807

15.7 Robust Estimation818

15.8 Markov Chain Monte Carlo824

15.9 Gaussian Process Regression836

16 Classification and Inference840

16.0 Introduction840

16.1 Gaussian Mixture Models and k-Means Clustering842

16.2 Viterbi Decoding850

16.3 Markov Models and Hidden Markov Modeling856

16.4 Hierarchical Clustering by Phylogenetic Trees868

16.5 Support Vector Machines883

17 Integration of Ordinary Differential Equations899

17.0 Introduction899

17.1 Runge-Kutta Method907

17.2 Adaptive Stepsize Control for Runge-Kutta910

17.3 Richardson Extrapolation and the Bulirsch-Stoer Method921

17.4 Second-Order Conservative Equations928

17.5 Stiff Sets of Equations931

17.6 Multistep，Multivalue，and Predictor-Corrector Methods942

17.7 Stochastic Simulation of Chemical Reaction Networks946

18 Two-Point Boundary Value Problems955

18.0 Introduction955

18.1 The Shooting Method959

18.2 Shooting to a Fitting Point962

18.3 Relaxation Methods964

18.4 A Worked Example：Spheroidal Harmonics971

18.5 Automated Allocation of Mesh Points981

18.6 Handling Internal Boundary Conditions or Singular Points983

19 Integral Equations and Inverse Theory986

19.0 Introduction986

19.1 Fredholm Equations of the Second Kind989

19.2 Volterra Equations992

19.3 Integral Equations with Singular Kernels995

19.4 Inverse Problems and the Use of A Priori Information1001

19.5 Linear Regularization Methods1006

19.6 Backus-Gilbert Method1014

19.7 Maximum Entropy Image Restoration1016

20 Partial Differential Equations1024

20.0 Introduction1024

20.1 Flux-Conservative Initial Value Problems1031

20.2 Diffusive Initial Value Problems1043

20.3 Initial Value Problems in Multidimensions1049

20.4 Fourier and Cyclic Reduction Methods for Boundary Value Problems1053

20.5 Relaxation Methods for Boundary Value Problems1059

20.6 Multigrid Methods for Boundary Value Problems1066

20.7 Spectral Methods1083

21 Computational Geometry1097

21.0 Introduction1097

21.1 Points and Boxes1099

21.2 KD Trees and Nearest-Neighbor Finding1101

21.3 Triangles in Two and Three Dimensions1111

21.4 Lines，Line Segments，and Polygons1117

21.5 Spheres and Rotations1128

21.6 Triangulation and Delaunay Triangulation1131

21.7 Applications of Delaunay Triangulation1141

21.8 Quadtrees and Octrees：Storing Geometrical Objects1149

22 Less-Numerical Algorithms1160

22.0 Introduction1160

22.1 Plotting Simple Graphs1160

22.2 Diagnosing Machine Parameters1163

22.3 Gray Codes1166

22.4 Cyclic Redundancy and Other Checksums1168

22.5 Huffman Coding and Compression of Data1175

22.6 Arithmetic Coding1181

22.7 Arithmetic at Arbitrary Precision1185

Index1195