topics: |  computer | algorithm | numerical

1 Preliminaries
  1.0 Introduction 1
  1.1 Program Organization and Control Structures 5
  1.2 Some C Conventions for Scientific Computing 15
  1.3 Error, Accuracy, and Stability 15
2 Solution of Linear Algebraic Equations
  2.0 Introduction 32
  2.1 Gauss-Jordan Elimination 36
  2.2 Gaussian Elimination with Backsubstitution 41
  2.3 LU Decomposition and Its Applications 43
  2.4 Tridiagonal and Band Diagonal Systems of Equations 50
  2.5 Iterative Improvement of a Solution to Linear Equations 55
  2.6 Singular Value Decomposition 59
  2.7 Sparse Linear Systems 71
  2.8 Vandermonde Matrices and Toeplitz Matrices 90
  2.9 Cholesky Decomposition 96
  2.10 QR Decomposition 98
  2.11 Is Matrix Inversion an $N^3$ Process? 102
3 Interpolation and Extrapolation
  3.0 Introduction 105
  3.1 Polynomial Interpolation and Extrapolation 108
  3.2 Rational Function Interpolation and Extrapolation 111
  3.3 Cubic Spline Interpolation 113
  3.4 How to Search an Ordered Table 117
  3.5 Coefficients of the Interpolating Polynomial 120
  3.6 Interpolation in Two or More Dimensions 123
4 Integration of Functions
  4.0 Introduction 129
  4.1 Classical Formulas for Equally Spaced Abscissas 130
  4.2 Elementary Algorithms 136
  4.3 Romberg Integration 140
  4.4 Improper Integrals 141
  4.5 Gaussian Quadratures and Orthogonal Polynomials 147
  4.6 Multidimensional Integrals 161
5 Evaluation of Functions
  5.0 Introduction 165
  5.1 Series and Their Convergence 165
  5.2 Evaluation of Continued Fractions 169
  5.3 Polynomials and Rational Functions 173
  5.4 Complex Arithmetic 176
  5.5 Recurrence Relations and Clenshaw's Recurrence Formula 178
  5.6 Quadratic and Cubic Equations 183
  5.7 Numerical Derivatives 186
  5.8 Chebyshev Approximation 190
  5.9 Derivatives or Integrals of a Chebyshev-approximated Function 195
  5.10 Polynomial Approximation from Chebyshev Coefficients 197
  5.11 Economization of Power Series 198
  5.12 Pad\'e Approximants 200
  5.13 Rational Chebyshev Approximation 204
  5.14 Evaluation of Functions by Path Integration 208
6 Special Functions
  6.0 Introduction 212
  6.1 Gamma Function, Beta Function, Factorials, Binomial Coefficients 213
  6.2 Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function 216
  6.3 Exponential Integrals 222
  6.4 Incomplete Beta Function, Student's Distribution, F-Distribution,Cumulative Binomial Distribution 226
  6.5 Bessel Functions of Integer Order 230
  6.6 Modified Bessel Functions of Integer Order 236
  6.7 Bessel Functions of Fractional Order, Airy Functions, SphericalBessel Functions 240
  6.8 Spherical Harmonics 252
  6.9 Fresnel Integrals, Cosine and Sine Integrals 255
  6.10 Dawson's Integral 259
  6.11 Elliptic Integrals and Jacobian Elliptic Functions 261
  6.12 Hypergeometric Functions 271
7 Random Numbers
  7.0 Introduction 274
  7.1 Uniform Deviates 275
  7.2 Transformation Method: Exponential and Normal Deviates 287
  7.3 Rejection Method: Gamma, Poisson, Binomial Deviates 290
  7.4 Generation of Random Bits 296
  7.5 Random Sequences Based on Data Encryption 300
  7.6 Simple Monte Carlo Integration 304
  7.7 Quasi- (that is, Sub-) Random Sequences 309
  7.8 Adaptive and Recursive Monte Carlo Methods 316
8 Sorting
  8.0 Introduction 329
  8.1 Straight Insertion and Shell's Method 330
  8.2 Quicksort 332
  8.3 Heapsort 336
  8.4 Indexing and Ranking 338
  8.5 Selecting the $M$th Largest 341
  8.6 Determination of Equivalence Classes 345
9 Root Finding and Nonlinear Sets of Equations
  9.0 Introduction 347
  9.1 Bracketing and Bisection 350
  9.2 Secant Method, False Position Method, and Ridders' Method 354
  9.3 Van Wijngaarden--Dekker--Brent Method 359
  9.4 Newton-Raphson Method Using Derivative 362
  9.5 Roots of Polynomials 369
  9.6 Newton-Raphson Method for Nonlinear Systems of Equations 379
  9.7 Globally Convergent Methods for Nonlinear Systems of Equations 383
10 Minimization or Maximization of Functions
  10.0 Introduction 394
  10.1 Golden Section Search in One Dimension 397
  10.2 Parabolic Interpolation and Brent's Method in One Dimension 402
  10.3 One-Dimensional Search with First Derivatives 305
  10.4 Downhill Simplex Method in Multidimensions 408
  10.5 Direction Set (Powell's) Methods in Multidimensions 412
  10.6 Conjugate Gradient Methods in Multidimensions 420
  10.7 Variable Metric Methods in Multidimensions 425
  10.8 Linear Programming and the Simplex Method 430
  10.9 Simulated Annealing Methods 444
11 Eigensystems
  11.0 Introduction 456
  11.1 Jacobi Transformations of a Symmetric Matrix 463
  11.2 Reduction of a Symmetric Matrix to Tridiagonal Form: Givens and Householder Reductions 469
  11.3 Eigenvalues and Eigenvectors of a Tridiagonal Matrix 475
  11.4 Hermitian Matrices 481
  11.5 Reduction of a General Matrix to Hessenberg Form 482
  11.6 The QR Algorithm for Real Hessenberg Matrices 486
  11.7 Improving Eigenvalues and/or Finding Eigenvectors by Inverse Iteration 493
12 Fast Fourier Transform
  12.0 Introduction 496
  12.1 Fourier Transform of Discretely Sampled Data 500
  12.2 Fast Fourier Transform (FFT) 504
  12.3 FFT of Real Functions, Sine and Cosine Transforms 510
  12.4 FFT in Two or More Dimensions 521
  12.5 Fourier Transforms of Real Data in Two and Three Dimensions 525
  12.6 External Storage or Memory-Local FFTs 532
13 Fourier and Spectral Applications
  13.0 Introduction 537
  13.1 Convolution and Deconvolution Using the FFT 538
  13.2 Correlation and Autocorrelation Using the FFT 545
  13.3 Optimal (Wiener) Filtering with the FFT 547
  13.4 Power Spectrum Estimation Using the FFT 549
  13.5 Digital Filtering in the Time Domain 558
  13.6 Linear Prediction and Linear Predictive Coding 564
  13.7 Power Spectrum Estimation by the Maximum Entropy (All Poles) Method 572
  13.8 Spectral Analysis of Unevenly Sampled Data 575
  13.9 Computing Fourier Integrals Using the FFT 584
  13.10 Wavelet Transforms 591
  13.11 Numerical Use of the Sampling Theorem 606
14 Statistical Description of Data
  14.0 Introduction 609
  14.1 Moments of a Distribution: Mean, Variance, Skewness, and So Forth 610
  14.2 Do Two Distributions Have the Same Means or Variances? 615
  14.3 Are Two Distributions Different? 620
  14.4 Contingency Table Analysis of Two Distributions 628
  14.5 Linear Correlation 636
  14.6 Nonparametric or Rank Correlation 639
  14.7 Do Two-Dimensional Distributions Differ? 645
  14.8 Savitzky-Golay Smoothing Filters 650
15 Modeling of Data
  15.0 Introduction 656
  15.1 Least Squares as a Maximum Likelihood Estimator 657
  15.2 Fitting Data to a Straight Line 661
  15.3 Straight-Line Data with Errors in Both Coordinates 666
  15.4 General Linear Least Squares 671
  15.5 Nonlinear Models 681
  15.6 Confidence Limits on Estimated Model Parameters 689
  15.7 Robust Estimation 699
16 Integration of Ordinary Differential Equations
  16.0 Introduction 707
  16.1 Runge-Kutta Method 710
  16.2 Adaptive Stepsize Control for Runge-Kutta 714
  16.3 Modified Midpoint Method 722
  16.4 Richardson Extrapolation and the Bulirsch-Stoer Method 724
  16.5 Second-Order Conservative Equations 732
  16.6 Stiff Sets of Equations 734
  16.7 Multistep, Multivalue, and Predictor-Corrector Methods 747
17 Two Point Boundary Value Problems
  17.0 Introduction 753
  17.1 The Shooting Method 757
  17.2 Shooting to a Fitting Point 760
  17.3 Relaxation Methods 762
  17.4 A Worked Example: Spheroidal Harmonics 772
  17.5 Automated Allocation of Mesh Points 783
  17.6 Handling Internal Boundary Conditions or Singular Points 784
18 Integral Equations and Inverse Theory
  18.0 Introduction 788
  18.1 Fredholm Equations of the Second Kind 791
  18.2 Volterra Equations 794
  18.3 Integral Equations with Singular Kernels 797
  18.4 Inverse Problems and the Use of A Priori Information 804
  18.5 Linear Regularization Methods 808
  18.6 Backus-Gilbert Method 815
  18.7 Maximum Entropy Image Restoration 818
19 Partial Differential Equations
  19.0 Introduction 827
  19.1 Flux-Conservative Initial Value Problems 834
  19.2 Diffusive Initial Value Problems 847
  19.3 Initial Value Problems in Multidimensions 853
  19.4 Fourier and Cyclic Reduction Methods for Boundary Value Problems 857
  19.5 Relaxation Methods for Boundary Value Problems 863
  19.6 Multigrid Methods for Boundary Value Problems 871
20 Less-Numerical Algorithms
  20.0 Introduction 889
  20.1 Diagnosing Machine Parameters 889
  20.2 Gray Codes 894
  20.3 Cyclic Redundancy and Other Checksums 896
  20.4 Huffman Coding and Compression of Data 903
  20.5 Arithmetic Coding 910
  20.6 Arithmetic at Arbitrary Precision 915