1 Preliminaries
1.0 Introduction
1.1 Program Organization and Control Structures
1.2 Some C++ Conventions for Scientific Computing
1.3 Imptementation of the Vector and Matrix Classes
1.4 Error, Accuracy, and Stability
2 Solution of Linear Algebraic Equations
2.0 Introduction
2.1 Gauss-Jordan Elimination
2.2 Gaussian Elimination with Backsubstitution
2.3 LU Decomposition and Its Applications
2.4 Tridiagonal and Band Diagonal Systems of Equations
2.5 Iterative Improvement of a Solution to Linear Equations
2.6 Singular Value Decomposition
2.7 Sparse Linear Systems
2.8 Vandermonde Matrices and Toeplitz Matrices
2.9 Cholesky Decomposition
2.10 QR Decomposition
2.11 Is Matrix Inversion an N3 Process?
3 Interpolation and Extrapolation
3.0 Introduction
3.1 Polyaomial Interpolation and Extrapolation
3.2 Rational Function Interpolation and Extrapolation
3.3 Cubic Spline Interpolation
3.4 How to Search an Ordered Table
3.5 Coefficients of the Interpolating Polynomial
3.6 Interpolation in Two or More Dimensions
4 Integration of Functions
4.O Introduction
4.1 Classical Formulas for Equally Spaced Abscissas
4.2 Elementary Algorithms
4.3 Romberg Integration
4.4 Improper Integrals
4.5 Gaussian Quadratures and Orthogonal Polynomials
4.6 Multidimensional Integrals
5 Evaluation of Functions
5.0 Introduction
5.1 Series and Their Convergence
5.2 Evaluation of Continued Fractions
5.3 Polynomials and Rational Functions
5.4 Complex Arithmetic
5.5 Recurrence Relations and Clenshaw's Recurrence Formula
5.6 Quadratic and Cubic Equations
5.7 Numerical Derivatives
5.8 Chebyshev Approximation
5.9 Derivatives or Integrals of a Chebyshev-approximated Function
3.10 Polynomial Approximation from Chebyshev Coefficients
5.11 Economization of Power Series
5.12 Pade Approximants
5.13 Rational Chebyshev Appmximation
5.14 Evaluation of Functions by Path Integration
6 Speclal Functions
6.0 Introduction
6.1 Gamma Function, Beta Function, Factorials. Binomial Coefficients
6.2 Incomplete Gamma Function, Error Function. Chi-Square Probability Function, Cumulative Poisson Function
6.3 Exponential Integrals
6.4 Incomplete Beta Function, Student's Distribution, F-Distribution Cumulative Binomial Distribution
6.5 Bessel Functions of Integer Order
6.6 Modified Bessel Functions of Integer Order
6.7 Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions
6.8 Spherical Harmonics
6.9 Fresnel Integrals, Cosine and Sine Integrals
6.10 Dawson's Inegral
6.ll Elliptic Integrals and Jacobian Elliptic Functions
6.12 Hypergeometric Functions
7 Random Numbers
7.O Introduction
7.1 Uniform Deviates
7.2 Transformation Method: Exponential and Normal Deviates
7.3 Rejection Method: Gamma. Poisson, Binomial Deviates
7.4 Generation of Random Bits
7.5 Random Sequences Based on Data Encryption
7.6 Simple Monte Carlo Integration
7.7 Quasi- (that is, Sub-) Random Sequences
7.8 Adaptive and Recursive Monte Carlo Methods
8 Sorting
8.0 Introduction
8.1 Straight Insertion and Shell's Method
8.2 Quicksort
8.3 Heapsort
8.4 Indexing and Ranking
8.5 Selecting the Mth Largest
8.6 Determination of Equivalence Classes
9 Root Finding and Nonlinear Sets of Equations
9.0 Introduction
9.l Bracketing and Bisection
9.2 Secant Method, False Position Method, and Ridders' Method
9.3 Van Wijngaarden-Dekker-Brent Method
9.4 Newton-Raphson Metkod Using Derivative
9.5 Roots of Polynomials
9.6 Newton-Raphson Mealod for Nonlinear Systems of Equations
9.7 Globally Convergent Methods for Nonlinear Systems of Equations
10 Minimization or Maximization of Functions
10.0 Introduction
10.1 Golden Section Search in One Dimension
10.2 Parabolic lnterpolation and Brent's Method in One Dimension
10.3 One-Dimensional Search with First Derivatives
10.4 Downhill Simplex Method in Multidimensions
10.5 Direction Set (Powell's) Methods in Multidimensions
10.6 Conjugate Gadient Methods in Multidimensions
10.7 Variable Metric Methods in Multidimensions
10.8 Linear Pro
10.9 Simulated Annealing Meathods
11 Eigensystems
11.0 Introduction
11.1 Jacobi Transformations of a Symmetric Matrix
11.2 Reduction of a Symmetric Matrix to Tridiagonal Form
Givens and Householder Reductions
11.3 Eigenvalues and Eigenvectors of a Tridiagonal Matrix
11.4 Hermitian Matrices
11.5 Reduction of a General Matrix to Hessenberg Form
11.6 The QR Algorithm for Real Hessenbery Matrices
11.7 Improving Eigenvalues and/or Finding Eigenvectors by
Inverse Iteration
12 Fast Fourier Transform
12.0 Introduction
12.1 Fourier Transform of Discretely Sampled Data
12.2 Fast Fourier Transform (FFT)
12.3 FFT of Real Functions, Sine and Cosine Transforms
12.4 FFT in Two or More Dimensions
12.5 Fourier Transforms of Real Data in Two and Three Dimensions
12.6 External Storage or Memory-Local FFTs
13 Fourier and Spectral Applications
13.0 Introduction
13.1 Convolution and Deconvolution Using the FFT
13.2 Correlation and Autocorrelation Using the FFT
13.3 Optimal (Wiener) Filterin with the FFT
13.4 Power Spectrum Estimation Using the FFT
13.5 Digital Filtering in ahe Time Domain
13.6 Linear Prediction and Linear Predictive Coding
13.7 Power Spectrum Estimation by the Maximum Entropy
(All Poles) Method
13.8 Spectral Analysis of Unevenly Sampled Data
13.9 Computing Fourier Integrals Using Ule FFT
13.10 Wavelet Transforms
13.11 Numerical Use of the Sampling Theorem
14 Statistical Deseription of Data
14.O Introduction
14.1 Moments of a Distribution: Mean, Variance, Skewness,
and So Forth
14.2 Do Two Distributions Have the Same Means or Variances?
14.3 Are Two Distributions Different?
14.4 Contingency Table Analysis of Two Distributions
14.5 Linear Correlation
14.6 Nonparametric or Rank Correlation
14.7 Do Two-Dimensional Distributions Differ?
14.8 Savitzky-Golay Smoothing Filters
15 Modeling of Data
15.0 Introduction
15.1 Least Squares as a Maximum Likelihood Estimator
15.2 Fitting Data to a Straight Line
15.3 Straight-Line Data with Errors in Both Coordinates
15.4 General Linear Least Squares
15.5 Nonlinear Models
15.6 Confidence Limits on Estimated Model Parameters
15.7 Robust Estimation
16 Integration of Ordinary Differential Equations
l6.0 Introduction
16.1 Runge-Kutta Method
16.2 Adaptive Stepsize Control for Runge-Kutta
16.3 Modified Midpoint Mealod
16.4 Richardson Extrapolation and the Bulirsch-Stoer Method
16.5 Second-Order Conservative Equations
16.6 Stiff Sets of Equations
l6.7 Multistep, Multivalue, and Predictor-Corrector Methods
17 Two Point Boundary Value Problems
17.O Introdnction
17.1 The Shooting Method
17.2 Shooting to a Fitting Point
17.3 Relaxation Methods
17.4 A Worked Example: Spheroidal Harmonics
17.5 Automated Allocation of Mesh Points
17.6 Handling Internal Boundary Conditions or Singular Points
18 Integral Equations and Inverse Theory
18.0 Introduction
18.1 Fredholm Equations of the Second Kind
18.2 Volterra Equations
18.3 Integral Equations with Singular Kernels
18.4 Inverse Problems and the Use of A Priori Information
18.5 Linear Regularization Methods
18.6 Backus-Gilbert Method
18.7 Maximum Entropy Image Restoration
19 Partial Differential Equations
19.0 Introduction
19.1 Flux-Conservative Initial Value Problems
19.2 Diffusive Initial Value Problems
19.3 Initial Value Problems in Multidimensions
19.4 Fourier and Cyclic Reduction Methods for Boundary
Value Problems
19.5 Relaxation Methods for Boundary Value Problems
19.6 Multigrid Methods for Boundary Value Problems
20 Less-Numerical Algorithms
20.0 Introduction
20.1 Diagnosing Machine Parameters
20.2 Gray Codes
20.3 Cyclic Redundancy and Other Checksums
20.4 Huffman Coding and Compression of Data
20.5 Arithmetic Coding
20.6 Arithmetic at Arbitrary Precision
References
Appendix A: Table of Funetion Declarations
Appendix B: Utility Routines and Classes
Appendix C: Convetring to Single Precision
Index of Programs and Dependencies
General Index
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