Fundamental Numerical Methods and Data Analysis

Table of Contents
List of Figures
List of Tables
Preface
Notes to the Internet Edition
1. Introduction and Fundamental Concepts
1.1 Basic Properties of Sets and Groups
1.2 Scalars, Vectors, and Matrices
1.3 Coordinate Systems and Coordinate Transformations
1.4 Tensors and Transformations
1.5 Operators

Chapter 1 Exercises
Chapter 1 References and Additional Reading
2. The Numerical Methods for Linear Equations and Matrices
2.1 Errors and Their Propagation
2.2 Direct Methods for the Solution of Linear Algebraic Equations
a. Solution by Cramer's Rule
b. Solution by Gaussian Elimination
c. Solution by Gauss Jordan Elimination
d. Solution by Matrix Factorization: The Crout Method
e. The Solution of Tri-diagonal Systems of Linear Equations
2.3 Solution of Linear Equations by Iterative Methods
a. Solution by The Gauss and Gauss-Seidel Iteration Methods
b. The Method of Hotelling and Bodewig
c. Relaxation Methods for the Solution of Linear Equations
d. Convergence and Fixed-point Iteration Theory
2.4 The Similarity Transformations and the Eigenvalues and Vectors of a Matrix



Chapter 2 Exercises
Chapter 2 References and Supplemental Reading
3. Polynomial Approximation, Interpolation, and Orthogonal Polynomials
3.1 Polynomials and Their Roots
a. Some Constraints on the Roots of Polynomials
b. Synthetic Division
c. The Graffe Root-Squaring Process
d. Iterative Methods
3.2 Curve Fitting and Interpolation
a. Lagrange Interpolation
b. Hermite Interpolation
c. Splines
d. Extrapolation and Interpolation Criteria
3.3 Orthogonal Polynomials
a. The Legendre Polynomials
b. The Laguerre Polynomials
c. The Hermite Polynomials
d. Additional Orthogonal Polynomials
e. The Orthogonality of the Trigonometric Functions

Chapter 3 Exercises
Chapter 3 References and Supplemental Reading
4. Numerical Evaluation of Derivatives and Integrals
4.1 Numerical Differentiation
a. Classical Difference Formulae
b. Richardson Extrapolation for Derivatives
4.2 Numerical Evaluation of Integrals: Quadrature
a. The Trapezoid Rule
b. Simpson's Rule
c. Quadrature Schemes for Arbitrarily Spaced Functions
d. Gaussian Quadrature Schemes
e. Romberg Quadrature and Richardson Extrapolation
f. Multiple Integrals
4.3 Monte Carlo Integration Schemes and Other Tricks
a. Monte Carlo Evaluation of Integrals
b. The General Application of Quadrature Formulae to Integrals

Chapter 4 Exercises
Chapter 4 References and Supplemental Reading
5. Numerical Solution of Differential and Integral Equations
5.1 The Numerical Integration of Differential Equations
a. One Step Methods of the Numerical Solution of Differential Equations
b. Error Estimate and Step Size Control
c. Multi-Step and Predictor-Corrector Methods
d. Systems of Differential Equations and Boundary Value Problems
e. Partial Differential Equations
5.2 The Numerical Solution of Integral Equations
a. Types of Linear Integral Equations
b. The Numerical Solution of Fredholm Equations
c. The Numerical Solution of Volterra Equations
d. The Influence of the Kernel on the Solution

Chapter 5 Exercises
Chapter 5 References and Supplemental Reading
6. Least Squares, Fourier Analysis, and Related Approximation Norms
6.1 Legendre's Principle of Least Squares
a. The Normal Equations of Least Squares
b. Linear Least Squares
c. The Legendre Approximation
6.2 Least Squares, Fourier Series, and Fourier Transforms
a. Least Squares, the Legendre Approximation, and Fourier Series
b. The Fourier Integral
c. The Fourier Transform
d. The Fast Fourier Transform Algorithm
6.3 Error Analysis for Linear Least-Squares
a. Errors of the Least Square Coefficients
b. The Relation of the Weighted Mean Square Observational Error to the Weighted Mean Square Residual
c. Determining the Weighted Mean Square Residual
d. The Effects of Errors in the Independent Variable
6.4 Non-linear Least Squares
a. The Method of Steepest Descent
b. Linear approximation of f(aj,x)
c. Errors of the Least Squares Coefficients
6.5 Other Approximation Norms
a. The Chebyschev Norm and Polynomial Approximation
b. The Chebyschev Norm, Linear Programming, and the Simplex Method
c. The Chebyschev Norm and Least Squares

Chapter 6 Exercises
Chapter 6 References and Supplementary Reading
7. Probability Theory and Statistics
7.1 Basic Aspects of Probability Theory
a. The Probability of Combinations of Events
b. Probabilities and Random Variables
c. Distributions of Random Variables
7.2 Common Distribution Functions
a. Permutations and Combinations
b. The Binomial Probability Distribution
c. The Poisson Distribution
d. The Normal Curve
e. Some Distribution Functions of the Physical World
7.3 Moments of Distribution Functions
7.4 The Foundations of Statistical Analysis
a. Moments of the Binomial Distribution
b. Multiple Variables, Variance, and Covariance
c. Maximum Likelihood

Chapter 7 Exercises
Chapter 7 References and Supplemental Reading
8. Sampling Distributions of Moments, Statistical Tests, and Procedures
8.1 The t, ?2 , and F Statistical Distribution Functions
a. The t-Density Distribution Function
b. The ?2 -Density Distribution Function
c. The F-Density Distribution Function
8.2 The Level of Significance and Statistical Tests
a. The "Students" t-Test
b. The ?2-test
c. The F-test
d. Kolmogorov-Smirnov Tests
8.3 Linear Regression, and Correlation Analysis
a. The Separation of Variances and the Two-Variable Correlation Coefficient
b. The Meaning and Significance of the Correlation Coefficient
c. Correlations of Many Variables and Linear Regression
d Analysis of Variance
8.4 The Design of Experiments
a. The Terminology of Experiment Design
b. Blocked Designs
c. Factorial Designs
Chapter 8 Exercises
Chapter 8 References and Supplemental Reading
Index

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