- Mathematics Preliminaries and Floating-Point Representation
1.1 Introduction
1.2 Mathematics Preliminaries
1.3 Floating-Point Representation
1.4 Loss of Significance
- Linear Systems
2.1 Naive Gaussian Elimination
2.2 Gaussian Elimination with Scaled Partial Pivoting
2.3 Tridiagonal and Banded Systems
- Nonlinear Equations
3.1 Bisection Method
3.2 Newton's Method
3.3 Secant Method
- Interpolation and Numerical Differentiation
4.1 Polynomial Interpolation
4.2 Errors in Polynomial Interpolation
4.3 Estimating Derivatives and Richardson Extrapolation
- Numerical Integration
5.1 Trapezoid Method
5.2 Romberg Algorithm
5.3 Simpson's Rule and Newton-Cotes Rule
5.4 Gaussian Quadrature Formulas
- Spline Functions
6.1 First-Degree and Second-Degree Splines
6.2 Natural Cubic Splines
6.3 B Splines: Interpolation and Approximation
- Initial-Value Problems
7.1 Taylor Series Methods
7.2 Runge-Kutta Methods
7.3 Adaptive Runge-Kutta and Multistep Method
7.4 Methods for First and Higher-Order Systems
7.5 Adams-Bashforth-Moulton Methods
- More on Linear Systems
8.1 Matrix Factorizations
8.2 Eigenvalues and Eigenvectors
8.3 Power Methods
8.2 Iterative Solution of Linear Systems
- Least Squares Methods and Fourier Series
9.1 Method of Least Squares
9.2 Orthogonal Systems and Chebyshev Polynomials
9.3 Examples of the Least Squares Principle
9.4 Fourier Series
- Monte Carlo Methods and Simulation
10.1 Random Numbers
10.2 Estimation of Areas and Volumes by Monte Carlo Techniques
10.3 Simulation
- Boundary Value Problems
11.1 Shooting Method
11.2 A Discretization Method
- Partial Differential Equations
12.1 Parabolic Problems
12.2 Hyperbolic Problems
12.3 Elliptic Problems
- Minimization of Functions
13.1 One-Variable Case
13.2 Multivariate Case
- Linear Programming Problems
14.1 Standard Forms and Duality
14.2 Simplex Method
14.3 Inconsistent Linear Systems