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1. Preliminaries
1.1. Mathematical Preliminaries
1.2. Error Analysis
1.3. Algorithms and Convergence
1.4. Big O Notation
2. Root-finding algorithms
2.1. Bracketing Methods
2.2. Fixed-point iteration
2.3. Newton methods
2.4. Error Analysis for Iterative Methods
2.5. Acceleration of Convergence
2.6. Zeros of Polynomials and Müller’s Method
3. Interpolation and Polynomial Approximation
3.1. Introduction
3.2. Vandermonde Method
3.3. Interpolation and Lagrange polynomial
3.4. Newton Polynomial
3.5. Linear Regression
4. Numerical Differentiation
4.1. Approximation of the First Derivative of a Function
4.2. Approximation of the second derivative of a function
4.3. Higher-order approximations
5. Numerical Integration
5.1. Introduction
5.2. Trapezoidal rule
5.3. Simpson’s rule
5.4. Midpoint rule
5.5. Gaussian quadrature
6. Initial-Value Problems for Ordinary Differential Equations
6.1. Introduction
6.2. The elementary theory of initial-value problem
6.3. First-order methods
6.4. Higher-Order Taylor Methods
6.5. Runge–Kutta methods
7. Numerical Linear Algebra
7.1. Gaussian elimination for Linear Systems of Equations
7.2. Matrix Factorization
7.3. Cholesky factorization
7.4. LDL decomposition
7.5. QR factorization
8. Regularization Methods
8.1. Savitzky-Golay Filter
9. References
Index
