Introduction
1.
Introduction to Numerical Analysis
1.1.
Importance of Numerical Methods
1.2.
How computers represent numbers
1.2.1.
Integers
1.2.2.
Floating Point Numbers
1.2.3.
Rounding Errors
1.2.4.
Overflow and Underflow
1.3.
Errors in Numerical Computation
1.4.
Stability and Convergence
2.
Root Finding Algorithms
2.1.
Bisection Method
2.2.
Newton-Raphson Method
2.3.
Secant Method
2.4.
Fixed Point Iteration
3.
Linear Algebraic Equations
3.1.
Gaussian Elimination
3.2.
LU Decomposition
3.3.
Matrix Inversion
3.4.
Iterative Methods
4.
Interpolation and Approximation
4.1.
Polynomial Interpolation
4.2.
Lagrange and Newton Forms
4.3.
Splines
4.4.
Least Squares Approximation
5.
Numerical Differentiation and Integration
5.1.
Finite Difference Methods
5.2.
Richardson Extrapolation
5.3.
Trapezoidal Rule
5.4.
Simpson’s Rule
5.5.
Gaussian Quadrature
6.
Ordinary Differential Equations (ODEs)
6.1.
Initial Value Problems
6.2.
Euler's Method
6.3.
Runge-Kutta Methods
6.4.
Stability and Stiffness
6.5.
Boundary Value Problems
7.
Partial Differential Equations (PDEs)
7.1.
Finite Difference Method
7.2.
Method of Lines
7.3.
Finite Element Method Basics
8.
Numerical Optimization
8.1.
Unconstrained Optimization
8.2.
Constrained Optimization
9.
Fast Fourier Transform (FFT) and Signal Processing
9.1.
Discrete Fourier Transform and FFT
9.2.
Applications in Signal Processing
10.
Special Topics
10.1.
Monte Carlo Methods
10.2.
Random Number Generation
10.3.
Machine Learning Basics
11.
Case Studies and Practical Applications
12.
Appendices
12.1.
Mathematical Preliminaries
12.2.
Programming Basics in Rust
12.3.
Advanced Topics in Numerical Analysis
Light
Rust
Coal
Navy
Ayu
Numerical Algorithms in Rust
Random Number Generation