Introduction

Numerical methods in computer science refer to mathematical algorithms and techniques used to solve problems involving numerical computations. These methods aim to approximate solutions to mathematical problems that may not have exact analytical solutions or are computationally expensive to solve directly. Numerical methods play a crucial role in various fields, including scientific computing, engineering, physics, finance, and more.

A typical chapter or course on numerical methods covers several fundamental topics. Here are some common topics you can expect to learn in a numerical methods chapter/course:

1. Numerical Approximation:

- Round-off errors and sources of error in numerical computations.

- Truncation errors and their impact on the accuracy of numerical methods.

- Error analysis and estimation techniques.

2. Root Finding:

- Bisection method.

- False Position method.

- Newton's method (Newton-Raphson method).

- Secant method.

- Fixed-point iteration method.

3. Interpolation and Extrapolation:

- Polynomial interpolation (Lagrange interpolation, Newton interpolation).

- Divided differences and Newton's divided difference interpolation.

- Error analysis in interpolation.

- Extrapolation techniques (e.g., Richardson extrapolation).

4. Numerical Integration:

- Newton-Cotes formulas (e.g., Trapezoidal rule, Simpson's rule).

- Composite integration rules.

- Gaussian quadrature.

- Error estimation in numerical integration.

5. Systems of Linear Equations:

- Gaussian elimination.

- LU decomposition.

- Iterative methods (e.g., Jacobi method, Gauss-Seidel method).

- Matrix factorizations (e.g., Cholesky decomposition).

6. Numerical Optimization:

- Unconstrained optimization methods (e.g., golden section search, Newton's method, gradient descent).

- Constrained optimization and linear programming.

- Multi-dimensional optimization.

7. Numerical Solutions of Differential Equations:

- Initial value problems and ordinary differential equations.

- Euler's method.

- Runge-Kutta methods.

- Finite difference methods.

- Boundary value problems.

8. Numerical Linear Algebra:

- Matrix operations (e.g., matrix multiplication, matrix inversion).

- Eigenvalue and eigenvector computations.

- Singular value decomposition.

In this unit we are going to learn about the mentioned topics briefly.