Numerical Analysis 1

Semester: 1

Target audience: 2nd-year engineering students in science and technology

Objectives:

This course serves as an introduction to scientific computing. Its objectives are to:

  • Present fundamental numerical methods that enable solving real-world engineering problems.
  • Identify the challenges associated with the numerical resolution of real-world problems.
  • Develop and apply discretization methods for continuous problems.
  • Master and implement fundamental techniques of numerical matrix analysis.
  • Apply basic numerical computation techniques effectively.

Course Content:

Chapter 1: Introduction to Numerical Analysis

1.1. Sources of Errors: Modeling errors, data errors, approximate values, error propagation, relative and absolute errors, floating-point arithmetic, IEEE-754 standard, rounding errors, truncation error, significant digits, risky operations.
1.2. Conditioning and Stability: Examples of numerical instabilities, problem conditioning.
1.3. Methods and Algorithms: Exact methods, approximate methods, iterative methods.

Chapter 2: Solving Nonlinear Equations

2.1. Functions of a Real Variable: Localization theorems and root separation.
2.2. Classical Methods: Bisection method, secant method, stopping criteria.
2.3. Iterative Methods: Fixed-point method, Newton’s method, order of convergence, stopping criteria.

Chapter 3: Solving Linear Systems

3.1. Direct Methods: Upper (or lower) triangular matrix, symmetric matrices (definitions and properties), Gaussian elimination method, LU factorization (Crout, Doolittle), Cholesky factorization (for positive definite symmetric matrices).
3.2. Numerical Algebra Vocabulary: Vector norms, matrix norms, matrix conditioning (definitions and properties), spectral radius, example of an ill-conditioned linear system.
3.3. Iterative Methods: Jacobi method, Gauss-Seidel method, relaxation methods, convergence analysis of iterative methods, stopping criteria.