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Numerical Methods

• 36 h - Lectures (18h), tutorial and practical work (18h)
• Experimental project : no

The main goals of these lectures are to introduce concepts of numerical methods that are typically encountered (and used) in science and engineering. The material is developed in tandem with Python which allows rapid prototyping and testing of the methods. The lectures are designed to be used in a computer classroom, but could be used in a lecture format with students doing computer exercises afterward.

• Description of the lectures:
  • Series expansions for functions
    1. Taylor series in one, two, and p dimensions
    2. Finite difference
       • 1-dimensional problem
       • 2-dimensional problem
    3. Second order linear partial differential equations in two independent variables
  • Matrices: fundamental definitions
    1. Matrix notation
    2. Arithmetic of matrices
    3. Determinant of a square matrix
    4. Inverse of a square matrix
    5. Eigenvalues and eigenvectors
    6. Solution of q linear equations in p unknowns
    7. Solution of q nonlinear equations in p unknowns (q>p)
  • Numerical solution of ordinary differential equations
    1. Two-point boundary value problems
    2. Initial value differential equations
       • Euler's method
       • Modified Euler's method
       • Method of Runge-Kutta
  • Numerical integration techniques
    1. Lagrange interpolation
    2. Numerical integration techniques
       • One-point formula: rectangle rule for integration
       • Two-point formula: trapezoid rule for integration
       • Three-point formula: Simpson's rule for integration
  • Roots of functions of a single variable
    1. Roots of polynomials
    2. Roots of any real function
       • Bisection method
       • False position method
       • Newton-Raphson method
    3. Simultaneous nonlinear functions
  • Minimization of a multivariable function
    1. Direct and inverse model
       • Validation of a model
       • Direct mode and inverse mode
    2. Optimization methods
    3. Linear least squares fitting of polynomials of degree p
    4. Minimization of a multivariable function
       • Gradient method
       • Quasi-Newton method
       • Simplex method
    5. Application to the Rosenbrock function
• Description of the tutorials:
  • TC1: Numerical integration
  • TC2: Satellite orbit
  • TC3: Heat equation
  • TC4: Kubelka-Munk theory
  • TC5: Optimization
  • TC6: Bessel function