Introduction To Numerical Analysis

Timetable

Module aim (aims)

Module aim (aims)
• to teach students to recognize the type of problems that require numerical techniques for their solution
• to show them some examples of the error propagation that can occur when numerical methods are applied
• to teach the students to approximate the solution to some problems that cannot be solved exactly.
• students are taught to use the numerical analysis methods in MS Excel.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

The course is dedicated to students with some knowledge of mathematics and MS Excel.

The course consists of 15h of the lecture and 30 h of the computer classes and 5 ECTS points are given for enrolling for both components of the course (a lecture and classes). 

Syllabus

Week 1: Round-off errors: absolute error, relative error, significant digits.
Week 2: Solutions of nonlinear equations in one variable: the bisection algorithm.
Week 3: The Newton-Raphson method, the secant method. Fixed point iteration.
Week 4: Interpolation and polynomial approximation: Taylor polynomials and Lagrange polynomial.
Week 5: Divided differences method.
Week 6: Numerical differentiation: forward and backward-difference formula.
Week 7: Three-point formula of numerical differentiation. The Richardson’s extrapolation.
Week 8: Numerical integration: trapezoidal rule and Simpson’s rule.
Week 9: Initial value-problem for differential equations: Euler’s method, the Runge-Kutta methods.
Week 10: Methods for solving linear systems: linear systems of equations, Cramer’s rule, Gaussian elimination.
Week 11: Approximation theory: least-squares approximation.
Week 12: Linear algebra, matrix inversion and the determinant of a matrix.
Week 13: The similarity transformations. Eigenvalues and eigenvectors.
Week 14: Iterative techniques in matrix algebra: Jacobi iterative method.
Week 15: Optimization.

Reading list

1) Numerical analysis, R. L. Burden, J. D. Faires
2) Fundamental numerical methods and data analysis, G. W. Collins