Introduction To Numerical Analysis

Timetable

Classes: Wednesdays, 11:30 - 13:00 in room 3.50, Faculty of Chemistry, Morasko Campus

Lectures: Wednesdays, 13:15- 14:00 in room 3.48, Faculty of Chemistry, Morasko Campus

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.

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