Numerical Methods

An introductory course on numerical methods for students in computer science and engineering. The aim is to present basics in numerical analysis and to teach how to proceed when choosing and applying numerical algorithms to certain computational problems. Individual projects beeing an important part of the course are designed to develop practical skills in implementation of selected numerical procedures chosen from representative classes (linear algebra, nonlinear equations, simulation of continuous dynamical systems).


Lecture contents
Number representation, error analysis. Computer representation of numbers, representation errors, floating-point arithmetic, condition number, numerical stability of algorithms.

Linear algebra selected algorithms. Vector and matrix norms. Systems of linear equations: conditioning, Gaussian elimination and LU factorization, LLT factorization, calculation of determinants and inverse matrices, basic iterative methods. Eigenvalues of symmetric matrices, the QR method. Singular value decomposition (SVD).

Iterative solution of nonlinear equations. Elementary methods. Finding roots of polynomials, the Muller method. Root polishing. Systems of nonlinear equations: the Newton algorithm.

Interpolation and approximation.: Lagrange and Newton formulae for polynomial interpolation. Spline function interpolation. Least-squares approximation, ortogonalisation. Pade approximation.

Ordinary differential equations. Runge-Kutta and Runge-Kutta-Fehlberg single-step algorithms. Accuracy estimation and automatic step-length correction. Multi-step predictor-corrector algorithms. Methods for stiff systems of ordinary differential equations.



Projects contents
Three computer-based individual projects are required to be completed:

1. Implementation, testing and validation of a finite method and an iterative method for solving systems of linear equations, error analysis.


2. Implementation, testing and validation of selected root-finding algorithms.



3. Implementation, testing and validation of an algorithm for solving a given set of ordinary differential equations (simulation of a continuous-time dynamical system).

Basic:
1. P. Tatjewski: „Numerical Methods", OWPW (Warsaw University of Technology Publishing House), 2014.
2. R. Z. Morawski, A. Miękina: "Solved problems in numerical methods – for students of electronics and information technology", OWPW (Warsaw University of Technology Publishing House), 2021.
3. J. H. Mathews, K. D. Fink: „Numerical Methods (using Matlab)”, Prentice Hall, 1999.

Complementary:
1. K. E. Atkinson: „An Introduction to Numerical Analysis”, John Wiley & Sons, 1989.
2. W.H. Press, S.A Teukolsky, W.T. Vetterling, B.P. Flannery: „Numerical Recipes in C”, Cambridge University Press 1995.