Course: 2022/2023

Numerical methods

(16493)

Requirements (Subjects that are assumed to be known)

Lineal Algebra, Programming, Calculus I, Calculus II

Using NUMERICAL METHODS (NM) to calculate approximate solutions of mathematical models
Study the stability and accuracy of NM.
Calculate numerical solution of systems of nonlinear equations.
Approximate the minimum of a function of several variables.
Developing, analyzing, and implementing finite difference methods.
Solving ordinary differential equations and systems by numerical integration methods.
Using the software environments to discuss the efficiency, pros and cons of different NM.

Skills and learning outcomes

Description of contents: programme

1. Fundamentals (floating point, errors, stability, algorithms...).
2. Numerical linear algebra: systems of linear equations, matrix factorization, diagonalization, least squares.
3. Numerical solution of equations and systems of nonlinear equations.
4. Numerical optimization.
5. Interpolation and approximation of functions.
6. Numerical differentiation and integration.
7. Fast Fourier Transform.

Learning activities and methodology

One of the purposes of this course is to provide the mathematical foundations of numerical methods, to analyze their basic theoretical properties (stability, accuracy, computational complexity), and demonstrate their performances on examples and counterexamples which outline their pros and cons. The primary aim is to develop algorithmic thinking-emphasizing on long-living computational concepts. Every chapter is supplied with examples, exercises and applications of the discussed theory. The course relies throughout on well tested numerical procedures for which we include codes and test files.
Students should write their own codes by studying and eventually rewriting the codes given by the Teacher in Aula Global. The personal codes should be run, tested and given up in ¿Aula Global¿ in the Computer Room classes.
Throughout the course we emphasize graphic 2D and 3D representations of solutions. Through this visual approach, students will have a chance to experience the meaning, i.e. to understand what a solution means and how it behaves.

Assessment System

- % end-of-term-examination 40
- % of continuous assessment (assigments, laboratory, practicals...) 60

Basic Bibliography

- [A] K. Atkinson. Elementary Numerical Analysis. John Wiley & Sons. 2004
- [BC] A. Belegundu and T. Chandrupatla. Optimization Concepts and Applications in Engineering. Cambridge University Press. 2011
- [BF] R. L. Burden, J. D. Faires. Numerical Methods. Brooks/Cole, Cengage Learning. 2003
- [DCM] S. Dunn, A. Constantinides and P. Moghe. Numerical Methods in Biomedical Engineering. Elsevier Academic Press. 2010
- [DH] Peter Deuflhard and Andreas Hohmann. Numerical Analysis in Modern Scientific Computing. An Introduction. Springer. 2003
- [FJNT] P.E. Frandsen, K. Jonasson, H.B. Nielsen, O. Tingleff. Unconstrained Optimization. IMM, DTU. 1999
- [QSG] A. Quarteroni, F. Saleri and P. Gervasio. Scientific computing with MATLAB and Octave. Springer. 2010
- [QSS] A. Quarteroni, R. Sacco and F. Saleri. Numerical Mathematics. Springer. 2007
- [T] Lloyd N. Trefethen. Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations. Freely available online. 1996

Additional Bibliography

- [HH] D. Higham and N. Higham. Matlab Guide. SIAM. 2017
- [K] C. Kelley. Iterative Methods for Optimization. SIAM (available online). 1999

The course syllabus may change due academic events or other reasons.