Course: 2023/2024

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. Solution of linear systems of equations.
3. Numerical solution of equations and systems of nonlinear equations.
4. Interpolation and approximation of functions.
5. Least squares problems.
6. Numerical optimization.
7. Numerical integration.
8. Numerical differentiation: numerical solution of ODEs.
9. Fast Fourier Transform.

Learning activities and methodology

This is a "hands on" course. In the large-group lectures the theoretical contents will be introduced, and they will be delivered in a blackboard standard room. However, the small-group lectures will take place in the computer Lab, and students are supposed to follow the explanations of the instructor performing in real time the exercises, examples and other proposed activities. Students must become acquainted with MATLAB.
The course will start learning how to program MATLAB. After and introduction to the course, every two weeks (as a general rule), one of the topics of the course will be discussed in the classroom and practices related to these topics will be proposed to the students. Usually the practices involve to solve a simple problem by writing the appropriate code.

Assessment System

- % end-of-term-examination 50
- % of continuous assessment (assigments, laboratory, practicals...) 50

Basic Bibliography

- [A] K. Atkinson. Elementary Numerical Analysis. John Wiley & Sons. 2004
- [BF] R. L. Burden, J. D. Faires. Numerical Methods. Brooks/Cole, Cengage Learning. 2003
- [QSG] A. Quarteroni, F. Saleri, P. Gervasio. Scientific computing with MATLAB and Octave. Springer. 2010
- [QSS] A. Quarteroni, R. Sacco, F. Saleri. Numerical Mathematics. Springer. 2007

- Cleve Moler · Numerical Computing with MATLAB : https://es.mathworks.com/moler/chapters.html

Additional Bibliography

- [BC] A. Belegundu, T. Chandrupatla. Optimization Concepts and Applications in Engineering. Cambridge University Press. 2011
- [BV] S. Boyd, L. Vanderberghe. Convex Optimization. Cambridge University Press. 2004
- [DCM] S. Dunn, A. Constantinides, P. Moghe. Numerical Methods in Biomedical Engineering. Elsevier Academic Press. 2010
- [DH] P. Deuflhard, A. 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
- [HH] D. Higham, N. J. Higham. Matlab Guide. SIAM. 2017
- [HJ] R. A. Horn, C. R. Johnson. Matrix Analysis, 2nd ed.. Cambridge University Press. 2013
- [H] N. J. Higham. Accuracy and Stability of Numerical Methods. SIAM. 1998
- [K] C. Kelley. Iterative Methods for Optimization. SIAM (available online). 1999
- [NW] J. Nocedal, S. J. Wright. Numerical Optimization, 2nd ed.. Springer. 2006
- [TB] L. N. Trefethen, D. Bau. [TB] L. N. Trefethen, D. Bau. SIAM. 1997

- Gilbert Strang · Lecture 26: Complex matrices; fast Fourier transform : http://https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/resources/lecture-26-complex-matrices-fast-fourier-transform/
- Lloyd N. Trefethen · Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations : https://www.math.hmc.edu/~dyong/math165/trefethenbook.pdf

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