Course: 2022/2023

Numerical Methods

(18262)

Requirements (Subjects that are assumed to be known)

Linear Algebra, 1st semester 1st year,
Differential Calculus, 1st semester 1st year,
Integral Calculus, 2nd semester 1st year,
Programming, 1st semester 1st year

Familiarizing with the basic concepts of numerical analysis: algorithms, stability, accuracy, and efficiency.
Interpolating data with different techniques: Lagrange, Hermite, piecewise, splines.
Calculating numerical approximations, choosing the most adequate algorithm for each application, in each of the following problems: quadrature and derivation, systems of linear and non-linear equations, linear least-squares.
Programming the studied algorithms and use other ready-made algorithms, available in MATLAB or other recognized software packages.
Relating real problems and their mathematical models.

Skills and learning outcomes

Description of contents: programme

1. Introduction: errors, algorithms and estimates
Sources of error, roundoff, truncation, propagation. Machine numbers, floating-point arithmetics. Taylor polynomials and error. Estimating and bounding errors. Optimal step. Interval arithmetics.
2. Nonlinear equations and nonlinear systems
Nonlinear equations: Mean-value theorem, number of roots in an interval. Bisection, Secant, Newton-Raphson. Fixed-point methods. Convergence order. Error analysis. Nonlinear systems. Accelerated, Taylor and interpolation methods.
3. Methods for linear systems of equations
Linear systems, stability: condition number. Triangular systems. Gaussian elimination. Pivoting. Computing determinants and matrix inverses. Orthogonalization methods and improved methods. Least-squares problems. Regression. Normal equations and QR method. Overdetermined systems. Fast Fourier Transform. Applications.
4. Polynomial interpolation: Lagrange, Hermite, piecewise, splines
Newton/Lagrange Interpolation, errors. Equispaced (or not) nodes. Runge's phenomenon. Hermite interpolation. Richardson's extrapolation. Splines. Natural cubic splines.
5. Numerical quadrature and differentiation
Numerical differentiation: back/forward, central, general, higher order. Errors. Numerical Integration: Newton-Côtes formulae. Errors. Adaptive integration.

Learning activities and methodology

LEARNING ACTIVITIES AND METHODOLOGY
THEORETICAL-PRACTICAL CLASSES. [44 hours with 100% classroom instruction, 1.67 ECTS]
Knowledge and concepts students must acquire. Student receive course notes and
will have basic reference texts to facilitate following the classes and carrying
out follow up work. Students partake in exercises to resolve practical problems
and participate in workshops and evaluation tests, all geared towards
acquiring the necessary capabilities.
TUTORING SESSIONS. [4 hours of tutoring with 100% on-site attendance, 0.15 ECTS]
Individualized attendance (individual tutoring) or in-group (group tutoring)
for students with a teacher.
STUDENT INDIVIDUAL WORK OR GROUP WORK [98 hours with 0 % on-site, 3.72 ECTS]
WORKSHOPS AND LABORATORY SESSIONS [8 hours with 100% on site, 0.3 ECTS]
FINAL EXAM. [4 hours with 100% on site, 0.15 ECTS]
Global assessment of knowledge, skills and capacities acquired throughout the
course.
METHODOLOGIES
THEORY CLASS. Classroom presentations by the teacher with IT and audiovisual
support in which the subject's main concepts are developed, while providing
material and bibliography to complement student learning.
PRACTICAL CLASS. Resolution of practical cases and problem, posed by the
teacher, and carried out individually or in a group.
TUTORING SESSIONS. Individualized attendance (individual tutoring sessions) or
in-group (group tutoring sessions) for students with a teacher as tutor.
LABORATORY PRACTICAL SESSIONS. Applied/experimental learning/teaching in
workshops and laboratories under the tutor's supervision.

Assessment System

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

Basic Bibliography

- [CM] Moler, C. B.. Numerical computing with MATLAB. SIAM. 2004
- [KA] Atkinson, K.. Elementary Numerical Analysis. John Wiley and Sons. 2004
- [MF] Mathews, J. H., Fink, K. D.. Numerical methods using Matlab, 3rd edition. Prentice-Hall. 1998
- [TB] Trefthen, L. N., Bau, D., III. Numerical Linear Algebra. SIAM. 1997
- [WS] Wen Shen. An Introduction to Numerical Computation. World Scientific. 2016

Additional Bibliography

- Sanz Serna, J. M.. Diez lecciones de cálculo numérico. Universidad de Valladolid. 2010
- [ABD] Aubanell, A., Benseny, A., Delshams, A.. Útiles básicos de cálculo numérico. Universitat Autònoma de Barcelona. 1993
- [HH] Higham, D., Higham, N.. MATLAB guide, 2nd edition. SIAM. 2005
- [QSS] Quarteroni, A., Sacco, R., Saleri, F.. Numerical mathematics. Springer. 2007

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