M05CM01 - Knowledge about the basic techniques of numerical calculation and its translation into algorithms or constructive methods of problem solving.

M05CM02 - Programming on computer in a structured language the numerical methods studied in the course and apply them effectively.

M05CM03 - Use libreries and packages as a support tool for own programs with the aim applying the numerical methods studied.

M05CM04 - Analyze the convenience of one or another numerical method to a specific problem based on the analysis of errors, the computational cost and other characteristics.

M05CM05 - Evaluate the results obtained after a computation process and chose the data to visualize them in an appropriate way to communicate verbally and in writing the conclusions.

1. NUMERICAL INTERPOLATION: Introduction to polynomial interpolation. Lagrange’s and Newton’s formulas. Hermite interpolation. Rational interpolation.

2. NUMERICAL INTEGRATION METHODS: Newton-Cotes quadrature. Richardson extrapolation. Romberg integration. Gauss quadrature.

3. INTRODUCTION TO NUMERICAL INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS: Problem formulation and reduction of order. Convergence, zero-stability and absolute stability. Explicit Euler’s method.

4. ONE STEP RUNGE-KUTTA METHODS: Introduction to Runge-Kutta methods. Consistency order and stability of Runge-Kutta methods.

5. LINEAR MULTISTEP METHODS: Introduction to linear multistep methods. Consistency order and stability of linear multistep methods. Predictor-corrector methods. Stability of predictor-corrector methods

6. BACKWARD DIFFERENCE METHODS: Backward differences. Adams methods.

7. STIFF SYSTEMS: Concepts and interpretation. Stability definition of stiff systems. Pade approximations for the exponential function. Numerical methods for stiff systems.

The theoretical content will be exposed in master lessons following basic references that appear in the bibliography and in the compulsory materials. These master lessons will be complemented with problem lessons (classroom practices), in which the students will be proposed to solve questions where the knowledge acquired in the theoretical lessons will be applied. In the seminars, topics and examples representative of the content of the subject will be developed; they will be generally provided in advance to the students to work on them, and will motivate the subsequent reflection and discussion in the session devoted to it. In addition, computer assignments will be carried out with the aim of achieving the competences of the subject.

- Written exam (%60 of the overall grade). Minimum score = 4,00 to be graded.

- Programming numerical methods (%20 of the overall grade)

- Resolution and presentation of individual work (problems and seminars) (%20 of the overall grade)



A minimum grade of 4 points in the written exam and in the computer assignments will be required in order to apply the above percentages.



Computer practices will be done in groups of two students. Each group will develop their practices autonomously, that is, without sharing their content with other groups. In case of detecting substantial parts of code with an analogous structure in different groups, these practices will automatically be invalidated. The delivery of the practices is obligatory to gain acces to an individual exam consisting on questions about computing. This trial will ensure the acquisition of the corresponding competences and that will be used to determine the grade.



Students who request it throughout the first 9 weeks from the beginning of classes in the second semester of the course, may substitute the continuous assessment by a "single assessment" that will ensure the acquisition of the competences of the subject. This evaluation may consist of one or more tests such as a written exam, an oral presentation of materials related to the content and skills of the subject, or a practical programming exam. Students must apply for the "single assessment" modality to the subject coordinator by a writing and signed document.



Students who do not appear on the official date of the examination of each call, will automatically be considered as absent and thus will be registered by the teachers.



During the assessment trials, as long as it has not been explicitly authorized in writing by the teaching staff of the subject, the use of books or notes, as well as telephone, electronic, computer devices or of another type of devices is forbidden. In individual tests, all kinds of collaboration and exchange of academic material between people is forbidden. In the case of detecting any irregularity or cases of dishonest or fraudulent practices, the provisions of the protocol on academic ethics and the prevention of dishonest or fraudulent practices will be applied in assessment tests and in academic work at the UPV / EHU. These regulations can be consulted at the link:



https://www.ehu.eus/es/web/estudiosdegrado-gradukoikasketak/akademia-araudiak



The aforementioned protocol refers to the sanctioning regime, "Decree of September 8, 1954, which approves the Regulations for academic discipline of the official Centers of Higher Education and Technical Education under the Ministry of National Education, which includes the actions considered infractions, and the possible sanctions to be imposed after their commission, "where in article 5.4, practices such as" The impersonation of persons in acts of teaching life and the falsification of documents are declared as "serious misconduct".



The evaluation tests will be carried out in person, as long as the circumstances allow it and there are no orders from the competent authorities to prevent it. In case that causes that prevent the performance of in-person assessment tests take place, then, computer resources will be used to carry out online tests of the same type, weight and conditions.

Notes published in the host Moodle/e-gela

Basic bibliography

J. Stoer, R. Bulirsch: Introduction to Numerical Analysis. Springer, 1983.



D. Kincaid, W. Cheney: Análisis Numérico. Las matemáticas del cálculo científico. Addison-Wesley, 1994.



E. Hairer, S.P. Nørsett, G. Wanner: Solving Ordinary Differential Equations I. Non Stiff Problems. Springer, 1987



S.D. Lambert: Computational Methods in Ordinary Differential Equations. John Wiley & Sons, 1973



S.D. Lambert: Numerical Methods for Ordinary Differential Systems. John Wiley & Sons, 1991.

In-depth bibliography

J.C. Butcher: The Numerical Analysis of Ordinary Differential Equations. John Wiley & Sons, 1987

E. Hairer, S.P. Nørsett, G. Wanner: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer, 1996