Numerical Computing in One Variable

CE1 - Understand and use mathematical language.
CE2 - Know rigorous demonstrations of some classical theorems in different areas of Mathematics.
CE3 - Devise proofs of mathematical results, formulate conjectures and imagine strategies to confirm or deny them.
CE4 - Identify errors in incorrect reasoning proposing demonstrations or counterexamples.
CE5 - Assimilate the definition of a new mathematical object, relate it to others already known, and be able to use it in different contexts.
CE6 - Knowing how to abstract the properties and substantial facts of a problem, distinguishing them from those that are purely occasional or circumstantial.
CE7 - Propose, analyze, validate and interpret models of simple real situations, using mathematical tools more
adequate for the purposes pursued.
CE8 - Plan and execute algorithms and mathematical methods to solve problems in the academic, technical, financial or social domain.
CE9 - Use computer applications for statistical analysis, numerical and symbolic calculation, graphic visualization, optimization and scientific software, in general, to experiment in Mathematics and solve problems.

The previous competences, as well as those described on page 5 of the memory of the degree on the link
http://www.usc.es/export/sites/default/gl/servizos/sxopra/memorias_grao…,
are dealt with in class and evaluated according to the system described in the "Assessment system" section.

UNIT I. Introduction to numerical analysis. Errors in numerical calculations (approx. 7 expositive hours).

UNIT II. Lagrange's polynomial interpolation: construction of the polynomial and error formula (approx. 7 expositive hours).

UNIT III. Introduction to numerical integration: simple and composite trapezoidal and Simpson rules; error formulae. Introduction to numerical differentiation (approx. 7 expositive hours).

UNIT IV. Approximation of the roots of a numerical equation: separation of roots, concepts of iterative method, order of convergence and local and global convergence. Description and analysis of the algorithms of dichotomy, functional iteration and Newton-Raphson (approx. 7 expositive hours).

Basic:
[1] Michael METCALF, John K. REID, Malcolm COHEN. FORTRAN 95/2003 explained. Oxford University Press, 2004.
[2] Juan Manuel VIAÑO REY. Lecciones de métodos numéricos 1.- Introducción general y análisis de errores. Tórculo edicións, 1995.
[3] Juan Manuel VIAÑO REY. Lecciones de métodos numéricos 2.- Métodos de resolución de ecuaciones numéricas no lineales. Tórculo edicións, 1997.
[4] Juan Manuel VIAÑO REY, Margarita BURGUERA GONZÁLEZ. Lecciones de métodos numéricos 3.- Interpolación. Tórculo edicións, 2000.

Complementary:
[1] Richard L. BURDEN, J. Douglas FAIRES. Numerical Analysis (7th edition). Brooks/Cole Thomson Learning, cop. 2001.
[2] Eugene ISAACSON, Herbert Bishop KELLER. Analysis of Numerical Methods. John Wiley, 1994.
[3] David KINCAID, Elliot Ward CHENEY. Análisis Numérico: las Matemáticas del Cálculo Científico. Addison-Wesley Iberoamericana, 1991.
[4] Alfio QUARTERONI, Fausto SALERI. Cálculo científico con Matlab y Octave. Springer-Verlag Italia, Milano, 2006. Available online.
[5] David M. YOUNG, Robert Todd GREGORY. A Survey of Numerical Mathematics. Addison-Wesley, 1973.

To know the basic techniques of numerical calculus and their translation to algorithms.

To be able to apply the basic numerical methods of resolution of numerical equations, polynomial interpolation, differentiation and integration.

To be able to programme in a computer the studied numerical methods.

The previous competences, as well as those described on page 5 of the degree guidelines on the link
http://www.usc.es/export/sites/default/gl/servizos/sxopra/memorias_grao…,
are worked in class and evaluated according to the description in the section "Assessment system".

Thematic website for virtual teaching. Guided realization of small computer programs in the practical classes. Carrying out work by the student to reinforce knowledge.

Programming assignments will be performed in FORTRAN language, with the possible aid of MATLAB.

The evaluation system contemplates, on the one hand, a continuous evaluation (AC) and, on the other, a final exam on the date set by the Faculty, this one with two qualifications: ET=exam of all contents contents of the subject, ER= repetition of the continuous evaluation. All exams will be indentical for all groups.

All marks (AC, ET, ER) must be understood on the 0-10 scale.

Continuous assessment (AC) consists of monitoring the laboratory classes by carrying out two tests in this regard. The score corresponding to the continuous evaluation is calculated by means of the arithmetic mean of the two tests carried out.

The final mark (CF) is calculated taking into account that this subject has to provide programming skills, which makes it mandatory, in order to pass it, to reach a certain level of programming. For this reason, the following formula is applied for the final mark:

CF = 0.70*ET+0.30* MAX{ER, AC} se AC>=3 or ER>5;
CF = MIN{ER,MAX{ET,0.70*ET+0.30*AC}} otherwise.

Only the AC mark will be kept for the second opportunity of the course.

It is recalled that in cases of fraudulent completion of the tests or proofs (plagiarism or improper use of technologies) the provisions of the Regulations for evaluating the academic performance of the student body and reviewing marks will apply.

On-site work work: 28 h expositive + 28 h interactive laboratory + 2 h tutorials = 58 h.

Personal work: 35 hours autonomous study + 21 hours of exercises + 21 hours of programming + 15 hours of recommended readings) = 192 hours.

Total: 150 hours.

To support a continued contact with the contents explained at class.

To do the proposed exercises.

Start doing practices from the first session.