ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Applied Mathematics
Areas: Applied Mathematics
Center Faculty of Mathematics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable
The study and application of numerical methods for solving systems of linear equations (with an introduction to methods for nonlinear systems) and the computation of eigenvalues and eigenvectors of a matrix. In addition, in the laboratory practices, the studied algorithms will be put into practice on a computer, by means of the elaboration of the corresponding programs in FORTRAN.
Topics:
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Unit 1.- Preliminaries: Generalities about matrices. Special matrices: triangular, tridiagonal, hermitian, symmetric, unitary, orthogonal, positive definite. Spectral radius and Rayleigh quotient. Vectors and matrices norms. The need of the numerical methods for solving linear systems of equations: direct and iterative methods. Matrix condition number. (6 hours).
Unit 2.- Direct methods for solving a linear system: Gaussian elimination method, LU decomposition. Partial pivoting strategy. Cholesky decomposition. Householder method and QR decomposition. Applications: computation of matrix determinants and inverses, (12 hours).
Unit 3.- Iterative methods for the solution of linear systems. The convergence of vector and matrix sequences. The Jacobi, Gauss-Seidel, relaxation and symmetric relaxation methods. Convergence theorems. (5 hours).
Unit 4.- Numerical approximation of eigenvalues and eigenvectors. Estimation of eigenvalues: Gerschgorin's theorem.
Condition number for the eigenvalue problem. Power iteration and inverse power iteration methods. (4 hours).
Unit 5.- Iterative methods for systems of nonlinear equations. Newton and discretized Newton methods. (1 hour).
Basic:
CIARLET, P. G. [1999]: Introducción á análise numérica matricial e á optimización. Servicio de Publicacións da USC.
ORTEGA, J. M. [1990]: Numerical análisis: a second course. SIAM.
QUARTERONI, A. - SACCO, R. - SALERI, F. [2007]: Numerical mathematics. Springer-Verlag (available online).
STOER, J. - BULIRSCH, R. [1993]: Introduction to numerical analysis. 2nd ed. Springer-Verlag (available online).
VIAÑO, J.M. [2025]: Lecciones de métodos numéricos 5: Sistemas lineales y autovalores (to appear).
Complementary:
ATKINSON, K. E. - HAN, W. [2004]: Elementary numerical analysis. John Wiley and sons.
AUBANELL, A. - BENSENY, A. - DELSHAMS, A. [1991]: Eines bàsiques de càlcul numeric: amb 87 problemes results. Manuals de la Universitat Autònoma de Barcelona.
GANDER, W. – GANDER M. J. – KWOK, F. [2014]: Scientific computing – An introduction using MAPLE and MATLAB. Springer (available online).
GOLUB, G. H. - VAN LOAN, C. [2013]: Matrix computations. 4th ed. The Johns Hopkins University Press.
HEATH, M. T. [2005]: Scientific computing: an introductory survey. 2nd ed. McGraw Hill.
HORN, R. A. - JOHNSON, C. R. [2013]: Matrix analysis. 2nd ed. Cambridge University Press.
KINCAID, D. - CHENEY, W. [1994]: Análisis numérico: las matemáticas del cálculo científico. Addison-Wesley Iberoamericana.
METCALF, M. - REID, J. - COHEN M. [2011]: Modern Fortran explained. Oxford University Press.
QUARTERONI, A. - SALERI, F. - GERVASIO, P. [2014]: Scientific computing with MATLAB. Springer (available online).
STEWART, D.E. [2023]: Numerical Analysis: A Graduate Course. Springer (available online).
TREFETHEN, Ll. N. - BAU, D. [1997]: Numerical linear algebra. SIAM.
WATKINS, D. S. [2010]: Fundamentals of matrix computations. 3rd ed. Wiley.
The skills listed in the Memoria de Verificación de Título do Grao en Matemáticas. Available in:
https://assets.usc.gal/sites/default/files/plan/2023-01/Matem%C3%A1tica…
In the following section we indicate the skills worked with greater emphasis according to the type of meeting.
- Lecture classes (CG1, CT5, CE1, CE2).
- Interactive laboratory classes (CE8, CE9).
- Tutorials (CG3, CG4, CT3, CE4).
- Throughout the semester, newsletters will be proposed including computer programming (in FORTRAN) and exercises related to the theory, in order to students consolidate the knowledge acquired in the subject and programming skills .
- Students will have a Virtual Course, with notes and various material as a complement to face-to-face teaching.
The expository and interactive teaching will be face-to-face. The tutorials will be face-to-face or via email.
To compute the final mark (FM), the examen evaluation (EE) and the continuous assessment qualification (CAC) will be taken into account.
- The exam has an overall score of 10 points (EE) and will be carried out in the following two sessions:
1. Written final exam (theory, questions and problems), rated at 7.5 points
2. Practical final exam (programming in FORTRAN), rated at 2.5 points.
- The continuous evaluation also has an overall score of 10 points (CAC), resulting from the 2 controls carried out within the time reserved for the subject.
To obtain the final mark, the following formula will be applied: FM = max {EE, 0.7 * EE + 0.3 * CAC}
The CAC mark will be maintained for the second evaluation opportunity.
For repeat students, the CAC mark from previous courses will not be maintained.
The evaluation tests will be analogous for the different groups.
In cases of fraudulent performance of exercises or tests (plagiarism or misuse of technologies), the provisions of the "Regulations for the evaluation of the academic performance of students" and the review of grades will apply.
Expository lectures: 28 hours
Interactive laboratory classes: 28 hours
Tutorials: 2 hours
Total hours with the teacher: 58
Self-study individual or in group: 30 hours
Programming / testing or other computer work: 52 hours
Writing exercises, conclusions or other works: 10 hours
Total hours of personal work: 92
- Daily study of the contents covered in the classes, complemented with the virtual course and the recommended bibliography.
- Resolution of the exercises and programming of the algorithms proposed in the bulletins, for which the Faculty's computer rooms are available.
- Use of the tutorial hours with the teachers to solve all kinds of doubts about the subject.
Jose Antonio Alvarez Dios
- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813353
- joseantonio.alvarez.dios [at] usc.es
- Category
- Professor: University Lecturer
Maria Luisa Seoane Martinez
Coordinador/a- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813181
- marialuisa.seoane [at] usc.es
- Category
- Professor: University Lecturer
Rafael Vazquez Hernandez
- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813134
- rafael.vazquez [at] usc.es
- Category
- Investigador/a Distinguido/a
| Tuesday | |||
|---|---|---|---|
| 16:00-17:00 | Grupo /CLE_01 | Galician | Classroom 02 |
| 19:00-20:00 | Grupo /CLE_02 | Spanish | Classroom 03 |
| Wednesday | |||
| 15:00-16:00 | Grupo /CLIL_03 | Spanish | Computer room 4 |
| 16:00-17:00 | Grupo /CLIL_03 | Spanish | Computer room 0 |
| 16:00-17:00 | Grupo /CLIL_08 | Spanish | Computer room 3 |
| 17:00-18:00 | Grupo /CLIL_08 | Spanish | Computer room 3 |
| 18:00-19:00 | Grupo /CLIL_06 | Spanish | Computer room 3 |
| 18:00-19:00 | Grupo /CLIL_02 | Galician | Computer room 4 |
| 19:00-20:00 | Grupo /CLIL_06 | Spanish | Computer room 3 |
| 19:00-20:00 | Grupo /CLIL_02 | Galician | Computer room 4 |
| Thursday | |||
| 15:00-16:00 | Grupo /CLIL_07 | Spanish | Computer room 4 |
| 16:00-17:00 | Grupo /CLIL_01 | Galician | Computer room 2 |
| 16:00-17:00 | Grupo /CLIL_07 | Spanish | Computer room 4 |
| 17:00-18:00 | Grupo /CLIL_01 | Galician | Computer room 2 |
| 18:00-19:00 | Grupo /CLIL_04 | Spanish | Computer room 2 |
| 18:00-19:00 | Grupo /CLIL_05 | Spanish | Computer room 4 |
| 19:00-20:00 | Grupo /CLIL_04 | Spanish | Computer room 2 |
| 19:00-20:00 | Grupo /CLIL_05 | Spanish | Computer room 4 |
| 05.19.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 05.19.2026 10:00-14:00 | Grupo /CLE_01 | Computer room 2 |
| 06.26.2026 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 06.26.2026 16:00-20:00 | Grupo /CLE_01 | Computer room 2 |