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 90 or MATLAB.
Topics (with time of expository lectures dedicated to each topic)
Unit 1.-Presentation of the subject -1 hour-.
Importance of large systems of linear equations and the calculation of eigenvalues in real problems of scientific and technical research and innovation.
Unit 2.- Generalities about matrices (review). - 4 hours -
Matrix operations. Block operations. Determinants. Inverse matrix. Special types of matrices: triangular, tridiagonal, hermitian, symmetric, unitary, orthogonal, positive definite, permutation matrices, diagonal dominant, Hadamard's theorem. Eigenvalues and eigenvectors: spectral radius, localization, Gerschgorin's theorem, matrix similarity and reduction, Schur's theorem (statement), diagonalization, Rayleigh quotient. Sparse matrices. Profile storage.
Unit 3.- Generalities about the resolution methods of L.S.E. - 2 hours -
Systems of linear equations (S.L.E. ): existence and uniqueness of solution. Large systems and operational cost. Direct methods and iterative methods. Easy to solve systems: triangular and interchangeable to triangular systems (forward and backward substitution methods). Classification of direct methods: transformation methods (elimination and orthogonalization) and factorization methods (LU, Cholesky and QR). Application to the calculation of the inverse of a matrix and to the calculation of determinants.
Unit 4.- Gaussian elimination method and variants for L.S.E. - 5 hours -
Pivotless Gaussian elimination method: process, formulas, operational cost, coding bases, matrix interpretation, sufficient conditions, relationship with A = LU factorization and associated method. Doolitle formulas for direct calculation of L and U. Preservation of the profile: application to tridiagonal matrices. Gaussian elimination method with partial pivoting: process, formulas, coding bases, matrix interpretation, relationship with PA = LU factorization and the associated method.
Unit 5.- Choslesky's method for L.S.E. with symmetric matrix and positive definite - 2 hours -
Cholesky factorization A = BB^T and associated method for L.S.E. : existence, formulas, operational cost, profile preservation, application to tridiagonal matrices, coding bases.
Unit 6.- Householder elimination method - 2 hours -
Elementary Householder matrices. Householder elimination method: process, formulas, operational cost, coding bases, matrix interpretation, relationship with factorization A = QR.
Unit 7.- Norms and condition number of matrix - 3 hours -
Matrix norms as vectors. Subordinate and compatible matrix norms with vector norms. Important examples: Schur's norm, norm 1, infinite norm, norm 2. Convergence of matrix sequences. Conditioning of a linear system and effect on the error propagation.
Unit 8.- Iterative methods for L.S.E. - 3 hours -
Consistent and convergent iterative methods. Characterization of convergence. Iterative methods associated with an A = M- N decomposition (Richardson, Jacobi, Gauss-Seidel and relaxation): description, formulas, coding bases, sufficient convergence conditions (cases of symmetric positive defined and diagonal estrictly dominant matrices).
Unit 9.- Numerical approximation of eigenvalues and eigenvectors - 4 hours -
General idea and classification of the methods. Iterated power method with Rayleigh variants for dominant eigenvalue: description, coding bases, convergence. Inverse power method: description, coding bases, convergence. Householder transformation method and QR factorization method.
Unit 10.- Iterative methods for systems of nonlinear equations - 2 hours -
General idea about iterative methods. Methods of fixed point, Newton exact and discretized Newton: description, coding bases, idea of convergence.
Basic bibliography
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.
KINCAID, D. - CHENEY, W. [1994]: Análisis numérico: las matemáticas del cálculo científico. Addison-Wesley Iberoamericana.
STEWART, D.E. [2023]: Numerical Analysis: A Graduate Course. Springer. - Available on line.
STOER, J. - BULIRSCH, R. [1993]: Introduction to numerical analysis. 2nd ed. Springer-Verlag. Avalilable on line
VIAÑO, J.M.: Análisis Numérico Matricial. Notas de Curso. USC. Available on line in the Virtual Course
Complementary bibliography
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.
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.
METCALF, M. - REID, J. - COHEN M. [2004]: Fortran 95/2003 explained. Oxford University Press.
QUARTERONI, A. [2003]: Scientific computing with MATLAB. Springer.
QUARTERONI, A. - SACCO, R. - SALERI, F. [2000]: Numerical mathematics. Springer.
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:
http://www.usc.es/export9/sites/webinstitucional/gl/servizos/sxopra/mem…
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, a newsletter will be proposed for each theory unit that will include computer programming (in Matlab and Fortran 90) 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 and will be complemented by the virtual course of the subject, in which the students will find various bibliographic materials. Students will carry out tasks for continuous assessment, as described in the corresponding section. 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 90 or MATLAB), 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 added in the case that the unexcused absences in the sessions in laboratory groups do not exceed 10% and will be maintained for the second evaluation opportunity.
The evaluation tests will be identical for the different groups.
The same instruments allow to evaluate the skills previously mentioned.
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.
Juan Manuel Viaño Rey
Coordinador/a- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813188
- juan.viano [at] usc.es
- Category
- Professor: University Professor
Maria Luisa Seoane Martinez
- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813230
- 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 | |||
|---|---|---|---|
| 17:00-18:00 | Grupo /CLE_02 | Galician | Classroom 03 |
| 18:00-19:00 | Grupo /CLE_01 | Spanish | Classroom 02 |
| Wednesday | |||
| 15:00-16:00 | Grupo /CLIL_04 | Galician | Computer room 4 |
| 16:00-17:00 | Grupo /CLIL_04 | Galician | Computer room 4 |
| 17:00-18:00 | Grupo /CLIL_05 | Spanish | Computer room 4 |
| 18:00-19:00 | Grupo /CLIL_03 | Spanish | Computer room 2 |
| 18:00-19:00 | Grupo /CLIL_05 | Spanish | Computer room 4 |
| 19:00-20:00 | Grupo /CLIL_03 | Spanish | Computer room 2 |
| Thursday | |||
| 15:00-16:00 | Grupo /CLIL_02 | Spanish | Computer room 4 |
| 16:00-17:00 | Grupo /CLIL_02 | Spanish | Computer room 4 |
| 17:00-18:00 | Grupo /CLIL_01 | Spanish | Computer room 3 |
| 18:00-19:00 | Grupo /CLIL_01 | Spanish | Computer room 3 |
| 18:00-19:00 | Grupo /CLIL_06 | Spanish | Computer room 4 |
| 19:00-20:00 | Grupo /CLIL_06 | Spanish | Computer room 4 |
| 05.26.2025 09:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 05.26.2025 09:00-14:00 | Grupo /CLE_01 | Computer room 2 |
| 06.27.2025 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 06.27.2025 16:00-20:00 | Grupo /CLE_01 | Computer room 2 |