This course focuses on the construction and analysis of numerical algorithms for solving some of the main linear algebra problems, such as QR factorization, linear systems, eigenproblems. These algorithms are the foundations of contemporary scientific computing and have applications in several fields of applied sciences and engineering.
Expected learning outcomes
Ability to construct and analyze the main algorithms of Numerical Linear Algebra. Development of implementation skills using Matlab and numerical verification of the theoretical results.
Lesson period: Second semester
(In case of multiple editions, please check the period, as it may vary)
1) Introduction. Vector, matrices, norms. Schur's decomposition. Singular Value Decomposition. 2) Iterative methods for solving linear systems: gradient and conjugate gradient methods; splitting methods. 3) Householder reflections and Givens rotations. QR factorization. 4) Eigenvalue and eigenvector approximation: the Rayleigh quotient and the power method, Jacobi method, QR algorithm.
Prerequisites for admission
To properly face the course, the Student should have a basic knowledge of Linear Algebra, Mathematical Analysis and Numerical Analysis.
The course will be given by means of traditional lessons, using the blackboard. Furthermore, for the lab lessons the software MATLAB will be used to implement the studied methods.
- N. Trefethen, D. Bau, Numerical Linear Algebra, SIAM, 1997. - P.G. Ciarlet, Introduction to Numerical Linear Algebra and Optimisation, Cambridge University Press, 1898.
Assessment methods and Criteria
The final examination consists of an oral exam. During the exam, the student will be required to illustrate aspects presented during the course (topics developed during the lab lessons are included), and will be required to solve problems regarding Numerical Linear Algebra in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.