Linear Algebra and Scientific Computing

Recommended Prerequisites

Mathematical Analysis, Linear Algebra, Basic Programming

Teaching Methods

Teaching methodology:
In the theoretical and practical classes the students actively solve problems. The aim of these classes is to consolidate concepts and results taught in the theoretical classes. Some practical classes take place in the Calculus Laboratory to program computational methods.

Learning Outcomes

This course gives an overview of numerical linear algebra  and scientific computing.  The course includes the main theoretical notions and algorithms used in the approximation of functions, in the numerical integration and in the matrix computations for solving linear systems, linear least squares problems or eigenvalue problems. It also addresses the issue of stability and accuracy in scientific computing and some challenges encountered in high-performance computing with the advent of new computer architectures.
The main competencies to be developed are: capacity for analysis and synthesis; competence in oral and written communication; problem-solving; competence in working in an international context; autonomous learning; adaptability to new situations; creativity; competence to investigate; critical thinking.

Work Placement(s)

No

Syllabus

1. Introduction. Foundations of matrix analysis and scientific computing.   Norms. Classes of matrices. Singular value decomposition. The MATLAB environment.
2. Approximation of functions and data. Polynomial and trigonometric interpolation and FFT. Approximation by splines. The least square method.
3. Numerical integration. Midpoint, trapezoidal and Simpson formulae. Interpolatory quadratures.
4. Linear systems. Linear systems and complexity. LU and Cholesky factorization. Conditioning and condition numbers. QR factorization. Conditioning of least squares algorithms. Incomplete and nonnegative factorizations. Exploring sparsity and structure. Iterative Methods; Gauss-Seidel and SOR; conjugate gradient. Preconditioning.
5. Eigenvalues and singular values. Rayleigh quotient. QR algorithm with shifts. Algorithms for the singular value decomposition.
6. Introduction to high-performance computing.

Head Lecturer(s)

Stéphane Louis Clain

Assessment Methods

Assessment
Resolution Problems: 40.0%
Exam: 60.0%

Bibliography

H. Pina, Métodos Numéricos, McGraw-Hill, 1995.
R. Kress, Numerical Analysis, Springer, 1997.
A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics 2nd edition, Springer, 2007.
A. Quarteroni, F. Saleri, Cálculo científico com MATLAB e Octave, Springer, 2007.
L.N. Trefethen, D. Bau, Numerical Linear Algebra, SIAM, 1997.
G.H. Golub, C.F. Van Loan, Matrix Computations 4th edition, John Hopkins University Press, 2013.