Numerical Linear Algebra

Code:

M560

Acronym:

M560

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2015/2016 - 2S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=2026
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Doctoral Program in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
IUD-M	4	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

This course gives an overview of numerical linear algebra and describes the main theoretical notions and algorithms used in matrix computations for solving linear systems, linear least squares problems and 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.

Learning outcomes and competences

The main skills to be developed are: capacity for analysis and synthesis; competence in oral and written communication and problem-solving; competence in working in an international context; autonomous learning; adaptability to new situations; creativity; competence to research; critical thinking.

Working method

Presencial

Program

Introduction. Matrix decompositions. Conditioning and stability. Floating point arithmetic. Error analysis.

Systems of equations. Gaussian elimination with pivoting strategies. Cholesky factorization. Stability analysis.
Large dimensional systems of equations: Sparse matrix techniques. Iterative methods based on Krylov subspaces: Conjugate Gradients, GMRES, biorthogonalization methods (BiCG and BICGstab). Convergence and spectral properties. Preconditioning

Eigenvalues. Reduction to Hessenberg or tridiagonal forms. Rayleigh quotient and inverse iteration. QR algorithm. Lanczos iteration (symmetric case) and Arnoldi iteration (non symmetric case).
Large scale eigenvalue problems: Iterative methods based on Krylov subspaces (Conjugate Gradients, GMRES, BiCG and BICGstab). 

 

Mandatory literature

Michael T. Heath ; SCIENTIFIC COMPUTING: AN INTRODUCTORY SURVEY, McGraw-Hill , 2002. ISBN: 0-07-239910-4
Demmel James W.; Applied numerical linear algebra. ISBN: 0-89871-389-7

Complementary Bibliography

Trefethen Lloyd N.; Numerical linear algebra. ISBN: 0-89871-361-7
Higham Nicholas J.; Accuracy and stability of numerical algorithms. ISBN: 0-89871-3552 (pbk)
Golub Gene H.; Matrix computations. ISBN: 0-8018-5414-8

Teaching methods and learning activities

The topics developed in this curricular unit set the fundamental core of Numerical Linear Algebra. The programmatic contents will be adressed to an advanced level.

The teaching strategy and methods adopted aim at engaging the student in the learning process and his personal development, and lead to the development of some generic competencies of instrumental, personal and systemic nature.

With the knowledge and comprehension of the matters taught in the theoretical classes, conditions exist for the development of competencies in problem solving, critical reasoning, applying in practice theoretical knowledge and, at a more advanced level, analysis and synthesis. Will be given strong emphasis on independent work to acquire knowledge and understanding of concepts and methods. In order to make our presentation concrete and appeling we will adopt the programming environment MatLab as computational tool.

Software

MATLAB

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Participação presencial	10,00
Trabalho escrito	45,00
Trabalho laboratorial	45,00
Total:	100,00

Calculation formula of final grade

0.45*R1+0.45*R2+0.1*P

R1 = report 1; R2 = report 2; P = participation. Both reports must include a theoretical study and a computational experiments.