Numerical Analysis II

Code:

M332

Acronym:

M332

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2013/2014 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Physics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:AST	0	Plano de Estudos a partir de 2008	3	-	7,5	-	202,5
L:CC	0	Plano de estudos de 2008 até 2013/14	3	-	7,5	-	202,5
L:F	0	Plano de estudos a partir de 2008	3	-	7,5	-	202,5
L:M	26	Plano de estudos a partir de 2009	3	-	7,5	-	202,5

Teaching language

Portuguese

Objectives

In this course, we present constructive methods of numerical resolution of fundamental problems of Algebra and Mathematical Analysis, such as: solving systems of linear and nonlinear equations, computation of inverse matrices and determinants, computation of eigenvalues and eigenvectors of matrices, computation of integrals and solving differential equations.

In each issue, consider the following aspects related to numerical methods presented: convergence, numerical stability, error control, construction of algorithms, implementation and experimentation in computer, processing of examples and interpretation of results 

Learning outcomes and competences

Students will acquire knowledge of the basic methods of numerical solution of fundamental mathematical problems in their theoretical, practical, computational and experimental aspects.

Working method

Presencial

Program

1. Norms and limits of vectors and matrices

1.1 Subordinate and induced matrices norms.

1.2 Convergence criteria of sequences and series of matrices.

1.3. Inversion of perturbed matrices. Condition number of a matrix.

1.4 Conditioning of systems of linear equations and of matrix inversion.

2. Numerical Resolution of Systems of Linear Equations

2.1 Transformations and elementary matrices.

2.2 Types of matrices.

2.3 Triangular systems.

2.4 Gauss’s Method.

2.5 Compact factorization methods.

3. Iterative Methods for Systems of Linear Equations

3.1 Jacobi and Gauss-Seidel methods.

4. Numerical Resolution of Systems of Nonlinear Equations

4.1 Simple iterative method.

4.2 Newton’s method.

5. Numerical Integration of Differential Equations

5.1 Existence and uniqueness of solutions (Picard aprroximants).

5.2 Euler’s methods, predictor-corrector methods, methods of Taylor and Runge-Kutta.

6. Eigenvalues and Eigenvectors of Matrices

6.1 Localization of eigenvalues: Gerschgorin theorems and Rayleigh coefficient.

6.2 Methods of direct and inverse powers. Deflation.

6.3 Jacobi’s method.

6.4 Tridiagonalization of matrices: Givens rotations, Householder reflections.

6.4 Eigenvalues and eigenvectors of tridiagonal matrices.

Mandatory literature

Brezinski Claude; Méthodes numériques directes de l.algèbre matricielle. ISBN: 2-7298-2246-1
Brezinski Claude; Méthodes numériques itératives. ISBN: 978-2-7298-2887-5
Pina Heitor; Métodos numéricos. ISBN: 978-972-592-284-2
Golub Gene H.; Matrix computations. ISBN: 0-8018-5414-8

Teaching methods and learning activities

The contact hours are distributed in lectures, practical classes and tutorial classes.

In the lectures are presented the contents of the program, making use of examples to illustrate the concepts covered. In the lectures takes place, also, the proposal of computing projects done by students in groups and the presentations by the students.

In practical classes take place the resolution of theoretical and practical exercises and the development in computer of the projects.

The computational projects contain three components: computer programs, written report and oral presentation with projection means.

In tutorial classes is made the orientation of reports and presentations of projects.

All resources are available for students at the unit’s web page.

The evaluation is continuous with final exam and includes the presentations of computing projects.

Software

MatLab, SciLab, Máxima ou Phyton

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	62,50
Trabalho laboratorial	37,50
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Elaboração de projeto	60,00
Total:	60,00

Calculation formula of final grade

- Three computational projects to accomplish in group (7.5 points).

- A final written evaluation test (12.5 points) and final exam (12.5 points).