Numerical Analysis II

Code:

M332

Acronym:

M332

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2012/2013 - 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	19	Plano de estudos a partir de 2009	3	-	7,5	-	202,5

Teaching language

Portuguese

Objectives

 Establish constructive methods of numerical solution of fundamental problems of algebra and mathematical analysis, such as solving systems of linear and nonlinear equations, the calculation of inverse matrices and determinants, eigenvalues and eigenvectors of matrices, integrals and differential equations.

Under each theme, carried out the study of sufficient conditions for convergence of the methods presented, its stability, error control, construction of algorithms, their implementation and experimentation in computer processing of samples and corresponding interpretation of results.

Learning outcomes and competences

The student should be capable of establishing and building methods, translate them in shaped algorithm form, making the corresponding computer implementation, testing and application in obtaining solutions for various types of mathematical problems.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Basics of Numerical Analysis, Linear Algebra and Mathematical Analysis.

Domain of a programming language.

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 Conditioning numerical problems.

1.3.1 Condition number of a matrix.

1.3.2 Conditioning of matrix inversion disturbed.

 1.3.2.1 Method of iterative refinement of the inverse

the correctness of the residue.

1.3.3 Conditioning systems of linear equations.

1.3.3.1 Method of iterative refinement by correcting the residue.

2. Numerical solution of systems of linear equations

2.0 General (types of methods and computational efficiency).

2.1 Transformations and elementary matrices.

2.2 Types of matrices.

2.3 Triangular systems  and reverse triangular matrices.

2.3.1 Economics of memory space.

2.4 Gauss.

2.4.1 PA = LU  factorization by Gaussian elimination.

2.4.2 Solving systems of linear equations, calculating the inverse of a matrix and the determinant via Gauss.

2.4.3 Gauss algorithm method and its computational efficiency (counting the number of operations).

2.4.4 Principle of the method of Gauss.

2.5 Methods of compact factorization.

3rd. Iterative Methods for Solving Systems of Linear Equations

3.1 Methods of Jacobi and Gauss-Seidel.

4th. Numerical Resolution of Systems of Nonlinear Equations

4.1 Simple iterative method.

4.2 Method of Newton.

5th. Calculation of Numeric Eigenvalues and Eigenvectors of Matrices

5.1 Location of eigenvalues: Gerschgorin theorems and coefficient of Rayleigh.

5.2 Method of direct and inverse powers. Deflation.

5.3 Method of Jacobi.

5.4 Tridiagonalização matrices: Givens rotations, Householder reflections.

6th. Numerical Integration of Ordinary Differential Equations

6.1 Existence and uniqueness of solutions.

6.2 Euler, Euler modified predictor-corrector methods, methods of Taylor and Runge-Kutta.            

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
Golub Gene H.; Matrix computations. ISBN: 0-8018-5414-8
Pina Heitor; Métodos numéricos. ISBN: 978-972-592-284-2

Teaching methods and learning activities

- Lectures with explanation of the issues, resolution of theoretical problem, proposals of computational projects and presentations (by students) thereof.

- Pratical Lectures: solving exercises, computer implementation of computational projects as well as preparing reports and preparing presentations.

Software

MatLab, SciLab, Máxima ou Phyton

keywords

Technological sciences

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
	Trabalho laboratorial		37,50	2012-12-12
	Exame		62,50
	Total:	-	100,00

Amount of time allocated to each course unit

Description	Type	Time (hours)	End date
	Trabalho laboratorial		2012-12-12
	Total:	0,00

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).