Numerical Analysis II

Code:

M332

Acronym:

M332

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2011/2012 - 1S

Active?	Yes
Web Page:	http://moodle.up.pt/course/view.php?id=2984
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	39	Plano de estudos a partir de 2009	3	-	7,5	-	202,5

Teaching language

Portuguese

Objectives

The main aim of this subject is is given a mathematical problem, to study sufficient conditions for the existence and unicity of its solution, to establish a constructive method to solve it, to study and control the errors involved, to give an algoritmh for the solution and to implement it in a computer and to study and interpret the numerical results.
The following fundamental mathematical problems will be treated: the solution of systems of linear or nonlinear equations, computation of eigenelements of a matrix and the integration of ordinary differential equations.

Program

Systems of linear equations
Norms and limits of vectors and matrices. Direct methods. Gauss elimination. Pivoting. Inversion of regular matrices.
Sensibility to error in data. Iterative refinement of the solution of a system and of the inverse of a matrix.

Systems of nonlinear equations
Fixed point methods. Newton method.

Numerical integration of differential equations
Existence theorems. One-step methods: Euler, predictor-corrector, Taylor and Runge-Kutta methods. Consistence and convergence of one-step methods.

Numerical integration
Orthogonal polynomials. Gauss-Legendre Integration.

Eigenvalues
Gerschgorin theorems. Rayleigh coefficient.
Power method. Deflaction. Jacobi method. Triadiagonalization of matrices: Givens rotations and Householder reflections.

Mandatory literature

000071361. ISBN: 972-8298-04-8
000081959. ISBN: 2-7298-2246-1
000087488. ISBN: 978-2-7298-2887-5
000040213. ISBN: 0-8018-5414-8

Teaching methods and learning activities

Lectures, problems and computational projects.

Software

Python, Scilab or Maxima

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	75,00
	Total:	-	0,00

Eligibility for exams

A minimum of 3.5 points in the practical classification.

Calculation formula of final grade

Theoretical classification (CT): Sum of the classifications of 2 tests ( 5 points each)
Practical classification (CP): sum of classifications obtained in 4 practical tests (2.5points each)
Final classification (CF): CT+CP