Numerical Analysis

Code:

EIG0052

Acronym:

AN

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2015/2016 - 2S

Active?	Yes
Responsible unit:	Mathematics Section
Course/CS Responsible:	Master in Engineering and Industrial Management

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEIG	105	Syllabus since 2006/2007	2	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

General:

The students will be able to choose the most efficient methods for the solution of each basic Numerical Analysis problem. The students are expected to understand the theorems and convergence conditions of each of the methods described, to be able to program them, to test them effectively on a computer and discuss the results obtained.

Specific:
For each chapter in the program the successful students
will be able to list the applicability conditions of the numerical methods and state the corresponding theorems of convergence;
they will be able to apply the methods, formula and algorithms taught to simple problems;
they will be able to describe the behavior of the methods, translate them into algorithms and
‘Matlab Functions’ as well as test them on examples comparing and analyzing the results;
they will explain the proofs of the theorems given and apply the proof techniques involved to other related situations;
they will be able to solve new problems with the numerical tools here taught and compare the performance of the various numerical methods in terms of speed and accuracy.

Learning outcomes and competences

For each chapter in the program the successful students
will be able to list the applicability conditions of the numerical methods and state the corresponding theorems of convergence;
they will be able to apply the methods, formula and algorithms taught to simple problems;
they will be able to describe the behavior of the methods, translate them into algorithms and
‘Matlab Functions’ as well as test them on examples comparing and analyzing the results;
they will explain the proofs of the theorems given and apply the proof techniques involved to other related situations;
they will be able to solve new problems with the numerical tools here taught and compare the performance of the various numerical methods in terms of speed and accuracy.

Working method

Presencial

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

The students are supposed to know the subjects taught in Linear Algebra and Mathematical Analysis as well as in Computer programming

Program

Chapter 1. Number systems and errors ; number systems on computers; representation of integers and floating point arithmetic; round-off error; absolute error and relative error, significant digits,  Taylor's formula and error estimation; error analysis.

Chapter 2. Linear systems of equations: Gaussian elimination. Round off errors and possible instability of the numerical methods, pivoting strategies. Solution of triangular systems. Tridiagonal systems. LU factorization; application to the computation of determinants and to the inversion of matrices.
Iterative methods: Jacobi and Gauss-Seidel; convergence theorems and algorithms. solution of triangular systems.

Chapter 3. Least squares approximation. Orthogonal polynomials. Curve fitting. Over-determined systems of equations.

Chapter 4. Non linear equations: general conditions for the solution, stopping criteria for iterative methods; some iterative methods: successive bisection, fixed point iteration, Newton's method, secant method. Convergence theorems and algorithms; polynomial equations. 

Chapter 5. Numerical integration: Newton-Cotes formulae (ex: Trapezoidal and Simpson rules); composite rules; numerical quadrature errors. Gaussian quadrature.

Chapter 6. Polynomial interpolation: finite differences; methods of Newton and Lagrange; error of the interpolating polynomial.

Chapter 7. Ordinary Differential equations: Euler s method for ODE of order 1; Taylor methods. Order of a method for ODE of order1. Runge-Kutta methods of order 2 and 4

Small computer projects using WINDOWS or UNIX and MATLAB.

Mandatory literature

Cleve Moler; Numerical Computing with Matlab , SIAM , 2004
John Mathews; Kurtis Fink ; ; Numerical Methods using Matlab , Prentice Hall , 1999
Maria Raquel Valença ; Métodos Numéricos , Livraria do Minho , 1993
Edite Fernandes; Computação Numérica, 1997. ISBN: ISBN: 972-96944-1-9
Maria Raquel Valença ; Análise Numérica, Universidade Aberta
Heitor Pina ; Métodos numéricos , McGraw Hill , 1995

Complementary Bibliography

Rosário, Pedro ; Núnez, José ; Pienda, Júlio; Comprometer-se com o estudar na universidade : cartas do Gervásio ao seu umbigo, Livraria Almedina, 2006
Mário Graça, Pedro Lima ; Matemática Experimental , IST Press , 2006 . ISBN: ISBN: 972-8469-52-7

Teaching methods and learning activities

Lectures with "Powerpoint". Small illustrating computer projects supervised by teachers in the computer room, with Matlab.

Software

Matlab

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation	Weight (%)
Participação presencial	0,00
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	106,00
Frequência das aulas	56,00
Total:	162,00

Eligibility for exams

Minimum requirements to be admitted to the exam:
registration and not to exceed the maximum number of absences allowed.

Calculation formula of final grade

50% of Test 1+50% of Test 2

At appeal test ("recurso") students who fail to pass can repeat the first or the second test (the best mark will be taken into account). However, they can take a final exam, which will cover all themes of the course unit.

 

The successful students can improve their grades at appeal test ("recurso"), taking a final exam covering all themes of the course unit.

Grade 20 is only possible with an oral exam.