Numerical Analysis

Code:

EIG0052

Acronym:

AN

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2020/2021 - 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
MIEGI	129	Syllabus since 2006/2007	1	-	6	42	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.


Small computer projects using WINDOWS or UNIX and MATLAB.

Mandatory literature

Maria Raquel Valença ; Análise Numérica, Universidade Aberta
Burden Richard L.; Numerical analysis. ISBN: 0-534-38216-9
Heitor Pina ; Métodos numéricos , McGraw Hill , 1995

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.

 

 

Observations

ALTERATION IN THE EVALUATION PROCESS DUE TO THE COVID19 CONDITION

The evaluation will be made recurring to a Final Examination. There will also be a Proof of Appeal to students who miss the Final Examination or do not obtain approval, as well as to those who wish to improve the final mark.