Numerical Analysis

Code:

EC0007

Acronym:

ANUM

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2018/2019 - 2S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=702
Responsible unit:	Mathematics Division
Course/CS Responsible:	Master in Civil Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEC	250	Syllabus since 2006/2007	1	-	6	60	160

Teaching language

Portuguese and english

Objectives

JUSTIFICATION:
Problem-solving with recourse to traditional analytical methods has a reduced expression, particularly when the problems are of a practical nature, generated by applications within the engineering. The analytical study is however not neglected and MIEC studies program includes disciplines of mathematical analysis and algebra where students acquire the knowledge on what all subsequent application techniques will rely. On the area of applied mathematics, numerical methods allow an important extension of kinds of problem-solving techniques, intervene in situations where the analytically is impracticable or where the nature of the problems,  the data with which you work or the type of solutions in vista, require numerical approaches.

OBJECTIVES:
- develop students’ ability to determine and analyse results obtained by calculation tools with approximation techniques.
- introduce the notion of stability methods and number of condition of problems.
- Acquaint students with the use of several techniques to solve different problems by studying their efficiency, applicability and stability.
- solve problems using the computer.
- Acquaint students with methodologies to choose and decide which numerical resolution method should be applied and the most efficient one.
Students should be able to discuss the numerical results obtained.

Learning outcomes and competences

SKILLS AND LEARNING OUTCOMES:
1. Technical knowledge: to have the ability to understand numerical problems, so that the base of application of real problems can be constituted. Critical conscience on the Engineering field
2. Personal and professional skills: To apply the knowledge, understanding and the skills to solve problems in new and uncertain situations. Learning skills: lifelong learning, in an autonomous and self oriented way.
3. Interpersonal skills: oral and written communication: be capable of communicating conclusions, knowledge, thoughts to specialists and non-specialists in a clear and unambiguous way, both in group and individual assignments.

Working method

Presencial

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

Mathamatical Analysis I and II, Algebra and Computatiion.

Program

1. Goal for numerical analysis.
2. Theory of errors: basic concepts, errors definitions and properties. Error propagation. Floating-point arithmetic (10%).
3. Solution of non-linear equations: iterative methods, bisection method, false position method, secant method, and simple iterative method. General conditions for the resolution and stopping criteria for iterative methods (25%).
4. Systems of linear equations (20%):
4.1. Direct methods: Gauss elimination, pivoting techniques;
4.2. Iterative methods: Jacobi, Gauss-Seidel, convergence theorems;
5. Polynomial approximation:
5.1. Polynomial interpolation: divided differences, Newton and Lagrange methods, interpolation error, and direct and inverted interpolation (20%).
5.2. Introduction to the least squares method (5%).
6. Numerical integration (20%):
6.1. Newton-Cotes formulas (e.g.: trapezes and Simpson);
6.2. Composed formulas;
6.3. Errors in numerical integration.

Scientific Content: 70%
Technological content: 30%

DEMONSTRATION OF THE SYLLABUS COHERENCE WITH THE CURRICULAR UNIT'S OBJECTIVES:
Problem-solving with recourse to traditional analytical methods has a reduced expression, particularly when the problems are of a practical nature, generated by applications within the engineering. The analytical study is however not neglected and MIEC studies program includes disciplines of mathematical analysis and algebra where students acquire the knowledge on what all subsequent application techniques will rely. On the area of applied mathematics, numerical methods allow an important extension of kinds of problem-solving techniques, intervene in situations where the analytically is impracticable or where the nature of the problems, the data with which you work or the type of solutions in vista, require numerical approaches. The numerical analysis thus extends the possibilities of math in solving engineering problems.

Mandatory literature

Quarteroni Alfio; Cálculo Científico com Matlab e Octave. ISBN: 978-88-470-0717-8 (Access to this content is enabled by Universidade do Porto)
Quarteroni, A. & Saleri, F; Scientific Computing with MATLAB and Octave, Springer, 2014. ISBN: 978-3-642-45366-3 (Access to this content is enabled by Universidade do Porto)
Ana Maria Faustino; Análise Numérica, FEUP, 2014. ISBN: 978-989-98632-5-5

Complementary Bibliography

Valença, Maria Raquel; Métodos numéricos
Burden, Richard L.; Numerical analysis. ISBN: 0-534-38216-9
Pina, Heitor; Métodos numéricos. ISBN: 972-8298-04-8
Fausett, Laurene V.; Applied numerical analysis using Matlab. ISBN: 0-13-319849-9
Kharab, Abdelwahab; An introduction to numerical methods. ISBN: 1-58488-281-6
Bradie, Brian; A friendly introduction to numerical analysis. ISBN: 0-13-191171-6

Teaching methods and learning activities

Concepts and techniques are presented according to Analysis and Algebra knowledge and, whenever possible, the theoretical exposition is supported by practical examples and graphic representations. Theoretical aspects are presented with enough detail to exhibit the applicability of formulas. Additionally, methods are analyzed and compared in what concerns their efficiency, accuracy and applicability. Students are encouraged to develop algorithms for the numerical methods. In practical classes, several case studies are solved on the computer using Matlab. This curricular unit is inserted in the Moodle platform, in order to enhance the discussion among all participants. In this platform, all students have access to every issue provided by the teachers and may strengthen their concepts by solving self-evaluation tests whose results are immediately commented. They may also use the forums to bring questions before all the community of Numerical Analysis.

DEMONSTRATION OF THE COHERENCE BETWEEN THE TEACHING METHODOLOGIES AND THE LEARNING OUTCOMES:
The theoretical issues are accurately presented in lectures and the use of computers help to achieve the objectives of the curricular unit, related to the use of different numerical techniques for solving engineering problems. It allows to provide working tools for subsequent curricular that use computation intensively. The use of Moodle also allows better interaction between all participants of the curricular unit and consolidates the concepts by doing self-assessment tests.

Software

Octave
Matlab

keywords

Physical sciences > Mathematics > Applied mathematics > Numerical analysis

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Designation	Weight (%)
Exame	75,00
Trabalho prático ou de projeto	5,00
Trabalho escrito	20,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	70,00
Frequência das aulas	56,00
Elaboração de projeto	8,00
Total:	134,00

Eligibility for exams

Achieving final classification requires compliance with attendance at the course unit, according to the MIEC assessment rules. It is considered that students meet the attendance requirements if, having been regularly enrolled, the number of absences of 25% for each of the classes’ types is not exceeded.

Calculation formula of final grade

Final mark (CF) is calculated according to:

CF = maxima {EX; AD}

 where,

EX – Exam Mark

AD = (10% x MP1+ 10% x MP2 +5% x MDL) + 75% x EX

 

MP1 - mark of the first  pratical mini test;

MP1 - mark of the second  pratical mini test;

MDL - classifications obtained in 4 Moodle activities evaluated in the following scale:

• 100% if the 4 activities with a positive note were completed within the established deadline.
• 75%  if 3 activities were carried out in the established period with a positive note.
• 50% if 2 activities were carried out in the established period with a positive note.
• 0% in all other cases.

 

All marks are quoted on a  0 to 20 scale.

The distributed evaluation obtained in previous courses, is not valid.

Special assessment (TE, DA, ...)

Final written exam.

Classification improvement

Final written exam.

Observations

Working time estimated out of classes: 4 hours