Numerical Analysis

Code:

L.EC007

Acronym:

AN

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2023/2024 - 2S

Active?	Yes
Responsible unit:	Department of Civil Engineering
Course/CS Responsible:	Bachelor in Civil Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EC	246	Syllabus	1	-	6	52	162

Teaching language

Portuguese

Objectives

OBJECTIVES:
Developing the ability to determine and analyse results obtained by calculating instruments using approximate techniques.
Introduce the notion of stability of methods and number condition of a problem.
Use several techniques for numerical problem solving studying its efficiency, applicability and stability.
Solve problems using computer;
Know how to choose and decide which method of numerical solution to apply and which is the most efficient.
Discuss the numerical results obtained.

Learning outcomes and competences

SKILLS AND LEARNING OUTCOMES:
Knowledge: Know and describe the fundamental concepts and numerical methods for solving engineering problems. Identify the main concepts associated to numerical solution and determination of approximate solutions.

Understanding: Identify and interpret the different techniques to be used in numerical problem solving.

Application: Develop skills for numerical problem solving. Apply knowledge and the ability to understand and solve problems in new and unfamiliar situations, in broad and multidisciplinary contexts.

Analysis: Analyse, discuss and critically interpret results, highlighting the potential of methods and their limitations.

Synthesis: Formulate and validate numerical solutions for solving non-linear equations, linear and non-linear systems, approximation of functions and integration.

Assessment: Criticise solutions and methodologies used. Be able to communicate their conclusions and their underlying knowledge and reasoning in a clear unambiguous manner.

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. Numerical Analysis and Scientific Computing. Theory of errors: basic concepts, errors definitions and properties. Error propagation. Direct and iterative methods. Iterative methods. Order of convergence of a succession of iterates.
2. Solution of non-linear equations: iterative methods, bisection method, Newton method and secant method. Convergence conditions and stopping criteria for iterative methods. Error estimation.
3. Systems of equations:
3.1. Direct methods for Linear Systems: Gauss elimination, pivoting techniques;
3.2. Iterative methods for Non Linear Systems: Newton Method
4. Approximation of functions and data:
4.1. Polynomial interpolation: Different constructions of the interpolating polynomial; Lagrange polynomials; interpolation error. Divided differences and derivatives.
4.2. Introduction to the least squares method.
5. Numerical integration:
5.1. Newton-Cotes formulas;
5.2. Composed formulas;
5.3. 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)

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
Ana Maria Faustino; Análise Numérica, FEUP, 2014. ISBN: 978-989-98632-5-5

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

Jupyter Notebook
Octave
Matlab

keywords

Physical sciences > Mathematics > Applied mathematics > Numerical analysis

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Designation	Weight (%)
Exame	70,00
Teste	30,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	70,00
Frequência das aulas	56,00
Total:	126,00

Eligibility for exams

Achieving final classification requires compliance with attendance at the course unit, according to the L.EC assessment rules. In regular classes, attendance will be registered.

Calculation formula of final grade

Distributed assessment is optional.

1. Calculation formula of final grade for grades greater than or equal to 7.5 in the Final Exam (EF). Final mark (CF) is calculated according to:

CF = maximum {EX; AD} where,

EX – Final Exam Mark, in situ

AD = 30% x MP + 70% x EX

MP is the classification of the distributed evaluation component, obtained by activities:

- Mark of positive participation in 3 Moodle activities (10%)
- Summative test mark (20%)

2. For exam grades lower than 7.5 the final mark is the exam mark.

To achieve a classification of 18 or above, a supplementary exam is required.

All marks are quoted on a scale from 0 to 20.

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

Special assessment (TE, DA, ...)

Final exam.

Classification improvement

Final exam.