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Numerical Analysis

Code:

INF32151

Sigla:

NA

Áreas Científicas
Classificação	Área Científica
OFICIAL	Matemática

Ocorrência: 2023/2024 - 2S

Ativa?	Yes
Página Web:	https://moodle.ips.pt/2324/course/view.php?id=1974
Unidade Responsável:	Departamento de Matemática
Curso/CE Responsável:	Informatics Engineering

Ciclos de Estudo/Cursos

Sigla	Nº de Estudantes	Plano de Estudos	Anos Curriculares	Créditos UCN	Créditos ECTS	Horas de Contacto	Horas Totais
INF	266	Plano de Estudos	1	-	6	75	162

Língua de trabalho

Portuguese

Objetivos

The main objective of the Curricular Unit is to familiarize the student with computational methods that numerically approximate solutions of common questions in calculus. It is intended that the student learns various numerical methods, its application, its implementation on a computational tool and how to choose the most appropriate among them, considering the problem to be solved.

Learning objectives:



1- Represent real numbers in different exact and rounded numerical representation systems.

2- Analyze in R^n the error associated with approximate values, and its propagation.

3- Carry out operations with matrices and apply them in the formulation of linear problems and error analysis.

4- Determine the set of solutions of linear matrix equations, through the inverse substitution algorithm and the Gaussian elimination method.

5- Identify Doolittle and Cholesky LU decompositions of square matrices, interpreting and applying the results.

6- Apply iterative methods to solve linear equations (in particular the Jacobi and Gauss-Seidel methods) and non-linear ones (in particular the Newton-Raphson and secant methods), analyzing their convergence.

7- Apply interval methods of bisection and false position in solving nonlinear equations.

8- Apply Lagrange, Newton and Gregory-Newton polynomial interpolation formulas to obtain approximate values ​​or zeros of a function.

9- Apply different quadrature rules, simple and compound, to obtain approximate values ​​of an integra

Resultados de aprendizagem e competências

Showing knowledge in:








Developing Skills in:





Acquisition of Competences in:

Modo de trabalho

Presencial

Pré-requisitos (conhecimentos prévios) e co-requisitos (conhecimentos simultâneos)

Integral and differential calculus with real functions in one real variável.

Programa

T1. NUMBER REPRESENTATION IN COMPUTER SYSTEMS AND ERRORS



T2. SYSTEMS OF LINEAR EQUATIONS







T3. NONLINEAR EQUATIONS




T4. POLINOMIAL INTERPOLATION




T5. NUMERICAL INTEGRATION

Bibliografia Obrigatória

Vários; Apontamentos editados pelo Departamento de Matemática.

Bibliografia Complementar

César Fernández; Sebenta de Análise Numérica (Available in the Moodle platform)
Atkinson KE; An introduction to numerical analysis, John Wiley & sons, 1990. ISBN: 0471624896
Correia dos Santos, F.M.; Fundamentos de Análise Numérica, Sílabo, 2002. ISBN: 9789726182863
Kharab, A.; Guenther, R.B.; An introduction to Numerical Methods - A Matlab approach, CRC press, 2018. ISBN: 9781315107042
Pina, H.; Métodos Numéricos, McGraw-Hill, 2010. ISBN: 9789728298043
Quarteroni, A.M.; Saleri, F.E.; Cálculo científico com Matlab e Octave, Springer Science & Business Media, 2007. ISBN: 9788847007185
Rao, S.S.; Applied numerical methods for engineers and scientists, Pearson, 2002. ISBN: 9780130894809
Scheid, F.; Análise Numérica, McGraw-Hill, 2000. ISBN: 972-9241-19-8

Métodos de ensino e atividades de aprendizagem

Student’s oriented study is performed through:




The objectives can be summarized as the knowledge of different algorithms, its properties, and all issues related to programming numerical methods on a computer. Thus theoretical sessions should serve for the solid presentation of the fundamentals. The presentation of contents, properties and exemples in theoretical-practical lessons, combined with the experimentation with these tools through specific exercises and Matlab programming should lead to the achivement of this course’s goals.

Exercise solving in theoretical-practical hours should serve for the assimilation of these notions, and its applications in different practical cases. Practical sessions with Matlab/Octave will develop the required skills in programming of scientific algorithms

Different supporting materials are available for the autonomous study: lecture notes, solved exercises, Matlab codes, formularies, web links, books (distributed at the classroom, Moodle page or IPS library)

Palavras Chave

Physical sciences > Mathematics > Computational mathematics
Physical sciences > Mathematics > Algorithms
Physical sciences > Mathematics > Mathematical analysis > Functions

Tipo de avaliação

Distributed evaluation with final exam

Componentes de Avaliação

Designation	Peso (%)
Teste	100,00
Total:	100,00

Componentes de Ocupação

Designation	Tempo (Horas)
Estudo autónomo	87,00
Frequência das aulas	75,00
Total:	162,00

Obtenção de frequência

The student can choose either the assessment by tests or the assessment by exam:

Test evaluation
This assessment consists of performing 2 (two) tests.
The final CF grade for the subject will be calculated using the arithmetic average of the two tests rounded to units and the conditions for approval are:

1. If CF is greater than or equal to 17, the student must take an oral exam. The final grade is given by the arithmetic average (rounded to the units) of CF and the classification obtained in this test. If the student deso not attend the oral exam, the final classification will be 16 points;


2. If CF is greater than or equal to 10 and less than 17, the student is approved with a final grade of CF, provided that the classification in both tests has been equal to or greater than 7.0.

Exam evaluation
The assessment based on the performance of exams follows the usual rules, that is, students who choose not to perform test evaluation, or who, having opted for it, have not been approved, may attend the regular exam periods.
If E is the classification obtained in the exam (rounded to the units), if E is greater than or equal to 17, the student will have to take an oral exam, obtaining as a final grade the average of the classifications of the referred oral exam and the written exam . If the student does not attend the oral exam, the final classification will be 16 points;
If E (rounded to the units) is greater than or equal to 10 and less than 17, the student is approved with a final grade of E.

Fórmula de cálculo da classificação final

Test evaluation
If (T1 and T2>=7) and T1+T2<=32, then CF=(T1+T2)/2
If (T1 and T2>=7) and T1+T2>32, CF=(((T1+T2)/2)+OE)/2

Exam evaluation
If E<17, CF=E
If E>=17, CF=(E+OE)/2

Avaliação especial (TE, DA, ...)

Students that intend to apply for specific evaluation rules that are recognised within the IPS normatives, must communicate (no later than 14 days prior to the test) to any of the Teachers these circunstances. They shall present documentary evidence that these specific conditions and rules apply to their particular case.

In the case that this comunication is not stablished, all testing conditions that imply a planning by the evaluator may not apply due to lack of objective conditions.

Observações

Specific rules for written tests and exams:



• The student must bring the formatted exam sheets to any written evaluation process (one cover sheet and 4 additional A4 sheets)
•The presentation of ID is compulsory on all evaluation tests and exams.
• Calculating machines are authorized in tests and exams, as long as they comply with the rules declared in OficioCircular /S-DGE/2016/1798
• Using, handling or even carrying of nonauthorized electronic equipments, different from the calculating machine, is prohibited during the tests/exams.



Moodle's Link : https://moodle.ips.pt/2324/course/view.php?id=1974

Password: an2324


Tutorial support in the teaching period:

Prof. Artur Miguel Cruz (Responsable teacher of the UC)
Thursdays 9h00-11h00
Frydays 13h30-14h30

Prof. César Fernandez 
Tuesdays 14:30h-16h30
Frydays 12h30-13h30


Tutorial support in the exams period will be determined after the corresponding exams schedule is approved.