Numerical and Computational Methods

Recommended Prerequisites

Computer Programming: Basic notions of algorithm, main structure of a computer program, including data type, input-output instructions, decision-making tasks, repetitive tasks and functions.

Linear Algebra: Basic notions of indicial notation; vectors and matrices; matrix operations; systems of linear algebraic equations, including the Gauss elimination method; invertible matrices; Determining the inverse matrix; determinants; eigenvalues and eigenvectors.

Teaching Methods

The theoretical lectures take the form of master classes where the problems are outlined, using examples, and the numerical methods are discussed. In theoretical-practical classes selected problems, which illustrate the issues discussed in the theoretical lectures, are analysed and solved. In all theoretical-practical classes, the students are invited to solve some problems on the subject under study at home. In practical classes students develop and implement algorithms to test the application of the numerical methods to problem previously discussed in the theoretical-practical class.

Learning Outcomes

Provide skills in the numerical analysis field to engineering students, through a significant theoretical background and an applied component focusing on the resolution of engineering problems, using iterative methods. The development and implementation of simple numerical algorithms allows the students to acquire the necessary awareness about the numerical difficulties that may arise and possible solutions that can be adopted to overcome those difficulties.

Work Placement(s)

No

Syllabus

1. Elementary notions of tensor analysis: notation, tensors and basic operations in Cartesian coordinates.

2. Nonlinear equations: general conditions for their solving; iterative methods; convergence conditions; stopping criteria.

3. Systems of linear equations: iterative methods; convergence conditions; stopping criteria.

4. Systems of nonlinear equations: Newton-Raphson iterative method.

5. Polynomial interpolation: general conditions for their evaluation; interpolation methods; Interpolation error.

6. Numerical integration: integration methods; numerical integration error.

7. Ordinary differential equations: methods for their resolution; error and its propagation; extension to systems of ordinary differential equations.

8. Programming of numerical methods: development of algorithms and their implementation.

Head Lecturer(s)

Marta Cristina Cardoso de Oliveira

Assessment Methods

Assessment
Resolution Problems: 10.0%
Exam: 90.0%

Bibliography

M.C. Oliveira, L.F. Menezes;, Diapositivos de apoio à disciplina, continuamente atualizados .

S.C. Chapra, R.P. Canale, Métodos Numéricos para Engenharia, Mc Graw Hill, 2008.

F. Correia dos Santos, J. Duarte, Nuno D. Lopes, Fundamentos de análise numérica: com python 3 e R, Edições Sílabo, 2ª Edição, 2019.

J. Simon, Excel programming: your visual blueprint for creating interactive spreasheets, Wiley Publishing, 2nd Edition, 2005.

José Alberto Rodrigues, Métodos Numéricos - Introdução, Aplicação e Programação, Colecção Matemática 20, Ed. Sílabo, 2003.

F. Correia dos Santos, Fundamentos de Análise Numérica, Colecção Matemática 19, Ed. Sílabo, 2002.

Heitor Pina, Métodos Numéricos, Ed. McGraw-Hill, 1995.

A.J.C. Varandas, J. Brandão, A.A.C.C. Pais; Introdução à programação FORTRAN e cálculo científico, Minerva, Coimbra, 1994.

Stephen J. Chapman, Introduction to fortran 90/95; McGraw-Hill, Boston, 1998 (Basic Engineering Séries and Tools).