Numerical Methods

Math analysis

In the lectures it is made a theoretical exposition of each topic which is complemented by studying practical examples. Problem solving related with the acquired knowledge is performed in the laboratory lessons. The Laboratory lessons are taught using the Matlab software to program the numerical methods studied

Provide the student with tools enabling him to gather one or more approximate solutions of problems whose resolution cannot be obtain using exact analytical form. Generic skills: Application of knowledge and understanding. Realization of judgment and decision making. Communication. Self-learning Mode. Specific Skills: Ability to use mathematical techniques. Develop the capacity of concept’s perception, abstract reasoning, interpretation of results and their application to problem solving. Understanding the details of the algorithms studied for the resolution of specific problems in the field of Engineering. Use the MATLAB software in the numerical treatment of the subjects and compare, with criticism, the results obtained by computational means with the ones obtained
analytically.

1. Brief introduction to the theory of errors
2. Nonlinear Equations Introduction and motivation. Roots of Nonlinear Equations Location of roots: Bisection and Newton methods. Stopping criteria. Computational aspects. Minimization of  functions: Gradient Method.
3. Polynomial interpolation Introduction and motivation. Lagrange interpolating polynomial. Newton’s interpolating polynomial (divided differences). Inverse polynomial interpolation. Interpolation  error.
4. Numerical integration Introduction and motivation. Newton-Cotes rules: trapezoidal and Simpson rules (simple and composite). Integration error. Computational aspects.
5. Numerical integration of initial value problems of first order Introduction and motivation. Euler’s method and Runge–Kutta methods of second and fourth order. Study of error. Computational aspects.
6. Methods of solving linear systems: iterative Methods of Jacobi and Gauss-Seidel.

NAO

• Atkinson, K. E., An Introduction to Numerical Analysis, John Wiley.
• Chapra, S. C. e Canale, R.P., Numerical Methods for Engineers, McGraw-Hill.
• Conte, S. D. e De Boor, C., Elementary Numerical Analysis, McGraw-Hill.
• Faires, J. D. e Burden, R., Numerical Methods (2nd Ed.),Brooks / Cole Publishing Company.
• Valença, M. R., Métodos Numéricos, Instituto Nacional de Investigação Científica.
• Batel Anjo, A. J., Fernandes, R. e Carvalho, A. S., Curso de MatLab, Principia
• J. A. Rodrigues (2003.) Métodos Numéricos, Introdução, Aplicação e Programação, Edições Sílado, 1ª edição.
• Deolinda Rasteiro, Apontamentos de apoio às aulas teóricas – acetatos- Métodos Numéricos.
• R. Rodrigues, P.M. Silva, P.M. Rosa, Matemática e Animações, em http://www2.isec.pt/~ppr/.