Computational Methods in Engineering
Code
10437
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
3.0
Teacher in charge
António Manuel Morais Fernandes de Oliveira
Weekly hours
3
Total hours
42
Teaching language
Português
Objectives
Students must be able to apply numerical methods for mathematical problems, such as, nonlinear equations, function approximations, integration, systems of linear equations and ordinary differential equations (ODE''''s).
Students must also be able to implement computational algorithms in order to solve the aforementioned problems.
Prerequisites
Students must have basic knowledge in Mathematical Analysis I (AM I) and Linear Algebra and Analytic Geometry (ALGA).
Subject matter
1. Introduction
1.1 Errors, significant digits.
1.2 Conditioning of a problem and stability of a method.
1.3 Introduction to a computational language for Numerical Analysis.
2. Polynomial approximation and interpolation
2.1 Interpolation and Lagrange polynomial.
2.2 Divided differences, Newton''''s interpolating polynomial.
2.3 Cubic spline interpolation.
2.4 Least squares approximation.
3. Numerical integration
3.1 Newton-Cotes quadrature formulae (single and composite rules).
3.2 Gaussian quadrature. Other integration methods.
4. Root Finding Methods for Nonlinear Equations
4.1 Bisection method.
4.2 Fixed point iteration method. Newton''''s method. Secant method.
5. Iterative Methods for Solving Linear Systems of Equations
5.1 Vector and matrix norms. Conditioning of a system of equations.
5.2 Eigenvalues and eigenvectors.
Gershgorin''''s theorem for estimating eigenvalues.
5.3 Iterative methods: general procedure.
5.4 Jacobi, Gauss-Seidel and Relaxation methods.
6. Numerical Solution of Ordinary Differential Equations (ODE''''s)
6.1 Euler''''s method.
6.2 Taylor methods.
6.3 Runge-Kutta methods.
Bibliography
- Atkinson K., An Introduction to Numerical Analysis, Wiley, Second Edition, 1989.
- Burden R. and Faires J. , Numerical Analysis, Brooks-Cole Publishing Company, 9th Edition, 2011.
- Conte S. e Boor C., Elementary Numerical Analysis: an algorithmic approach, Mc Graw Hill, 1981.
- Isaacson E. and Keller H., Analysis of Numerical Methods, Dover, 1994.
- Martins, M. F. and Rebelo M., Introdução à Análise Numérica, Casa das Folhas, 1997.
- Pina H., Métodos Numéricos, Mc Graw Hill, 1995.
- Valença M. R., Métodos Numéricos, Livraria Minho, Terceira Edição, 1993.