Numerical Analysis
Code
10541
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Magda Stela de Jesus Rebelo
Weekly hours
4
Teaching language
Português
Objectives
The student must be able to apply numerical methods for mathematical problems, such as, non linear equations, approximation of functions, integration, systems of equations and ordinary differential equations.
The student must also be able to implement computational algorithms in order to solve the aforementioned problems.
Prerequisites
Students must have basic knowledge in mathematical analysis (AMI) and linear algebra (ALGA).
Subject matter
1. Errors
Absolute error, relative error, significant digits. Condition number. Numerical algorithms stability.
2. Polynomial approximation and interpolation
Polynomial interpolation: Lagrange and Newton formulas, cubic Spline interpolation.
Least squares approximation.
3. Numerical integration
Newton-Cotes integration formulas, Romberg method, Gaussian integration.
4. Rootfinding for nonlinear equations
Bissection method, fixed-point iteration, Newton method, modified Newton method.
5. Linear systems
Vector norms and induced matrix norms.
Eigenvectors and eigenvalues. Gershgorin theorem.
Iterative methods: general procedure, Jacobi method, Gauss-Seidel method, SOR method.
6. Numerical solution of ODE
Euler methods, Taylor methods for higher orders, Runge-Kutta methods
Bibliography
- Atkinson K., An Introduction to Numerical Analysis, Wiley, Second Edition, 1989
- Burden R. e 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. e Keller H., Analysis of Numerical Methods, Dover, 1994
- Martins, M. F. e 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