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# Computational Methods in Engineering

3831

### Department

Departamento de Matemática

6.0

### Teacher in charge

Luís Manuel Trabucho de Campos, Nuno Filipe Marcelino Martins

5

84

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. Introduction

- Floating point arithmetic. Error and error propagation. Conditioning of a problem and stability of a method.

2. Non linear equations.

- Bissection method

- Fixed point and Newton methods.

- Order of convergence.

3. Interpolation and approximation of functions.

- Polynomial interpolation: Lagrange polynomial; finite and divided differences. Newton polynomial. Chebyshev nodes of interpolation.

- Linear and cubic splines.

- Discrete least squares approximation.

4. Derivatives and integration.

- First order derivatives: forward, backward and centered differences. Higher order derivatives.

- Newton-Cotes formulas. Composite rules. Gaussian quadrature. Romberg method.

5.  Numerical methods in linear algebra.

- Direct methods for systems of linear equations. Gauss method and LU factorisation.

- Vector norms and induced matrix norms. Conditioning of a linear system.

- Eigenvalues and eigenvectors. Gershgorin theorem.

- Iterative methods for linear systems: Jacobi, Gauss-Seidel and SOR methods.

- Determination of eigenvalues and eigenvectors: Direct power method.

6. Numerical methods for ordinary differential equations.

- Taylor, Runge-Kutta and multistep methods

7. Systems of  non linear equations.

### Bibliography

ATKINSON, K., An Introduction to Numerical Analysis, Wiley, 1989.

BURDEN, R.; FAIRES, D., Numerical Analysis (8th Edition) - Brooks-Cole Publishing, 2004.

KINCAID D., CHENEY W., Numerical Analysis: mathematics of scientific computing, Brooks-Cole, 2002.

MARTINS, M. F., Introdução à Análise Numérica, Casa das Folhas, 1997.

VALENÇA, M. R., Análise Numérica, Universidade Aberta, 1996.