Numerical Analysis
Code
10541
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Elsa Estevão Fachadas Nunes Moreira
Weekly hours
4
Total hours
56
Teaching language
Português
Objectives
We will illustrate several numerical methods for the computer solution of certain classes of mathematical problems. We will show how to use these methods in order to solve nonlinear equations, linear systems, integrate and construct accurate approximations for the solution of differential equations.
Prerequisites
Basic knowledge in analysis and linear algebra
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 program for Numerical Analysis.
2. Polynomial approximation and interpolation
2.1 Interpolation and Lagrange polynomial
2.2 Divided differences, interpolating polynomial of Newton.
2.3 Cubic Spline interpolation.
2.4 Least squares approximation.
3. Numerical integration
3. 1 Newton-Cotes integration formulas (Single and composite rules)
3.2 Gaussian integration. Other integration methods.
4. Root finding for nonlinear equations
4.1 Bisection method.
4.2 Fixed-point iteration method. Newton method. Secant method.
5. Iterative methods for solving linear systems of equations
5.1 Norms of vectors and matrices. Conditioning of a system.
5.2 Eigenvalues and eigenvectors. Gershgorin theorem.
5.3 Iterative methods: general procedure.
5.4 Jacobi, Gauss-Seidel and relaxation methods.
6. Numerical solution of ordinary differential equations
6.1 Euler methods.
6.2 Taylor methods.
6.3 Runge-Kutta methods.