Numerical Analysis B
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.
General characterization
Code
12686
Credits
6.0
Responsible teacher
Nuno Filipe Marcelino Martins
Hours
Weekly - 4
Total - 56
Teaching language
Português
Prerequisites
Basic knowledge in analysis (calculus) and linear algebra
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.
-
Pina H., Métodos Numéricos, Mc Graw Hill, 1995
- Santos, F. Correia dos; Duarte, Jorge; Lopes, Nuno D.,
Fundamentos de Análise Numérica (Com Python 3 e R), Edições Sílabo, 2019 (2ª edição).
Teaching method
Available soon
Evaluation method
Available soon
Subject matter
1.Introduction
1.1 Errors, significant digits.
1.2 Conditioning of a problem and stability of a method.
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.