Numerical Analysis
Code
10541
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Nadir Arada
Weekly hours
4
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. Errors
Absolute error, relative error, significant digits. Condition number. Numerical algorithms stability.
2. Rootfinding for nonlinear equations
Bissection method, fixed-point iteration, Newton method, modified Newton method, secant method.
3. Polynomial approximation and interpolation
Polynomial interpolation: Lagrange and Newton formulas, Chebyshev polynomials, cubic Spline interpolation, least squares approximation.
4. Numerical integration
Newton-Cotes integration formulas, Romberg method, Gaussian integration.
5. Linear systems
Direct methods: Gaussian elimination, LU and Choleski factorizations. 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
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Quarteroni A., Saleri F., Scientific Computing with MATLAB and Octave, Series: Texts in Computational Science and Engineering , Vol. 2 Springer, 2006
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Quarteroni A., Sacco R., Saleri F., Numerical Mathematics, Springer, 2000
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Burden R. e Faires J. , Numerical Analysis, Brooks-Cole Publishing Company, 9th Edition, 2011.
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Pina H., Métodos Numéricos, Mc Graw Hill, 1995