Numerical Analysis A

Objectives

Students 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. Students must also be able to implement computational algorithms in order to solve the aforementioned problems.

General characterization

Code

12595

Credits

3.0

Responsible teacher

António Manuel Morais Fernandes de Oliveira

Hours

Weekly - 3

Total - 39

Teaching language

Português

Prerequisites

Students must have basic knowledge in Mathematical Analysis I (AM I) and Linear Algebra and Analytic Geometry (ALGA).

Bibliography

Available soon

Teaching method

Available soon

Evaluation method

Available soon

Subject matter

1. Errors

Absolute error,  relative error, significant digits. Condition number. Numerical algorithms stability. Introduction to a CAS (Computer Algebra System) code, suited to solve numerical problems arising in subsequent items.    

2. Polynomial approximation and interpolation

Polynomial interpolation: Lagrange and Newton formulae, cubic spline interpolation.

Least squares approximation. 

3. Numerical integration

Simple and composite Newton-Cotes integration formulas, Gaussian integration.

4. Rootfinding for nonlinear equations

Bissection method,  fixed-point iteration, Newton method, secant 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.