3

2018-2019

01001275

Mathematics

Portuguese

Face-to-face

SEMESTRIAL

6.0

Compulsory

1st Cycle Studies

Mathematical Analysis, Linear Algebra, Basic Programming.

The begining of the lecture consists in the presentation of concepts and theoretical results. In the second part of the lecture, students are asked to solve application problems in order to consolidate their understanding of concepts and results presented in the first part. Some classes take place in the Calculus Laboratory to program computational methods.

During this course students should acquire skills needed to analyze and to solve numerically mathematical problems studied within the course as well as to interpret the solutions obtained. It has also intent to develop the critical analysis capacity for results and for argumentation as well as the algorithmic reasoning.

The main competencies to be developed are: calculus; to use computational tools; to generalize and to formalize in mathematical abstract context; to formalize and to solve problems; to model mathematically problems from real context or to use mathematical models in real problems; independent learning; knowledge of mathematical results; strong and clear written and oral skills; individual initiative; critical thinking.

1. Errors and their propagation

2. Direct methods for systems of linear equations

Gauss elimination methods. QR factorization (Householder transformations). Singular value decomposition. Least mean square problem.

3. Iterative methods for systems of linear equations

Stationary methods: Jacobi, Gauss-Seidel, SOR. Non stationary methods: conjugate gradient method.

4. Polynomial interpolation

Lagrange interpolation. Hermite interpolation. Splines.

5. Non linear equations

Bisection method. Fixed point method. Newton-Raphson method.

José Augusto Mendes Ferreira

Assessment

*Resolution Problems: 35.0%*

*Exam: 65.0%*

A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Springer, 2000

G. Golub, V. Loan, Matrix Computations, John Hopkins University Press, 1996

H. Pina, Métodos Numéricos, McGraw-Hill, 1995

L. N. Trefethen, D. Bau, Numerical Linear Algebra, SIAM, 1997

R. Kress, Numerical Analysis, Springer, 1997