Numerical Mathematics I

Recommended Prerequisites

Mathematical Analysis, Linear Algebra, Basic Programming.

Teaching Methods

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.

Learning Outcomes

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.

Work Placement(s)

No

Syllabus

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.

Head Lecturer(s)

Gonçalo Nuno Travassos Borges Alves Pena

Assessment Methods

Assessment
Resolution Problems: 35.0%
Exam: 65.0%

Bibliography

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.