Numerical Mathematics II

Recommended Prerequisites

Differencial and integral calculus,  Linear Algebra, Numerical Mathematics I.

Teaching Methods

The begining of the lecture  consists in the presentation of concepts and theoretical results in detail and rigourously. In the second part of the lecture, students are asked to solve a couple of problems related to what has been presented in the first part. Some of these problems consist of theoretical results that were presented in the begining but not proven. In general, these problems are then solved in the blackboard by one of the students, that will explain to the class the details of the proof.  

Learning Outcomes

The main goal of this curricular unit is the development of numerical methods as tools to solve problems in different subjects, such as optimization, linear algebra, approximation theory and differential equations. The main competencies to be developed, apart from the computational tools, concern the knowledge of theoretical mathematical results, which develops the ability of generalization and abstraction, efficient formulation and resolution of new problems. Additionally, mathematical models of real physical situations will be given, in particular related to the problems assigned during the semester. These working problems require independent learning and critical thinking and a rigorous and clear text needs to be written.

Work Placement(s)

No

Syllabus

1. Non linear systems of equations and optimization without restrictions.

2. Numerical differentiation.

3. Numerical integration.

4. Aproximation theory of functions.

5. Differential equations with boundary conditions.

6. Differential equations with initial values.

Head Lecturer(s)

Adérito Luís Martins Araújo

Assessment Methods

Continuous assessment
Resolution Problems: 30.0%
Frequency: 70.0%

Final assessment
Exam: 100.0%

Bibliography

C. F. Van Loan,  Introduction to Scientific Computing: A Matrix-Vector Approach Using Matlab, The Matlab Curriculum Series, Prentice-Hall, 1997.

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Philadelphia: SIAM, 1996.

J. Nocedal, S. J. Wright, Numerical Optimization, Springer-Verlag, 1999.

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

M. Mori,The Finite Element Method and Its Applications, MacMillan Publishing Company, 1989.

R. S. Quarteroni, F. Saleri, Numerical Mathematics, Texts in Applied Mathematics, 37,  Springer-Verlag, 2000.