Numerical Analysis I
Code
10979
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Elvira Júlia Conceição Matias Coimbra
Weekly hours
6
Teaching language
Português
Objectives
The main objective in Numerical Analysis is concerned with the development of methods for approximation, in an efficient manner and using arithmetic operations, solutions to mathematically expressed problems. This course is intended to familiarize the student with the commonly numerical methods used in solution of equations in one variable, interpolation and approximation, differetiation and numerical integration, numerical methods in linear Algebra and initial-value problems for ordinary differential equations. It is also very important the study of basic aspects of numerical computation, namely the question related to rounding and truncation errors and the sensivity of the solution of a ill conditioned problem to slight changes in the data and the unstable methods.
Subject matter
1.Introduction
2. Errors (complements). Stability.
Ill-conditioned problems and well-conditioned problems. Unstable methods.
3. Polynomial interpolation and cubic splines interpolation. Differences. Newton divided differences formula. Hermite interpolation. Cubic spline interpolation.
4. Least squares approximation
Least squares problems. Choice of basis functions. Orthogonal polyomials.
5. Differentiation and Integration
Numerical differentiation. Numerical . Numerical integration. Newton-Cotes formulas. Romberg integration. Gaussian methods.
6. Nonlinear Equations (complements)
Convergence results- Fixed points, local convergence and monotone convergence. Newto´s method
7. Numerical methods in Linear Algebra. Systems of linear equations (complements). Vector norms and matrix norms. Error estimates. Matrix methods. condition number of linear systems. Iterative methods. Jacobi´s method and Gauss-Seidl method. Convergence theory
Bibliography
1. Hacques, G. - Mathematiques pour l''''informatique III- Algorithmique Numerique- Armand Colin.
2. Vandergraft, J.S. - Introduction to Numerical Computations- Academic Press.
3. Forsythe, G.; Malcolm, M.A.; Moler, C.B.- Computer Methods for Mathematical Computations- Prentice-Hall.
4. Burden, R.; Faires, J.D.;Reynolds, A.C.- Numerical Analysis-Wadsworth International Student Edition.
5. Coimbra, E. - Splines Cúbicos- Notas de lições para alunos do segundo ano das licenciaturas da F.C.T.- Departamento de Matemática da F.C.T. da U.N.L.
6. Coimbra, E. - Integração Adaptativa- Notas de lições para alunos do segundo ano da Licenciatura em Matemática da F.C.T.- Departamento de Matemática da F.C.T. da U.N.L.
7. Fox, L.; Mayers, D.F.- Computing Methods for Scientists and Engineers- Clarendon Press.
8. Freitas, A.C. -Introdução à Análise Numérica, Volume I- U.L.M.
Teaching method
Theoretical subjects are presented and explained to the students in a theoretical class (3h/week). These subjects are applied by students, solving problems in the pratical classes (3h/week).
During lab classes, students are grouped in teams and develop programming assignments whose objective is to consolidate the concepts addressed in lectures. After concluding each assignment, students should discuss with the professor, the behaviour of the developed programs and the respective connection to the concepts, learned throught the course.
Evaluation method
Evaluation components:
- Two tests, testing the knowledge of both theoretical and practical concepts, or a final exam.
- Two computational projects.
The final classification (NF) is the weighted mean of the classification of the two tests (NT) (or in alternative the mark on the final exam) and the pratical mark (NP) obtained by the arithmetical mean of the classification in two programming projects.
NF=0.7NT+0.3NP