# Numerical Analysis I

### Code

7807

### Academic unit

Faculdade de Ciências e Tecnologia

### Department

Departamento de Matemática

### Credits

7.0

### 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.

### Prerequisites

A background in linear algebra, calculus, and numerical methods is sufficient.

### 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.

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### 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

The main objective of the lecture courses is the exposition of the basic concepts, fundamental principles and methods of Numerical Analysis and its applications. At every stage we introduce illustrative examples and we present the detailed proofs of the essential theorems. In order to motivate the students some questions are previously introduced and these questions are discussed in the subsequent lectures. The references have been chosen on the basis of being the most generally available sources. The practical hours are devoted the the solution of selected problems. During the practical hours the students are organized in groups of two or three members and these groups implement and run some computer programs. The teacher reserve some extra hours, intended to help the students in better understanding the theoretial subjects and their applications.

### Evaluation method

The final mark is the weightedmean of the written final examination (80%) and the practical mark (20%). The practical is the arithmetical mean of the mark obtained in the computer programs