# Numerical Analysis

## Objectives

We will illustrate several numerical methods for the computer solution of certain classes of mathematical problems. We will show how to use these methods in order to solve nonlinear equations, linear systems, integrate and construct accurate approximations for the solution of differential equations.

## General characterization

10478

3.0

##### Responsible teacher

Magda Stela de Jesus Rebelo

Weekly - 3

Total - 39

Português

### Prerequisites

Basic knowledge in analysis and linear algebra

• ## Pina H., Métodos Numéricos, Mc Graw Hill, 1995

### Teaching method

The course works with theoretical-practical classes (TP), in which the successive topics of the chair program will be explained and discussed. In order to consolidate a given subject, exercises related to each of the topics covered will be solved. In order to implement some of the methods covered, TP classes will be taught in the Lab where there are computers with wxMaxima software installed.

Students must attend, at least 2/3 of TPs classes or attend the previous year.

The evaluation of the course consists of two tests that address the knowledge acquired in TPs classes and / or a computational work to be prepared in group.

### Evaluation method

Important:

In order to be evaluated, the student must attend at least 2/3 of the TP classes.

Evaluation Methods:

1-Continuous evaluation

The continuous evaluation consists of two tests during the semester and or a computational work.

•   If the student chooses the evaluation which consists of two test and a computational work the final grade, FG, is given by

FG ​​= 0.35 × NT1 + 0.55 × NT2 + 0.1 × NTC,

where NT1, NT2 are the grades of the Test 1 and Test 2, respectively and NTC is the garde of the computational work.

•   If the student chooses the evaluation which consists of two test, the final grade, FG, is given by

FG ​​= 0.4 × NT1 + 0.6 × NT2,

where NT1, NT2 are the grades of the Test 1 and Test 2, respectively.

The student is approved if the final grade is greater than or equal to 9.5.

The first test lasts for 1h30m and the second test lasts for 2h ..

2-Final exam evaluation.

The student is approved if the grade of the final exam, FE, is greater than or equal to 9.5 and the final grade is given by
FG = max {FE, 0.9xFE + 0.1xNTC}

The final exam lasts for 3h.

## Subject matter

1.Introduction

1.1 Errors, significant digits.

1.2 Conditioning of a problem and stability of a method.

1.3 Introduction to a computational program for Numerical Analysis.

2. Polynomial approximation and interpolation

2.1 Interpolation and Lagrange polynomial

2.2 Divided differences, interpolating polynomial of Newton.

2.3 Cubic Spline interpolation.

2.4 Least squares approximation.

3. Numerical integration

3. 1 Newton-Cotes integration formulas (Single and composite rules)

3.2 Gaussian integration. Other integration methods.

4. Root finding for nonlinear equations

4.1 Bisection method.

4.2 Fixed-point iteration method. Newton method. Secant method.

5. Iterative methods for solving linear systems of equations

5.1 Norms of vectors and matrices. Conditioning of a system.

5.2 Eigenvalues and eigenvectors. Gershgorin theorem.

5.3 Iterative methods: general procedure.

5.4 Jacobi, Gauss-Seidel and relaxation methods.

6. Numerical solution of ordinary differential equations

6.1 Euler methods.

6.2 Taylor methods.

6.3 Runge-Kutta methods.

## Programs

Programs where the course is taught: