# Numerical Analysis A

## Objectives

Students must be able to apply numerical methods for mathematical problems, such as, non linear equations, approximation of functions, integration, systems of equations and ordinary differential equations. Students must also be able to implement computational algorithms in order to solve the aforementioned problems.

## General characterization

##### Code

12595

##### Credits

3.0

##### Responsible teacher

Elsa Estevão Fachadas Nunes Moreira

##### Hours

Weekly - 3

Total - 39

##### Teaching language

Português

### Prerequisites

Students must have basic knowledge in Mathematical Analysis I (AM I) and Linear Algebra and Analytic Geometry (ALGA).

### Bibliography

**1.** Atkinson, K., An Introduction to Numerical Analysis, Wiley, 1989.**2.** Burden, R.; Faires, D., Numerical Analysis (9th. Edition), Brooks-Cole Publishing, 2011.**3.** Conte, S.; Boor, C., Elementary Numerical Analysis: Analgorithmic approach, McGraw-Hill, 1981.**4.** Isaacson, E.; Keller, H., Analysis of Numerical Methods, Dover, 1994.**5.** Pina, H.; Métodos Numéricos, Escolar Editora, 2010.**6.** Santos, F. Correia dos; Duarte, Jorge; Lopes, Nuno D., Fundamentos de Análise Numérica (Com Python3 e R), Edições Sílabo, 2019 (2ª edição).

### Teaching method

`The theory is explained to students during theoretical-practical classes with full demonstration`

(or presentation of an outline of the demonstration) of the main theoretical results and illustrative

application examples displayed. In practical classes, some problems proposed in a list of exercises

are solved and commented by the teacher, also giving the students the opportunity to work on solving

them.

### Evaluation method

**Regulation**

**Knowledge Assessment**

**Numerical Analysis A**

This document regulates the process of assessing knowledge of the curricular unit (UC)

Numerical Analysis A.

In any omitted situation, the Knowledge Assessment Regulation of the Faculty of Sciences and

Technology of the Universidade Nova de Lisboa, revised on July 31, 2020, applies.

It should be noted that all students were informed of these assessment methods in the presentation

class of the respective theoretical-practical shift where they were enrolled, taught at the beginning of

October 2021.

**1. Frequency**

In this semester (odd of the academic year 2022/23) all students enrolled in the

Curricular Unit (UC), with the exception of those with a special status, are required to obtain

frequency in theoretical-practical classes, which consists of the presence at, at minus 2/3 of the

expected total number of classes in the shift in which they are enrolled.

**2. Evaluation**

All exams (exam/test/computer work) are classified from 0 to 20 values.

A student obtains approval if the final grade in the UC is greater than or equal to 9.5 values.

This classification, as well as its calculation formula, is explained in the following items.

**2.1 Period of Continuous Assessment**

The evaluation during the semester consists of two written tests lasting 1.5 hours

and a optional computational work (project evaluation), to be carried out in groups of 4 or 5

students, in the wxMaxima language.

Let NT1 and NT2 be the 1st test and 2nd test classifications respectively and NTC be the computational work classification.

If the student chooses to perform the computational work, the final grade (NF) in the UC is given by:

NF = 0.45 × NT1 + 0.45 × NT2 + 0.1 × NTC ,

`If the student chooses not to perform the computational work, the final grade (NF) in the UC is given by:`

NF = 0.5 × NT1 + 0.5 × NT2,

`a minimum classification of 9.5 in NF is required for approval in the curricular unit. `

If NF < 9.5 values, the student fails the UC (they can take the test at the time of appeal).

If NF >= 9.5 values the student obtains approval in the UC with the classification NF rounded to the

units.

**2.2 Season of Appeal**

Any student who has not yet passed the UC can take the test ( Appeal Exam).

This test lasts for 3 hours.

Let ER be the grade of the Appeal Exam, the UC final grade is given by:

NF = max{ER grade ,0.90 × ER grade + 0.10 ×TC grade}

rounded to units.

`If NF >= 9.5 the student is approved, and NF is rounded to the units.`

** 2.3 Special Season**

Assessment made in the same way as in the Appeal Period.

## 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 Direct methods: Gauss Method; pivoting strategies.

5.3 Iterative methods: general procedure.

5.4 Jacobi and Gauss-Seidel 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: