# 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

12595

3.0

##### Responsible teacher

António Manuel Morais Fernandes de Oliveira

Weekly - 3

Total - 39

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 solvingthem.`

### Evaluation method

`RegulationKnowledge AssessmentNumerical 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 2021/22) all students enrolled in the Curricular Unit (UC), with the exception of those with a special status, are required to obtainfrequency in theoretical-practical classes, which consists of the presence at, at minus 2/3 of theexpected total number of classes in the shift in which they are enrolled. 2. EvaluationAll 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 AssessmentThe evaluation during the semester consists of a single written test (single test) lasting three hoursand a mandatory computational work (project evaluation), to be carried out in groups of 4 or 5 students, in the wxMaxima language (or alternatively Maple). Let NT be the test classification (single test) and NTC be the computational work classification. The final grade (NF) in the UC is given by: NF = 0.85 × NT + 0.15 × NTC , a minimum classification of 9.5 in NT 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 theunits. 2.2 Season of AppealAny student who has not yet passed the UC can take the test (Resource Exam).This test lasts for 3 hours.If the classification of this test, NER, is lower than 9.5, the student fails.If NER ≥ 9.5, the final grade, NF, will be given by: NF = 0.85 × NER + 0.15 × NTC,rounded to units. 2.3 Special SeasonAssessment made in the same way as in the Appeal Period.`

## Subject matter

1. Errors

Absolute error,  relative error, significant digits. Condition number. Numerical algorithms stability. Introduction to a CAS (Computer Algebra System) code, suited to solve numerical problems arising in subsequent items.

2. Polynomial approximation and interpolation

Polynomial interpolation: Lagrange and Newton formulae, cubic spline interpolation.

Least squares approximation.

3. Numerical integration

Simple and composite Newton-Cotes integration formulas, Gaussian integration.

4. Rootfinding for nonlinear equations

Bissection method,  fixed-point iteration, Newton method, secant method.

5. Linear systems

Vector norms and induced matrix norms.

Eigenvectors and eigenvalues. Gershgorin theorem.

Iterative methods: general procedure, Jacobi method, Gauss-Seidel method, SOR method.

6. Numerical solution of ODE''s

Euler methods, Taylor methods for higher orders, Runge-Kutta methods.

## Programs

Programs where the course is taught: