Numerical Analysis

Code:

L.EGI007

Acronym:

A N

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 2S

Active?	Yes
Web Page:	https://sigarra.up.pt/feup/pt/ucurr_geral.ficha_uc_view?pv_ocorrencia_id=538791
Responsible unit:	Mathematics Section
Course/CS Responsible:	Bachelor in Industrial Engineering and Management

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EGI	121	Syllabus	1	-	4,5	39	121,5

Teaching language

Portuguese

Objectives

General Objectives:
Understand the most applicable and efficient numerical resolution methods for each fundamental problem in Numerical Analysis, as well as the conditions for their applicability and the corresponding convergence theorems. Students are expected to conduct practical application tests on a computer, analyze the obtained results, and gain hands-on experience in numerical programming by implementing some of these methods in MATLAB.

Specific Objectives:
For each chapter of the program, students should be able to:

• Identify the applicability conditions of the methods and state the corresponding convergence theorems.
• Apply the given methods, formulas, and algorithms to simple concrete problems.
• Describe how the methods work, translate them into algorithms and subprograms (functions) in MATLAB, and test them using examples, comparing and analyzing the results.
• Explain the proofs of the given theorems and apply the described techniques to related situations.
• Solve new problems using the provided numerical tools and compare the performance of various numerical methods in terms of speed and reliability.

It is important to highlight that students must master MATLAB's language, but even more crucially, they should be proficient in writing pseudocode (algotithmic language), which serves as the foundation of any programming language.

Remark: during tests and exams, students will not have access to a computer, so they must have a strong command of MATLAB and algorithmic language (pseudocode). For assignments, students are required to use MATLAB.

Learning outcomes and competences

For each chapter of the program, students should be able to:

• Identify the applicability conditions of the methods and state the corresponding convergence theorems.
• Apply the given methods, formulas, and algorithms to simple concrete problems.
• Describe how the methods work, translate them into algorithms and subprograms (functions) in MATLAB, and test them using examples, comparing and analyzing the results.
• Explain the proofs of the given theorems and apply the described techniques to related situations.
• Solve new problems using the provided numerical tools and compare the performance of various numerical methods in terms of speed and reliability.

It is important to highlight that students must master MATLAB's language, but even more crucially, they should be proficient in writing pseudocode (algotithmic language), which serves as the foundation of any programming language.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Students are expected to be familiar with the topics covered in Linear Algebra, Mathematical Analysis I and II, and Computer Programming. It is assumed that they already have a solid understanding of the MATLAB software.


MATLAB instalation procedure:
https://www.up.pt/portal/pt/updigital/software/comunidade/

Program

Chapter 1 – Rounding Errors and Their Propagation: potential instability of numerical methods; origin of rounding errors; number systems in computers: fixed-point and floating-point systems.

Chapter 2 – Nonlinear Equations: general conditions for solving; stopping criteria for iterative methods; calculation methods: bisection, Newton, secant, simple iterative method (fixed point). Convergence theorems; estimation and bounding of truncation errors; order of convergence.

Chapter 3 – Systems of Equations: iterative methods for solving systems of equations. Jacobi Method, Gauss-Seidel Method, Newton’s Method. Convergence conditions.

Chapter 4 – Function Approximation: interpolating polynomial; Lagrange interpolation formula; Newton interpolation formula (divided differences); least squares.

Chapter 5 – Numerical Integration: Newton-Cotes formulas (e.g., Trapezoidal and Simpson's rules); composite formulas; numerical integration errors.

Chapter 6 – Initial Value Problems: Existence and uniqueness of the solution; numerical methods; convergence.

Mandatory literature

Maria Raquel Valença ; Análise Numérica, Universidade Aberta
Burden Richard L.; Numerical analysis. ISBN: 0-534-38216-9

Complementary Bibliography

Heitor Pina ; Métodos numéricos , McGraw Hill , 1995

Teaching methods and learning activities

The theoretical presentations, given on the blackboard, in PowerPoint presentations, or in videos, are strongly based on Analysis and Algebra, always accompanied, when possible, by practical examples that serve as motivation. Other examples are also suggested for testing on a computer to observe their practical behavior, which is then explained in light of the theory learned.

In practical classes, students carry out exercises on paper and, in some classes held in rooms equipped with computers and appropriate software, they implement the algorithms learned in theoretical lessons.

Outside of class, students will develop medium-complexity programming projects in Matlab (two practical assignments, which together account for 10% of the final grade). It is worth highlighting that students have 106 hours of independent study, which should be considered in their study planning. Relying exclusively on solving past-year tests is not a recommended practice on its own.

Software

Matlab

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation	Weight (%)
Participação presencial	0,00
Teste	90,00
Trabalho escrito	10,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	106,00
Frequência das aulas	56,00
Total:	162,00

Eligibility for exams

Attendance in 75% of practical classes

Calculation formula of final grade

Calculation Formula - final grade - continuous assessment:
45% of the first test + 45% of the second test + 10% (two assignments to be completed throughout the semester)

Resit: In the resit exam, students who have not yet passed may only take a global test, which is worth 100%.

Grade Improvement: Students who have passed can improve their grade in the resit exam by taking a final exam covering all the material, which is worth 100%. The maximum grade of 20 will only be awarded after an oral exam.

Important Note: During continuous assessment, students must achieve a minimum grade of 7 in each test. Otherwise, they will have to take a global exam during the resit period.

If students have, for example, a minimum grade of 7 or higher in the first test, and a grade lower than 7 in the second test, but the average of the two tests is 9.5 or higher, they will not be approved in the course and will need to take the final exam in the resit period, which is worth 100%.

Examinations or Special Assignments

Throughout the semester, students will have to complete two practical assignments involving MATLAB programming. The first assignment will be carried out at the beginning of the semester, while the second assignment will be done midway through the semester. Students will have to defend their work orally.

These two assignments are worth 10% of the final grade.

The first assignment is worth 2.5% (0.5 points out of 20)
The second assignment is worth 7.5% (1.5 points out of 20)

Special assessment (TE, DA, ...)

Only final exam.

Classification improvement

Students who have passed can improve their grade in the makeup exam only in a final exam covering the entire subject matter worth 100%. The maximum score of 20 will be awarded only through an oral exam.

Observations

The material used in the course will be made available in the course content (menu on the right). Students should also consult the recommended bibliography.