Numerical Analysis

Code:

L.EC007

Acronym:

AN

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2025/2026 - 2S

Active?	Yes
Web Page:	http://moodle.up.pt
Responsible unit:	Department of Civil and Georesources Engineering
Course/CS Responsible:	Bachelor in Civil Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EC	210	Syllabus	1	-	6	52	162

Teaching language

Portuguese

Objectives

OBJECTIVES:

Understand the limitations of numerical algorithms. Determine error estimates. Understand error propagation. Solve non-linear equations. Approximate data by interpolation and least squares. Approximate, derive, and integrate functions using numerical methods. Develop the ability to determine and analyze results obtained by scientific calculation instruments using approximate techniques. Acquire theoretical and practical knowledge of numerical techniques in solving engineering problems, namely: applying numerical techniques to model physical problems; solving engineering problems using numerical methods; implementing numerical algorithms; comparing different methods in solving numerical problems; choosing and deciding which numerical solving method to apply and which is the most efficient; evaluating and discussing the numerical results obtained; using numerical computing tools, MATLAB or Python, to determine numerical solutions

Learning outcomes and competences

SKILLS AND LEARNING OUTCOMES:
Knowledge: Know and describe the fundamental concepts and numerical methods for solving engineering problems. Identify the main concepts associated to numerical solution and determination of approximate solutions.

Understanding: Identify and interpret the different techniques to be used in numerical problem solving.

Application: Develop skills for numerical problem solving. Apply knowledge and the ability to understand and solve problems in new and unfamiliar situations, in broad and multidisciplinary contexts.

Analysis: Analyse, discuss and critically interpret results, highlighting the potential of methods and their limitations.

Synthesis: Formulate and validate numerical solutions for solving non-linear equations, linear and non-linear systems, approximation of functions and integration.

Assessment: Criticise solutions and methodologies used. Be able to communicate their conclusions and their underlying knowledge and reasoning in a clear unambiguous manner.

Working method

Presencial

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

Mathamatical Analysis 1 and 2, Algebra and Computation.

Program

1. Numerical Analysis and Scientific Computing
  1.1 Goal for numerical analysis. 
  1.2 Theory of errors: basic concepts, errors definitions and properties.
  1.3 Error propagation.
  1.4 Direct methods for solving linear systems. Doolittle LU decomposition without and with pivoting.
  1.5 Iterative methods. Order of convergence of a succession of iterates.
2. Solution of non-linear equations: iterative methods
  2.1 Bisection method
  2.2 Newton method.
  2.3 Secant method.
  2.4 Convergence conditions and stopping criteria for iterative methods.
  2.5 Error estimation.
  2.6 Iterative methods for Non Linear Systems: Newton Method
3. Approximation of functions and data
  3.1. Polynomial interpolation.
    3.1.1. Different constructions of the interpolating polynomial.
    3.1.2 Lagrange polynomials.
    3.1.3 Interpolation error.
    3.1.4 Divided differences and derivatives
  3.2. Introduction to the least squares method. Normal equations.
4. Numerical integration
  4.1. Newton-Cotes formulas.
  4.2. Simple formulae and Compose formulae. Properties
  4.3. Errors in numerical integration.

Scientific Content: 70%
Technological content: 30%

DEMONSTRATION OF THE SYLLABUS COHERENCE WITH THE CURRICULAR UNIT'S OBJECTIVES:

Understand the limitations of numerical algorithms. Determine error estimates. Understand error propagation. Solve non-linear equations. Approximate data by interpolation and least squares. Approximate, derive and integrate functions using numerical methods. Develop the ability to determine and analyze results obtained by scientific calculation instruments using approximate techniques. Acquire theoretical and practical knowledge of numerical techniques in solving engineering problems, namely: applying numerical techniques to model physical problems; solving engineering problems using numerical methods; implementing numerical algorithms; comparing different methods in solving numerical problems; choosing and deciding which numerical solving method to apply and which is the most efficient; evaluating and discussing the numerical results obtained; using numerical computing tools, MATLAB or Python, to find numerical solutions.

Mandatory literature

Quarteroni Alfio; Cálculo Científico com Matlab e Octave. ISBN: 978-88-470-0717-8 (Access to this content is enabled by Universidade do Porto)
Quarteroni, A. & Saleri, F; Scientific Computing with MATLAB and Octave, Springer, 2014. ISBN: 978-3-642-45366-3 (Access to this content is enabled by Universidade do Porto)

Complementary Bibliography

Valença, Maria Raquel; Métodos numéricos
Burden, Richard L.; Numerical analysis. ISBN: 0-534-38216-9
Pina, Heitor; Métodos numéricos. ISBN: 972-8298-04-8
Fausett, Laurene V.; Applied numerical analysis using Matlab. ISBN: 0-13-319849-9
Kharab, Abdelwahab; An introduction to numerical methods. ISBN: 1-58488-281-6
Bradie, Brian; A friendly introduction to numerical analysis. ISBN: 0-13-191171-6
Ana Maria Faustino; Análise Numérica, FEUP, 2014. ISBN: 978-989-98632-5-5

Teaching methods and learning activities

DEMONSTRATION OF THE COHERENCE BETWEEN THE TEACHING METHODOLOGIES AND THE LEARNING OUTCOMES:

The concepts and techniques are presented using knowledge of Analysis and Algebra, and, whenever possible, the theoretical exposition is accompanied by practical examples, graphical representations, and the use of scientific computing. The theoretical aspects are presented rigorously to demonstrate the applicability of the methods. In addition, justified comments are made comparing the various methods in terms of efficiency, accuracy of the results obtained, and applicability. Students are encouraged to design, implement, and apply algorithms. Special attention is paid to the analysis of results, with a focus on error estimates. Analysis and discussion of the solutions and methodologies used are valued. In practical classes, various case studies are solved using the computer and the MATLAB (or Python) programming language. In order to facilitate dialogue between all participants in the teaching/learning process, the course is hosted on the Moodle platform. On this page, students have access to all the content provided by the teachers and can consolidate concepts through self-assessment tests with immediate, commented evaluation. They can also use the forums to ask questions, which can be viewed by the entire Numerical Analysis community. In addition, explanatory videos and problem-solving demonstrations using Artificial Intelligence tools are used, promoting more interactive learning and keeping up to date with emerging technologies. Theoretical and theoretical-practical classes offer activities that encourage students to study independently. They are also encouraged to apply their knowledge and ability to understand and solve problems in new situations, in broad and multidisciplinary contexts.

Software

Jupyter Notebook
Octave
Matlab
Python

keywords

Physical sciences > Mathematics > Applied mathematics > Numerical analysis

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Designation	Weight (%)
Exame	70,00
Teste	30,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	110,00
Frequência das aulas	52,00
Total:	162,00

Eligibility for exams

According to the regulations and directives of the L.EC Director:

Approval of the curricular unit implies compliance with the attendance condition, with it being considered that a student meets this condition if, having been regularly enrolled, they do not exceed the maximum number of absences corresponding to 25% of the scheduled in-person classes for each type. In addition to the cases provided for by statute in the rules in force at FEUP, students who obtained a final grade in the UC equal to or greater than 6 points in the immediately preceding academic year are exempt from the attendance requirement for the curricular unit.

Calculation formula of final grade

The formula of calculation of the final classification for grades higher or equal to 6 in the Final Examination is:

CF = maximum { EF; 0.7xEF + 0.2xTS + 0.1xQZ}

where,

EF - final exam classification, onsite Test 
TS - grade of the Summative, onsite Test
QZ - Average of the marks in 3 online activities (quizzes)

For exam grades lower than 6, the final grade is the exam grade EF.


The classification of the distributed assessment obtained in previous years is not valid.

Special assessment (TE, DA, ...)

Final exam.

Classification improvement

Final exam.

Observations

The assessments for this course are dane on computers in the Moodle environment. The use of calculators is not permitted.

According to the regulations and directives of the L.EC Director:

During any assessment period, possession of any electronic devices (e.g., mobile phones, tablets, headphones, smartwatches, etc.) is strictly prohibited.
It is the responsibility of the student to anticipate this situation before the start of the assessment period.