Numerical Methods with Python Programming
3rd Edition
This Short Advanced Programme (SAP) is designed to introduce undergraduate and postgraduate students to fundamental numerical methods. The programme is intended for students with basic Python programming skills and a foundational understanding of differentiation and integration. Participants should also be familiar with using a computer, including editing files and downloading and installing software.
The SAP aims to provide students with a solid foundation in elementary numerical analysis, enabling them to develop general computational skills for solving numerical problems. In addition, students will gain practical knowledge and techniques applicable to engineering and scientific applications.
The programme uses the Python ecosystem provided through the Anaconda distribution (www.anaconda.com). Practical sessions will be delivered using Jupyter Notebook or Google Colab. Students who plan to use their own laptops during the classes are expected to have the Anaconda distribution installed before the programme begins.
Jointly coordinated by the Polytechnic University of Cávado and Ave (IPCA), the Polytechnic University of Leiria (IPLeiria), and the Technological University of the Shannon (TUS), the Short Advanced Programme (SAP) Numerical Methods with Python Programming, 3rd Edition introduces undergraduate and postgraduate students to the fundamentals of numerical methods. The programme is intended for students with basic Python programming skills and a foundational knowledge of differentiation and integration.
The programme provides students with a solid grounding in elementary numerical analysis while developing practical knowledge and techniques applicable to engineering and scientific problems.
Teaching is based on the Python ecosystem through the Anaconda distribution and is supported by Jupyter Notebook and Google Colab, enabling students to develop and test numerical methods in an interactive computing environment.
Build the analytical and digital skills needed to tackle real-world challenges through numerical methods, Python programming, and international collaboration.
15th of February 2027 to 21st of May 2027
English
Academic recognition will be ensured by each home institution. The SAP (3 ECTS) may be recognised as part of the study programme and recorded in the Diploma Supplement.
All RUN-EU 2.0 Undergraduate and Postgraduate Students
Belgium > Portugal (Barcelos): 862 EUR
Spain > Portugal (Barcelos): 764 EUR
Austria > Portugal (Barcelos): 862 EUR
Ireland > Portugal (Barcelos): 862 EUR
Finland > Portugal (Barcelos): 1133 EUR
Portugal (Leiria) > Portugal (Barcelos): 581 EUR
The Netherlands > Portugal (Barcelos): 862 EUR
Romania > Portugal (Barcelos): 948 EUR
Czech Republic > Portugal (Barcelos): 948 EUR
Programme at a glance
In this welcome and opening session, the participants will do their individual presentation.
Afterwards professors will present the objectives, program, and schedule of the course.
In this lecture we will start reviewing Python data types, functions, and control structures.
During the lecture several examples will be presented, and students will have the opportunity to solve programming exercises related to these topics.
In this lecture we will continue overviewing Python programming language topics such as lists, tuples, and traditional Python modules, namely, the modules math, random and numpy.
During the lecture several examples will be presented, and students will have the opportunity to solve programming exercises related to these topics.
In this lecture we will finish our overview of Python programming language with dictionaries and graphics. During the lecture several examples will be presented, and students will have the opportunity to solve programming exercises related to these topics.
The lecture will start with an introduction to the topic of nonlinear equations and their solution by numerical methods.
After learning how to locate roots graphically, two numerical methods to find numerical solutions of a nonlinear equation in one unknown are presented: the bisection method and the fixed-point method.
Examples will be given for each method, and students will have the opportunity to solve some exercises.
The lecture will start with an overview of the previous lecture and afterwards, the Newton-Raphson method for solving nonlinear equations in one unknown will be presented.
Some examples will be given, and students will have the opportunity to solve exercises by this topic.
The lecture will continue with the implementation of the algorithms corresponding to the numerical methods studied using Python and their use in practical applications.
In this lesson we will continue to explore the methods we have studied and implemented using practical applications.
In this lecture we will extend the Python implementations of the numerical methods studied and investigate additional practical applications.
The lecture will start with an introduction to polynomial interpolation. Afterwards, the undetermined coefficients method, the Lagrange interpolation method, divided differences, and the Newton interpolation method based on divided differences will be presented. Examples of the application of these methods will be studied, and students will have the opportunity to solve exercises related to these topics.
In this lesson we will continue to explore the methods we have studied and implemented using practical applications.
The lecture will start with the presentation of the theoretical concepts of inverse interpolation and Hermite interpolation, together with illustrative examples.
Students will work on the solution of exercises involving practical applications.
The lecture will conclude with Python implementations of these methods and their application to practical problems.
In this lesson we will continue to explore the methods we have studied and implemented using practial applications.
The lecture will start with an introduction to function approximation and discrete least squares. Afterwards, we will study linear and polynomial models, as well as linearization techniques for exponential and power models. Finally, more general linear models will be presented.
Examples of the application of these methods will be studied, and students will have the opportunity to solve exercises involving practical applications.
In this lecture we will develop Python programming implementations of general models. The implemented scripts are applied to solve practical applications.
The lecture will start with an overview of the previous lecture, and we will continue implementing the discrete least-squares methods studied in Python.
Afterwards, we will study the theoretical concepts and examples of multiple linear regression.
Students will have the opportunity to solve application exercises on these topics.
The lecture will conclude with the implementation of these numerical techniques in Python and their application to practical problems.
In this lecture we will continue developing Python programming implementations of the numerical methods studied and of their use in practical applications.
The aim of this lecture is to present some numerical methods for approximating definite integrals.
The lecture will start with an introduction to the topic of numerical integration, followed by the presentation of the trapezoidal, Simpson and Gauss-Legendre rules, illustrated through examples.
The students will have the opportunity to apply these concepts by solving the exercises provided.
During this team work session, students will continue working on the implementation in Python of the trapezoidal, Simpson, and Gauss–Legendre rules and applying them to practical problems.
In this lecture, we will continue the study of numerical methods for numerical integration.
Students will have the opportunity to apply these concepts by solving the exercises provided.
In this teamwork session, students will continue working on the implementation in Python of the numerical integration methods studied and exploring their use in practical applications.
The aim of this lecture is to numerical methods for approximating the solution of initial-value problems for ordinary differential equations.
Some theoretical concepts on this topic will be presented, together with the Euler method and the Runge-Kutta methods.
We will study examples and solve exercises involving the application of these numerical methods.
In this teamwork session, students will continue working on the implementation in Python of the Euler and Runge–Kutta methods and exploring their use in practical applications. Different ways of presenting the numerical solutions will be explored, and the results obtained will be discussed.
This session is devoted to practical work with examples and exercises using all the topics covered in the previous lectures.
This teamwork session is devoted to practical work involving examples and exercises covering all the topics studied in the previous lectures.
In this closing session, participants will have the opportunity to provide an individual evaluation of the SAP lectures and contents, and to identify the positive and negative aspects of the SAP.
Afterwards, the professors will present the Face-to-Face component (“Erasmus Week”), which will take place at IPCA in Barcelos from 17 to 21 May 2027.
Monday:
9:00 A.M. - 10:30 A.M. - Team Building: Opening Session Presentation
10:30 A.M. - 12:00 A.M. - Team Building: Defining Groups
2:00 P.M. - 5:00 P.M. - Team Work. Coaching: Project development and python programming with coaching support.
Tuesday:
9:00 A.M. - 12:00 A.M. - Team Work. Coaching: Project development and python programming with coaching support.
2:00 P.M. - 5:00 P.M. - Team Work. Coaching: Project development and python programming with coaching support.
Wednesday:
9:00 A.M. - 12:00 A.M. - Team Work. Coaching: Project development and python programming with coaching support.
2:00 P.M. – 3:30 P.M. - Team Work. Coaching: Preparation of Presentation
3:30P.M. – 5:00 P.M.- Team Work. Coaching: Project Presentation
Thursday:
9:00 A.M. – 12:00 A.M. - Enterprise Visits
2:00 P.M. – 5:00 P.M.- Cultural Visits
Friday:
9:00 A.M. – 10:00 A.M. - Group Learning Reflection
10:00 A.M. – 12:00 A.M. - Project Presentation
Learning outcomes
By the end of this Short Advanced Programme, learners will be able to:
Python Reviewing:
- Data types
- Control structures
- Functions and modules
- Data visualisation and graphics
- Working with Python libraries and modules
- Practical examples and exercises
Solutions of Equations in One Variable:
- Graphical methods for root finding
- Bisection method
- Fixed-point iteration
- Newton–Raphson method
- Practical examples and applied problems
Polynomial Interpolation:
- Method of undetermined coefficients
- Lagrange interpolation
- Divided differences
- Newton interpolation
- Inverse interpolation
- Hermite interpolation
- Practical examples and applied problems
Discrete Least Squares Approximation:
- Linear and polynomial regression models
- Linearisation techniques
- Exponential and power models
- General least squares models
- Multiple linear regression
- Practical examples and applied problems
Numerical Integration:
- Trapezoidal rule
- Simpson's rule
- Gauss–Legendre rule
- Practical examples and applied problems
Ordinary Differential Equations:
- Initial-value problems
- Euler's method
- Runge–Kutta methods
- Practical examples and applied problems
Selection criteria
1. Motivation: Quality of the applicant's motivation for participating in the programme and its expected impact on their academic or professional development.
2. Academic Background: Relevance of the applicant's academic background and current programme of study to the objectives of the SAP.
3. Academic and Scientific Interests: Alignment of the applicant's academic, research or professional interests with the topics covered by the programme.
4. Representation of Subject Areas: Preference will be given to achieving a broad representation of academic disciplines and fields of study.
5. Balanced Institutional Participation: Consideration will be given to achieving balanced participation among the RUN-EU member institutions.
Involved organisations and persons
Polytechnic University of Cávado and Ave
Lead Organisation, Host Organisation- Teresa Abreu (Lead Instructor)
Technological University of the Shannon
Partner Organisation- Sajjad Sajjadi (Instructor)
University of Leiria and Oeste
Partner Organisation- Ana Cristina Lemos (Instructor)
- Carlos Campos (Instructor)
Funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Education and Culture Executive Agency (EACEA). Neither the European Union nor EACEA can be held responsible for them. Grant Agreement Number: 101124674
