2025
Numerical Analysis I
Name: Numerical Analysis I
Code: MAT14224L
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Regime de Frequência: Presencial
Presentation
Numerical analysis refers to a series of computational tools for solving mathematical problems, the exact solution of which is difficult or even impossible from an analytical point of view. The CU seeks to introduce students to the most important topics of the theory of numerical methods.
Sustainable Development Goals
Learning Goals
Introduction to the theory of numerical methods with an emphasis on applications.In addition to learning theoretical results that allow them to develop the capacity for generalization and mathematical abstraction, it is intended to familiarize students with the fundamentals of the development of computational tools, which, in turn, will allow them to efficiently formulate and solve practical problems.
Contents
- Floating point systems, errors, conditioning, convergence, stability.
- Nonlinear equations. Methods of bisection, Newton, fixed point iteration.
- Systems of linear equations. Direct methods: Gauss elimination, triangular factorizations, inverse matrix and determinant calculation. Matrix norms and condition numbers. Iterative methods: Jacobi, Gauss-Seidel, conjugate gradients.
- Systems of nonlinear equations.
- Interpolation and approximation of functions. Lagrange, Newton, Hermite polynomials, Chebyshev interpolation, splines, least squares approximation.
- Numerical derivation and integration. First and second order derivatives. Newton-Cotes quadrature rules, composite rules.
- Eigenvalues and eigenvectors. Localization and calculation.
- Implementation of some algorithms in an interactive numerical and symbolic calculation system.
- Nonlinear equations. Methods of bisection, Newton, fixed point iteration.
- Systems of linear equations. Direct methods: Gauss elimination, triangular factorizations, inverse matrix and determinant calculation. Matrix norms and condition numbers. Iterative methods: Jacobi, Gauss-Seidel, conjugate gradients.
- Systems of nonlinear equations.
- Interpolation and approximation of functions. Lagrange, Newton, Hermite polynomials, Chebyshev interpolation, splines, least squares approximation.
- Numerical derivation and integration. First and second order derivatives. Newton-Cotes quadrature rules, composite rules.
- Eigenvalues and eigenvectors. Localization and calculation.
- Implementation of some algorithms in an interactive numerical and symbolic calculation system.
Teaching Methods
The teaching process will be organized based on theoretical and practical-laboratory sessions. Theoretical sessions are predominantly given on the board and with the projection of resources with JupytherLab. Theoretical concepts are illustrated by practical examples. In practical-laboratory classes, the active use of computational resources and the implementation of the most important numerical algorithms are foreseen with Python language through JupytherLab.
Assessment
There will be two in-class assessments (F1 and F2) and examinations (regular, resit, special and extraordinary).
In addition to the in-class assessments and examinations, a computing project will be carried out in the form of an individual Practical Laboratory Test (PLT) completed on a computer in the classroom.
The PLT will be held on a single occasion and the mark obtained will be taken into account in any assessment scheme.
For the continuous assessment scheme, the marks from the first and second assessments and the PLT will be taken into account, with the final mark calculated according to the formula:
(2xF1 + 2xF2 + TPL)/5 (*)
In none of the assessments may the mark be less than 7.5 marks, and the average must be greater than or equal to 9.5 marks.
To pass under the continuous assessment system, the mark obtained using formula (*),
rounded to the nearest whole number, must be equal to or greater than 10 marks.
The mark for any examination (Ex) will be calculated using the formula,
(4 × Ex + PLT) / 5 (**)
where Ex must be equal to or greater than 9.5.
Formula (**) will apply to all types of examination: regular, resit, special or extraordinary.
A student will pass the course unit when the marks obtained using formulas (*) or (**), rounded to the nearest whole number, are 10 or above.
In addition to the in-class assessments and examinations, a computing project will be carried out in the form of an individual Practical Laboratory Test (PLT) completed on a computer in the classroom.
The PLT will be held on a single occasion and the mark obtained will be taken into account in any assessment scheme.
For the continuous assessment scheme, the marks from the first and second assessments and the PLT will be taken into account, with the final mark calculated according to the formula:
(2xF1 + 2xF2 + TPL)/5 (*)
In none of the assessments may the mark be less than 7.5 marks, and the average must be greater than or equal to 9.5 marks.
To pass under the continuous assessment system, the mark obtained using formula (*),
rounded to the nearest whole number, must be equal to or greater than 10 marks.
The mark for any examination (Ex) will be calculated using the formula,
(4 × Ex + PLT) / 5 (**)
where Ex must be equal to or greater than 9.5.
Formula (**) will apply to all types of examination: regular, resit, special or extraordinary.
A student will pass the course unit when the marks obtained using formulas (*) or (**), rounded to the nearest whole number, are 10 or above.
Teaching Staff
- Marília da Conceição Valente Oliveira Pires [responsible]
Accreditations and Certifications
