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Computational Mathematics

Code: M3010     Acronym: M3010     Level: 300

Classification Keyword
OFICIAL Mathematics

Instance: 2019/2020 - 1S Ícone do Moodle

Active? Yes
Responsible unit: Department of Mathematics
Course/CS Responsible: First Degree in Mathematics

Cycles of Study/Courses

Acronym No. of Students Study Plan Curricular Years Credits UCN Credits ECTS Contact hours Total Time
L:B 0 study plan from 2016/17 3 - 6 56 162
L:CC 1 Plano de estudos a partir de 2014 2 - 6 56 162
L:F 2 study plan from 2017/18 2 - 6 56 162
L:G 0 study plan from 2017/18 2 - 6 56 162
L:M 31 Plano estudos a partir do ano letivo 2016/17 2 - 6 56 162
L:Q 0 study plan from 2016/17 3 - 6 56 162

Teaching Staff - Responsibilities

Teacher Responsibility
Manuel Augusto Fernandes Delgado
Maria Zélia Ramos Alves da Rocha

Teaching - Hours

Theoretical and practical : 4,00
Type Teacher Classes Hour
Theoretical and practical Totals 1 4,00
Manuel Augusto Fernandes Delgado 2,00
Maria Zélia Ramos Alves da Rocha 2,00

Teaching language

Suitable for English-speaking students


Computational Algebra module:
To introduce basic concepts of Computational Algebra, along with Gröbner basis.

Numerical Linear Algebra Module:
Study constructive methods of numerical resolution of the following problems of Linear Algebra: systems of equations, inverse of matrices and determinants, focusing on the aspects of conditioning and stability, convergence, error control, construction of algorithms, implementation and experimentation in computer in the MATLAB language and processing of study cases.

Learning outcomes and competences

Computational Algebra module:

Students should acquire knowledge  on some basic concepts of Computational Algebra, as well as to have contact with Gröbner basis.

Numerical Linear Algebra module:

Students should acquire the knowledge of the fundamental methods of Numerical Linear Algebra in their theoretical, practical, computational and experimental aspects. 

Working method


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

Computational Algebra module:
It is expected that the student has good knowledge of abstract algebra. In particular the student should know the division algorithm for polynomials in one variable, the Euclidean algorithm and how to calculate the greatest common divisor of two polynomials in one variable.

Numerical Linear Algebra Module:
Fundamental notions of Linear Algebra.
Basic notions of any programming language.


Computational Algebra module:

  • Motivation: affine varieties and polynomial ideals.

  • Gröbner bases: polynomial ideals, monomial orders and multivariate division with remainder, monomial ideals and Hilbert basis theorem, Gröbner bases and S-polynomials, Buchberger's algorithm.

Numerical Linear Algebra Module:

  • Introduction to MATLAB

  • The MATLAB environment. Random, Hilbert and Pascal matrices, the command gallery. Linear algebra: norms, condition numbers, the operator \, Gauss and Cholesky factorizations, the lu and chol commands. Programming. Graphs.

  • Numerical resolution of linear systems, inverse of matrices and determinants: vector and matrix norms, matrix series, conditioning, condition numbers, triangular systems and inverses, direct methods of Gauss and Cholesky; iterative methods of Jacobi and Gauss-Seidel.

Mandatory literature

Pina Heitor; Métodos numéricos. ISBN: 978-972-592-284-2
Cox David; Ideals, varieties, and algorithms. ISBN: 0-387-97847-X ((4th edition))

Complementary Bibliography

Brezinski Claude; Méthodes numériques itératives. ISBN: 978-2-7298-2887-5
Brezinski Claude; Méthodes numériques directes de l.algèbre matricielle. ISBN: 2-7298-2246-1
Gathen Joachim von zur; Modern computer algebra. ISBN: 0-521-82646-2

Teaching methods and learning activities

Computational Algebra modue:

The course material and examples will be presented by the teacher. Some time is to be reserved for the resolution of exercises by the students with the advice of the teacher.

Numerical Linear Algebra module:
In the theoretic-practical classes are presented the contents of the syllabus with illustrative examples followed by the resolution of theoretical, practical and computational exercises implemented in the MATLAB language. 




Physical sciences > Mathematics > Algorithms
Physical sciences > Mathematics > Computational mathematics

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation Weight (%)
Exame 50,00
Teste 50,00
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

Course registration.

Calculation formula of final grade

Regular Exam: arithmetic mean between the grades obtained in the test at the end of the Computational Algebra Module and the grade obtained in the (theoretical-practical computer based) test at the enf of the Numerical Linear Algebra Module.

Make-up Exam: arithmetic mean between the grades of the Computational Algebra Module and the Numerical Linear Algebra Module,  which will be ontained through an exam and where any of the two parts can be substituted by the corresponding test.

Special assessment (TE, DA, ...)

Computational Algebra/Geometry module: exam.

Numerical Linear Algebra module: exam.

Classification improvement

Computational Algebra module: exam.

Numerical Linear Algebra module: exam.
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