Computational Mathematics

Code:

M3010

Acronym:

M3010

Level:

300

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2017/2018 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor 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	Official Study Plan	3	-	6	56	162
L:CC	0	Plano de estudos a partir de 2014	2	-	6	56	162
			3
L:F	0	Official Study Plan	2	-	6	56	162
			3
L:G	1	study plan from 2017/18	3	-	6	56	162
L:M	34	Official Study Plan	2	-	6	56	162
			3
L:Q	0	study plan from 2016/17	3	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

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

Learning outcomes and competences

Computational Algebra module:



Numerical Linear Algebra module:

Working method

Presencial

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

Computational Algebra module:


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

Program

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:



Numerical Linear Algebra module:

Software

MatLab
SageMath

keywords

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

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.