Computational Mathematics

Code:

M3010

Acronym:

M3010

Level:

300

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2018/2019 - 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	2	Plano de estudos a partir de 2014	2	-	6	56	162
			3
L:F	1	Official Study Plan	2	-	6	56	162
			3
L:G	1	study plan from 2017/18	3	-	6	56	162
L:M	48	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 (1st part):
Introduction to basic concepts of Computational Algebra, in particular to Gröbner basis and the arithmetic of multivariate polynomials.

Learning outcomes and competences

Computational Algebra module (1st part):



Numerical Linear Algebra module (2nd part):

Working method

Presencial

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

Computational Algebra module (1st part):


Numerical Linear Algebra Module (2nd part):
Fundamental notions of Linear Algebra.
Basic notions of any programming language.

Program

Computational Algebra module (1st part):

• 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 (2nd part):

• Python environments. Random, Hilbert and Pascal matrices. Linear algebra: norms, condition numbers,  Gauss and Cholesky factorizations. 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 module (1st part):



Numerical Linear Algebra module (2nd part):

Software

Python

keywords

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

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Teste	85,00
Trabalho prático ou de projeto	15,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	106,00
Frequência das aulas	52,00
Total:	158,00

Eligibility for exams

Course registration.

Calculation formula of final grade

Final classification = t1 + t2 + tc1 + tc2
t1 = 1st test score quoted to 10 (1st part)
t2 = 2st test score quoted to 6 (2nd part)
tc1 = 1st computational work quoted to 1,5
tc1 = 2st computational work quoted to 1,5

NOTE: tc1 and tc2 are obtained during class time.

SECOND SEASON EXAM:
Final classification = er1 + er2 + tc1 + tc2
er1 = 1st exam score quoted to 10
er2 = 1st exam score quoted to 7
tc1, tc2 = tc1 and tc2 are obtained during class time.

(1) The second season exam consists of two parts corresponding to the division of matter for the tests.

(2) In the second season exam, the student can choose one or two of its parts. If he/she submits it for correction, it will replace the corresponding classification(s) obtained in the test(s).

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.

Observations

Contact:
Christian Lomp
FCUP-DM,
Office FC1 369,
Email: clomp@fc.up.pt