Numerical Analysis

Code:

L.EMAT011

Acronym:

A N

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2021/2022 - 1S

Active?	Yes
Web Page:	http://consultoriodigitalmatematica.pt.vu/
Responsible unit:	Department of Metallurgical and Materials Engineering
Course/CS Responsible:	Bachelor in Materials Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EMAT	37	Syllabus	2	-	6	52	162

Teaching language

Portuguese

Objectives

The promotion of logical reasoning, methods of analysis and the theoretical development of mathematical concepts is fundamental to support the study of the majority of course units along this programme of studies.
This UC aims to ensure the acquisition of solid knowledge in numerical techniques for solving engineering problems, which are of vital importance, as well as to familiarize them with the most varied methods and their implementation, advantages and disadvantages of its application in solving numerical problems. It is intended that students develop numerical manipulation capabilities as well as independent and analytical thinking and ability to apply mathematical concepts to solve practical problems. The students will be able to choose the most efficient methods for the solution of each basic Numerical Analysis problem. The students are expected to understand the theorems and convergence conditions of each of the methods described, to be able to program them, to test them effectively and discuss the results obtained.
The students are supposed to know the subjects taught in Linear Algebra and Mathematical Analysis.

The scientific component is 100%.

Learning outcomes and competences

At the end of this, students should be able to: 
- List the applicability conditions of the numerical methods and state the corresponding theorems of convergence; 
- Apply the methods, formula and algorithms taught to simple problems; 
- Describe the behavior of the methods, translate them into algorithms as well as test them on examples comparing and analyzing the results; 
- Explain the proofs of the theorems given and apply the proof techniques involved to other related situations; 
- Solve new problems with the numerical tools here taught and compare the performance of the various numerical methods in terms of speed and accuracy.

Working method

Presencial

Program

1. Number systems and errors ; number systems on computers; representation of integers and floating point arithmetic; round-off error; absolute error and relative error, significant digits, Taylor's formula and error estimation; error analysis.

2. Linear systems of equations: Gaussian elimination. Round off errors and possible instability of the numerical methods, pivoting strategies. Solution of triangular systems. Tridiagonal systems. LU factorization; application to the computation of determinants and to the inversion of matrices.
Iterative methods: Jacobi and Gauss-Seidel; convergence theorems and algorithms. solution of triangular systems.

3. Least squares approximation. Orthogonal polynomials. Curve fitting. Over-determined systems of equations.

4. Non linear equations: general conditions for the solution, stopping criteria for iterative methods; some iterative methods: successive bisection, fixed point iteration, Newton's method, secant method. Convergence theorems and algorithms; polynomial equations.

5. Numerical integration: Newton-Cotes formulae (ex: Trapezoidal and Simpson rules); composite rules; numerical quadrature errors. Gaussian quadrature.

6. Polynomial interpolation: finite differences; methods of Newton and Lagrange; error of the interpolating polynomial.

7. Ordinary Differential equations: Euler s method for ODE of order 1; Taylor methods. Order of a method for ODE of order1. Runge-Kutta methods.

Mandatory literature

Conte, S. D.; Elementary numerical analysis. ISBN: 0-07-012447-7
Burden, Richard L.; Numerical analysis. ISBN: 0-53491-585-X
W. Cheney, R. Kincaid; Numerical Mathematics and Computing, Brooks Cole
A bibliografia de referência básica e obrigatória é disponibilizada pelo docente nos conteúdos da unidade curricular

Complementary Bibliography

Steven Chapra | Raymond Canale; Métodos Númericos para a Engenharia, McGrawHill. ISBN: 978-85-86804-87-8
Heitor Pina; Métodos numéricos. ISBN: 972-8298-04-8

Teaching methods and learning activities

In class concepts are presented and important results associated with an emphasis on geometric interpretations and practical applications. In order to clarify the definitions and theorems presented, several exercises are solved and illustrative applications are presented. The aim is to, whenever possible, the participation of students, not only in solving the exercises, but also in introducing new concepts. It remains to enhance the resolution of individual exercises and the guidance should be in the study of discipline and clarify questions that may arise in proposal exercises.

keywords

Physical sciences > Mathematics > Applied mathematics > Numerical analysis

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation	Weight (%)
Participação presencial	0,00
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	102,00
Frequência das aulas	60,00
Total:	162,00

Eligibility for exams

Gets frequency to this UC, in the current school year, every student that: - Regularly registered in UC and does not exceed the maximum number of absences. All students duly registered in UC, can perform the tests and examinations that are proposed and used for evaluation.
Regarding the assessment there are three different times, and they were the following:
1) First Test (T1) (Item 1, 2 and 3 of the program);
2) Second Test (T2) (3, 5, 6, and 7 of the program);
3) Exam resource (E) (all matter) - to be marked.
Remarks:
a) classifications of tests from previous years are not considered.
b) if a student misses one test, the rating assigned to this test, to calculate the final average, is zero values.

Calculation formula of final grade

For the final grade of the UC student must have or be exempt from such frequency. In these conditions, any student can choose to get approved by tests or final exam resource (E). If a student does not obtain approval for tests, he can still do the test resource.

The final grade of UC is (scale 0-20):
- Arithmetic average of T1 and T2;
- The grade of the exam recourse (E).

Special assessment (TE, DA, ...)

Students who are under special statutes or having the or who have had previous years frequency, are exempt frequency. The approval can be obtained by performing the tests (T1 and T2) or exam resource (E), the final classification is done according to the previous point.

Classification improvement

Students wishing to undertake improvement of classification may submit the evaluation defined for the UC according with existing regulations