Numerical Analysis

Code:

L.EM015

Acronym:

A N

Keywords
Classification	Keyword
OFICIAL	Computational Methods

Instance: 2022/2023 - 1S

Active?	Yes
Web Page:	https://moodle.up.pt/
Responsible unit:	Mathematics Section
Course/CS Responsible:	Bachelor in Mechanical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EM	255	Syllabus	2	-	6	45,5	162

Teaching language

Portuguese

Objectives

General: 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 on a computer and discuss the results obtained.

Specific: For each chapter in the program the successful students will be able to list the applicability conditions of the numerical methods and state the corresponding theorems of convergence;

they will be able to apply the methods, formula and algorithms taught to simple problems; they will be able to describe the behavior of the methods, translate them into algorithms and ‘Matlab Functions’ as well as test them on examples comparing and analyzing the results; they will explain the proofs of the theorems given and apply the proof techniques involved to other related situations; they will be able to solve new problems with the numerical tools here taught and compare the performance of the various numerical methods in terms of speed and accuracy.

Learning outcomes and competences

For each chapter in the program the successful students
will be able to list the applicability conditions of the numerical methods and state the corresponding theorems of convergence;
they will be able to apply the methods, formula and algorithms taught to simple problems;
they will be able to describe the behavior of the methods, translate them into algorithms and
‘Matlab Functions’ as well as test them on examples comparing and analyzing the results;
they will explain the proofs of the theorems given and apply the proof techniques involved to other related situations;
they will be able to 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

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

The students are supposed to know the subjects taught in Linear Algebra and Mathematical Analysis as well as in Computer programming

Program

Chapter 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. Chapter 2. Polynomial interpolation: finite differences; methods of Newton and Lagrange; error of the interpolating polynomial. Hermite interpolation. Numerical differentiation. Chapter 3. Numerical integration. Newton Cotes formulae. Composite rules. Gaussian quadrature. Chapter 4. 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. Chapter 5. 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.  Chapter 6. Least squares approximation. Orthogonal polynomials. Curve fitting. Over-determined systems of equations. Chapter 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 of order 2 and 4 .

Mandatory literature

Cleve Moler; Numerical Computing with Matlab , SIAM , 2004
John Mathews; Kurtis Fink ; ; Numerical Methods using Matlab , Prentice Hall , 1999
Maria Raquel Valença ; Métodos Numéricos , Livraria do Minho , 1993
Maria Raquel Valença ; Análise Numérica, Universidade Aberta
Heitor Pina ; Métodos numéricos , McGraw Hill , 1995
Edite Fernandes; Computação Numérica, 1997. ISBN: ISBN: 972-96944-1-9

Teaching methods and learning activities

Lectures with "Powerpoint, illustrating examples and discussion of results of small computer projects Lab work: conclusion and presentation of home work small projects supervisedby teachers in the computer room, with Matlab.   1000 characters

Software

Matlab

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Designation	Weight (%)
Exame	75,00
Participação presencial	5,00
Teste	20,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	120,00
Frequência das aulas	45,50
Total:	165,50

Eligibility for exams

Minimum requirements to be admitted to the exam:
see NORMAS GERAIS DE AVALIAÇÂO in FEUP.

Calculation formula of final grade

Type of assessment: Avaliação distribuída com exame final

Observations

e-learning at https://moodle.up.pt/