Numerical Analysis

Code:

EM0016

Acronym:

AN

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2010/2011 - 2S

Active?	Yes
Web Page:	http://moodle.up.pt/
E-learning page:	http://moodle.up.pt/
Responsible unit:	Mathematics Section
Course/CS Responsible:	Master in Mechanical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEIG	80	Syllabus since 2006/2007	2	-	6	56	160

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.

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

Chapter 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.

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

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

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

Small computer projects using WINDOWS or UNIX and MATLAB.

Mandatory literature

Cleve Moler; Numerical Computing with Matlab , SIAM , 2004
John Mathews; Kurtis Fink ; Numerical Methods using Matlab , Prentice Hall , 1999
Maria Raquel Valença ; Análise Numérica, Universidade Aberta, 1996. ISBN: 9726741955
Heitor Pina ; Métodos numéricos , McGraw Hill , 1995
Maria Raquel Valença ; Métodos Numéricos , Livraria do Minho , 1993

Complementary Bibliography

Mário Graça, Pedro Lima; Matemática Experimental, IST Press, 2006. ISBN: 972-8469-52-7
Rosário, Pedro ; Núnez, José ; Pienda, Júlio ; Comprometer-se com o estudar na universidade : cartas do Gervásio ao seu umbigo, Livraria Almedina, , 2006

Teaching methods and learning activities

Lectures with "Powerpoint". Small illustrating computer projects supervised by teachers in the computer room, with Matlab.

Software

Matlab

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	56,00
	Exame	2,00
	Exame	3,00
	Trabalho escrito	36,00
	Total:	-	0,00

Amount of time allocated to each course unit

Description	Type	Time (hours)	End date
	Estudo autónomo	25
	Estudo autónomo	20
	Estudo autónomo	20
	Total:	65,00

Eligibility for exams

Minimum requirements to be admitted to the exam:
registration and not to exceed the maximum number of absences allowed.

Calculation formula of final grade

NT = mark of test, Nex = mark of exam, NF =final classification, NDO= classification after oral exam

Nprov=0,75xNEx+0,25xNT,
if Nprov <=16 then NF=Nprov else NF=max(16,NDO)

Examinations or Special Assignments

The students will do small computer projects and self evaluation tests available on the e-learning platform Moodle, from GATIUP, which do not count directly to the final classification. The final exam will include questions about knowledge acquired in these projects and quizzes.
The classifications will be obtained in a partial quiz (25%) and in the final exam (75%). For students with more than 16 the exam will be completed with an oral exam.

Special assessment (TE, DA, ...)

Final exam.
Students with a special status who only attended to the final exam or recurso exam (resit) and who did not attend to the test, will be assessed based on the final exam.

Classification improvement

Final exam.