Numerical Analysis I

Code:

M231

Acronym:

M231

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2010/2011 - 2S

Active?	Yes
Web Page:	http://moodle.up.pt/course/view.php?id=1992
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	Plano de estudos a partir de 2008	3	-	7,5	-
L:CC	0	Plano de estudos de 2008 até 2013/14	3	-	7,5	-
L:F	0	Plano de estudos a partir de 2008	3	-	7,5	-
L:G	0	P.E - estudantes com 1ª matricula anterior a 09/10	3	-	7,5	-
		P.E - estudantes com 1ª matricula em 09/10	3	-	7,5	-
L:M	56	Plano de estudos a partir de 2009	2	-	7,5	-
L:Q	0	Plano de estudos Oficial	3	-	7,5	-

Teaching language

Portuguese

Objectives

The main aim of this subject is given a mathematical problem, to study sufficient conditions for the existence and unicity of its solution, to establish a constructive method to solve it, to study and control the errors involved, to give an algoritmh for the solution and to implement it in a computer and to study and interpret the numerical results.
The following mathematical problems will be treated: propagation of errors in variable values to the evaluation of functions of those variables, computation of the sum of a series, solution of nonlinear equations, direct methods to solve a system of linear equations, methods for approximation: polynomial interpolation and least squares approximation, and numerical computation of derivatives and integrals.

Program

Error theory
Types of errors. Absolute and relative error. Rounding error and truncation error. Propagation of the error. Computation of the sum of a convergent series.
Nonlinear equation
Root finding methods: bisection method, fixed point method , Newton method and variants.
Systems of linear equations
Direct methods. Gauss elimination. Pivoting.
Polynomial interpolation
Lagrange method. Error in interpolation. Aitken-Neville method. Divided differences. Newton method. Inverse interpolation.
Approximation
Least squares polynomial approximation of a set of points. Generalized least squares approximation. Least squares approximation of a function defined in an interval.
Numerical differentiation and integration
Newton-cotes formulas. Simple and composite rules of rectangles, trapezium and Simpson. Truncation errors. Numerical differentiation formulas.

Mandatory literature

M. Redivo-Zaglia; Calcolo Numerico - metodi ed algoritmi, 2005. ISBN: 88-87331-49-9
A. Quarteroni, R. Sacco e F. Saleri; Numerical Mathematics, 2000. ISBN: 0-387-98959-5
E. M. Fernandes; Computação numérica, 1998. ISBN: 972-96944-1-9
H. Pina; Métodos Numéricos, 1995. ISBN: 972-8298-04-8

Teaching methods and learning activities

Lectures, problems and computational projects.

Software

Python, Scilab or Maxima

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	75,00
	Total:	-	0,00

Eligibility for exams

A minimum of 3.5 points in the practical classification.

Calculation formula of final grade

Theoretical classification (CT): Sum of the classifications of 4 tests ( 3 points each) or a final examination (12 points)
Practical classification (CP): sum of classifications obtained in 4 practical tests (2 points each)
Final classification (CF): CT+CP

Special assessment (TE, DA, ...)

One final examination (theoretical and practical).