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# Numerical Methods

 Code: EEC0016 Acronym: MNUM

Keywords
Classification Keyword
OFICIAL Mathematics

## Instance: 2011/2012 - 1S

 Active? Yes E-learning page: http://moodle.fe.up.pt/ Responsible unit: Department of Electrical and Computer Engineering Course/CS Responsible: Master in Electrical and Computers Engineering

### Cycles of Study/Courses

Acronym No. of Students Study Plan Curricular Years Credits UCN Credits ECTS Contact hours Total Time
MIEEC 370 Syllabus (Transition) since 2010/2011 2 - 6 63 162
Syllabus 2 - 6 63 162

### Teaching - Hours

 Lectures: 2,00 Recitations: 2,00
Type Teacher Classes Hour
Lectures Totals 2 4,00
Aníbal Castilho Coimbra de Matos 2,00
Carlos Alberto Costa Mendonça e Moura 2,00
Recitations Totals 12 24,00
Manuel Joaquim da Silva Oliveira 6,00
Joaquim José de Amaral Vieira e Costa 12,00
Ana Cristina Costa Aguiar 2,00
Carlos Alberto Costa Mendonça e Moura 4,00

Portuguese

### Objectives

1- BACKGROUND
Electrical and computer engineers make an increasing use of numerical models and tools to solve problems. A solid knowledge of those tools is required for design engineers.
2- SPECIFIC AIMS
To endow students with skills so that they can rigorously apply numerical techniques to solve engineering problems. Topics covered include: finite precision representation of numbers, iterative methods to solve nonlinear equations, function approximation, numerical methods for linear algebra, numerical integration and numerical solution of ordinary differential equations.
3- PREVIOUS KNOWLEDGE
Basic courses on Calculus.
4- PERCENT DISTRIBUTION
Scientific component:70%
Technological component:30%
5- LEARNING OUTCOMES
Knowledge and Understanding- Understanding the limitations of finite precision, understanding the approximate nature of numerical tools.
Engineering analysis- Analysis of convergence of iterative methods.
Transferable skills- Knowledge of iterative methods to find numerical solutions of equations and systems of equations; knowledge of numerical approximation techniques.

### Program

I. Error theory: exact and approximate value: approximation error; Absolute and relative error; Representation of real numbers: significant figures; Propagation of errors in the calculation of functions; Truncation error.
II. Non-linear equations: Direct and iterative methods; Separation of zeros of functions; Methods of successive bisection, false position, simple iterative, Newton and secant; Errors and convergence; Polynomial zeros.
III. Systems of non-linear equations: fixed-point method; Newton’s method; Errors and convergence; Modifications of Newton’s method.
IV. Function approximation: least squares method; Method extension; Approximation in vector spaces.
V. Polynomial interpolation: interpolation polynomial; Divided differences and finite differences; Interpolation error; Double and inverse interpolation; Segmented polynomial interpolation (splines).
VI. Numerical integration: Rules of simple and composite integration; Trapezium rule and Simpson’s rule; Romberg integration; Gaussian quadrature;
VII. Linear equations systems: Gaussian elimination; Pivoting techniques; Iterative methods: Jacobi and Gauss-Sidel; Error and residual of an approximate solution; Relationship between them.
VIII. Integration of differential equations: Euler method; Taylor method; Truncation error; Consistency; Predictor-corrector method; Runge Kutta, Milne and Adams methods.

### Mandatory literature

A. Matos; Apontamentos de Análise Numérica, 2005
Burden, Richard L.; Numerical analysis. ISBN: 0-53491-585-X
Conte, S. D.; Elementary numerical analysis. ISBN: 0-07-012447-7
E. Fernandes; Computação Numérica, Universidade do Minho

### Complementary Bibliography

Pina, Heitor; Métodos numéricos. ISBN: 972-8298-04-8
W. Cheney, R. Kincaid; Numerical Mathematics and Computing, Brooks Cole

### Teaching methods and learning activities

Theoretical classes: presentation and discussion of the program; presentation of examples; answering to students’ questions
Theoretical-practical classes: programming of methods and techniques of Numerical Analysis using “Matlab”; resolution and discussion of exercises.

Matlab

### keywords

Physical sciences > Mathematics > Applied mathematics > Numerical analysis
Physical sciences > Mathematics > Applied mathematics > Numerical analysis

### Evaluation Type

Distributed evaluation with final exam

### Assessment Components

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

### Amount of time allocated to each course unit

Description Type Time (hours) End date
Study Estudo autónomo 78
Preparation for exams Estudo autónomo 25
Total: 103,00

### Eligibility for exams

To be admitted to exams, students:
- cannot miss more classes than allowed by the rules (practical and theoretical-practical);
- have to reach a minimum grade of 8 (out of 20) in programming assignments and exercises;
Students, who attended this course last year, do not need to attend practical and theoretical-practical classes. Students have to choose at the beginning of the semester and the decision is irreversible.

### Calculation formula of final grade

Final Grade (N) of students who attended practical and theoretical-practical classes will be based on grade of assignments and problems (P) and on the average grade of the two tests (E). It will be based on the following formula:

N = (0.1+0.01xE)xP + (0.9-0.01xE)xE

For P it will only be taken into account the 4 best programming assignments (out of 5) and 4 best problems (out of 5).

Students, who want to keep their continuous assessment grade from last year, their Final Grade (N) will be based on the same formula, being the grade of practical classes substituted by last year continuous assessment grade.

Students can only reach a higher grade than 18 (out of 20) if they attend an oral exam.

### Examinations or Special Assignments

See Special Assessment

### Special assessment (TE, DA, ...)

Students with a working student status or other students, who choose not to attend practical and theoretical-practical classes, will be assessed in a different way:
- the grade of practical classes will be substituted by a programming assignment or an extra exercise (P);
- final grade will be based on the following formula:
N = (0.1+0.01xE)xP + (0.9-0.01xE)xE.

And the average grade of the two tests.

### Classification improvement

Final grade will be based on the following formula:
N = (0.1+0.01xE)xP + (0.9-0.01xE)xE

E- Exam
P- Programming assignment (to be done at the day of the exam)
Students can only reach a higher grade than 18 (out of 20) if they attend an oral exam.

### Observations

Students, who want to keep their last year continuous assessment grade, cannot enrol in practical and theoretical-practical classes.
All assessment components are graded from 0 to 20.
The assessment of the practical component is based on the average grade of the programming assignments and problems solved in class. It will be taken into account the 4 best results. If students miss a class, in which it will take place an exercise/assignment, they will earn a 0 in that exercise/assignment.