Numerical Methods
Keywords |
Classification |
Keyword |
OFICIAL |
Mathematics |
Instance: 2010/2011 - 1S
Cycles of Study/Courses
Teaching language
Portuguese
Objectives
This course unit aims to endow students with skills, so that they can rigorously apply numerical techniques to solve engineering problems. They have to:
- understand the fundaments of numerical methods
- know how to apply these methods, by using:
- programming
-calculators
-computer applications
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.
Software
Matlab
keywords
Physical sciences > Mathematics > Applied mathematics > Numerical analysis
Physical sciences > Mathematics > Applied mathematics > Numerical analysis
Evaluation Type
Distributed evaluation without final exam
Assessment Components
Description |
Type |
Time (hours) |
Weight (%) |
End date |
Attendance (estimated) |
Participação presencial |
52,00 |
|
|
Study for Finals |
Exame |
32,00 |
|
2011-01-08 |
|
Total: |
- |
0,00 |
|
Amount of time allocated to each course unit
Description |
Type |
Time (hours) |
End date |
Study |
Estudo autónomo |
78 |
2010-12-18 |
|
Total: |
78,00 |
|
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.