Computational Methods in Engineering
Objectives
We will illustrate several numerical methods for the computer solution of certain classes of mathematical problems. We will show how to use these methods in order to solve nonlinear equations, linear systems, integrate and construct accurate approximations for the ordinary differential equations.
General characterization
Code
10437
Credits
3.0
Responsible teacher
António Manuel Morais Fernandes de Oliveira
Hours
Weekly - 3
Total - 39
Teaching language
Português
Prerequisites
Students must have basic knowledge in Mathematical Analysis (AM I) and Linear Algebra (ALGA).
Bibliography
- Atkinson K., An Introduction to Numerical Analysis, Wiley, Second Edition, 1989.
- Burden R.; Faires J. , Numerical Analysis, Brooks-Cole Publishing Company, 9th Edition, 2011.
- Conte S.; Boor C., Elementary Numerical Analysis: An algorithmic approach, Mc Graw Hill, 1981.
- Isaacson, E.; Keller, H., Analysis of Numerical Methods, Dover, 1994.
- Pina H., Métodos Numéricos, Escolar Editora, 2010.
- Santos, F. Correia dos; Duarte, Jorge; Lopes, Nuno D., Fundamentos de Análise Numérica (Com Python 3 e R), Edições Sílabo, 2019 (2ª edição).
Teaching method
The Curricular Unit operates with Theoretical-Practical (TP) classes, in which the successive topics of the course program will be explained and discussed. In order to consolidate the material given, exercises will be solved in classes relating to each of the topics covered. In order to implement some of the methods covered, classes will be taught in a Computational Laboratory where computers are present. Alternatively, students can take their laptop where they will be advised to install software recommended by the teacher.
In order to be assessed, students will have to attend at least 2/3 of the classes taught or have attended in the previous year (for more details see assessment methods).
The assessment of the course consists of two tests that cover the knowledge acquired in the course classes and a computational work (optional) to be prepared in groups.
Evaluation method
This document regulates the knowledge assessment process of the Curricular Unit (UC) Computational Methods in Engineering. In any omission, the Knowledge Assessment Regulations of the Faculty of Science and Technology of Universidade Nova de Lisboa, dated 17-11-2020, applies (see here).
1. Attendance
All students enrolled at the UC are assigned attendance, that is, they are admitted to the exam (and tests). In classes, registration of students present will be made.
To attend a shift Tp (Theoretical-practical ), students must be registered for it (registration provided via CLIP).
2. Evaluation
All evaluation elements are classified from 0 to 20 values. A student obtains approval if the final grade in the UC is greater than or equal to 9.5 values.
For any evaluation (test or exam) the student must have his identification document.
For tests or exams, the student may use a calculating machine, which may be of a graphical type. Students may not use any other electronic material that allows them to perform mathematical calculations.
In the assessment, the student can choose between continuous evaluation or by appeal exam.
2.1 Continuous evaluation
The evaluation during the semester consists of two tests lasting one hour and a half and a computational work to be done in groups of 3 or 4 students. Carrying out the computational work is optional.
Let NT1 and NT2 be the grades of tests 1 and 2, respectively, and NTC the grade of computational work. The student may choose to do or not to do the computational work. If a student does not attend one of the assessments, this assessment element will be rated 0 values.
If the student performs the computational work,the grade of continuous assessment (NAC) is given by:
NAC = 0.45 × NT1 + 0.45 × NT2 + 0.10 × NTC.
If the student does not perform the computational work, the grade of continuous assessment (NAC) is given by
NAC = 0.5 × NT1 + 0.5 × NT2
If NAC < 9.5 values, the student fails the continuous assessment (may go to the appeal exam).
If NAC ≥ 9.5 values, the student obtains approval in the UC, with the NAC classification rounded to the nearest units.
2.2 Examination Appeal
Any student who has not yet passed the UC can take the appeal exam.
The appeal exam lasts for 3 hours.
If the CEA, exam grade, is lower than 9.5 values, the student fails.
If CEA ≥ 9.5 values and the student has performed computational work, the appeal grade, AG will be given by:
AG = max {CEA, 0.90 × CEA + 0.10 × NTC};
If CEA ≥ 9.5 values and the student has not performed computational work, the appeal grade, AG will only be CEA, rounded to the nearest units.
3. Grade improvement
Students wishing to take the exam to improve their grade must, in advance, request such an improvement from the academic services.
The grade improvement exam is graded in a similar manner to the Season of Appeal.
If the result is higher than that already obtained in the UC, it will be taken as a final grade. Otherwise, there is no grade improvement, maintaining the previous grade.
4. Special exam
Only students registered for the purpose in the Academic Office are able to attend the special period. The special season exam classification is performed in a similar way to the Appeal Season.
18th February 2024
Subject matter
1. Errors
Absolute error, relative error and significant digits.
Condition number.
Numerical algorithms stability.
2. Polynomial Approximation and Interpolation
Polynomial interpolation: Lagrange and Newton formulas; cubic Spline interpolation.
Least squares approximation.
3. Numerical Integration
Newton-Cotes integration formulae.
Gaussian integration.
4. Rootfinding for Nonlinear Equations
Bissection method, fixed-point iteration, Newton method and Secant method.
5. Linear systems
Vector norms and matrix norms.
Conditioning of a linear system.
Eigenvalues and eigenvectors. Finding eigenvalues (Gershgorin Theorem).
Iterative methods: general procedure, Jacobi method, Gauss-Seidel method and SOR method.
6. Numerical solution of ODE''s
Taylor methods. Runge-Kutta methods.