# Computational Methods in Engineering

## Objectives

Students must be able to apply numerical methods for mathematical problems, such as, nonlinear equations, function approximations, integration, systems of linear equations and ordinary differential equations.

Students must also be able to implement computational algorithms in order to solve the aforementioned problems.

## General characterization

10437

3.0

##### Responsible teacher

Maria Cecília Marques Rodrigues

Weekly - 3

Total - 39

Português

### Prerequisites

Students must have basic knowledge in mathematical analysis (AMI) and linear algebra (ALGA).

### Bibliography

• Atkinson K., An Introduction to Numerical Analysis, Wiley, Second Edition, 1989.
• Burden R. e Faires J. , Numerical Analysis, Brooks-Cole Publishing Company, 9th Edition, 2011.
• Conte S. e Boor C., Elementary Numerical Analysisan algorithmic approach, Mc Graw Hill, 1981
• Isaacson E. e Keller H., Analysis of Numerical Methods, Dover, 1994
• Martins, M. F. e Rebelo M., Introdução à Análise Numérica, Casa das Folhas, 1997
• Pina H., Métodos Numéricos, Mc Graw Hill, 1995
• Valença M. R., Métodos Numéricos,Livraria Minho, Terceira Edição, 1993

### Evaluation method

1.Frequency

Approval in the Course Unit (U.C) requires obtaining or be dismissed frequency.

Students with Student Worker status are released from obtaining frequency.

Any student without frequency must be registered in one of the practical classes.

Frequency will be attributed to any student who does not, unjustifiably, miss more than 1/3 of the number of lessons.

In case of absences justification, the proof of justification must be delivered on the first day the student returns to class.

2. Evaluation

Assessment are classified from 0 to 20 values. A student obtains approval if has frequency and if the final grade at U.C. is greater than or equal to 9.5 values.

The student who obtains a final grade at U.C. equal to or higher than 17.5 values can choose to get the final grade of 17 values or take a complementary evaluation for grade defence. If the student does not take this evaluation will have the final grade of 17 values.

For any evaluation (test or exam) the student must have his Citizen Card or official identification document.

For tests or exams, the student may use a calculating machine, which may be graphical. Other electronic material is forbidden.

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, in wxmaxima language. Let NT1 and NT2 be the grades of tests 1 and 2, respectively, and NTC the classification of computational work. The student may choose to do the computational work or not. If a student does not attend one of the assessments, this assessment element will be rated 0.

If the student performs the computational work, the grade of continuous assessment (Nav) is given by:

Nav = 0.45 × NT1 + 0.45 × NT2 + 0.10 × NTC.

If the student does not perform the computational work, the grade of continuous assessment (Nav) is

Nav = 0.5 × NT1 + 0.5 × NT2

If Nav <9.5 values the student fails, the continuous assessment (may go to the appeal exam if he has frequency);

If 9.5 ≤ Nav <17.5 values the student obtains approval in U.C. with Nav classification rounded to units;

If Nav ≥ 17.5 values the student can choose to get the final grade of 17 values or take a complementary test for grade defence.

2.2 Examination Appeal

An appeal exam may be submitted to any student not yet approved in the course with the Frequency. The appeal exam lasts for 3 hours.

If the CR exam grade is lower than 9.5, the student fails;

If CR ≥ 9.5 and the student has performed computational work, the appeal grade; NR will be given by:

NR = max {CR, 0.90 × CR + 0.10 × NTC};

If CR ≥ 9.5 and the student has not performed computational work, the appeal grade; NR will only be CR.

If 9.5 ≤ CR <17.5 values the student obtains approval in the course with the classification NR rounded to the units;

If NR ≥ 17.5 values the student can choose to get the final grade of 17 values or take a complementary test for grade defence.

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 U.C, it will be taken as a final grade. Otherwise, there is no grade improvement, maintaining the previous grade.

4. Special exam

The special season exam classification is carried out analogously to the Examination appeal.

## Subject matter

1. Errors

Absolute error,  relative error, 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 formulas,  Gaussian integration.

4. Rootfinding for nonlinear equations

Bissection method,  fixed-point iteration, Newton method, Secant method.

5. Linear systems

Vector norms and induced matrix norms.

Eigenvectors and eigenvalues. Gershgorin theorem.

Iterative methods:  general procedure, Jacobi method, Gauss-Seidel method, SOR method.

6. Numerical solution of ODE

Euler methods, Taylor methods for higher orders, Runge-Kutta methods.

## Programs

Programs where the course is taught: