Numerical Methods

Recommended Prerequisites

Mathematical Analysis I, Mathematical Analysis II, Linear Algebra.

Teaching Methods

Theoretical classes will be based on the presentation and explanation of concepts and methods (using media) complemented with the resolution of some illustrative problems of the exercises sheets.
The practical-theoretical classes has two components:
• Resolution of important examples to improve the understanding of the theoretical lectures (30 min maximum);
• Discussion of problems proposed at the exercises sheets, being the students stimulated to solve them individually or in group, under the supervision of the professor (1h00);  
There is the possibility of two types of evaluation:
• Evaluation only with final exam
• Continuous assessment (maximum number of absences: 30% of the predicted classes). Two tests (T1 and T2, 37.5% each) to be held during the theoretical classes (75 min each), and one Test (T3) to be realized at the end of the semester (25%). The test T3 and exam (EX) will take place at the same time. Students can either attend to the test T3 or to the exam (EX).

Learning Outcomes

This course aims to make students aware of the importance that currently have numerical methods in solving different engineering problems (especially large size problems or when there is no analytic solution) and to present the mathematical foundations of a range of well-established numerical algorithms used in its resolution.

At the end of the course, the students should be able to:

- Understand the capabilities and the limitations of computer arithmetic. Differentiate between the errors caused by computer arithmetic and those caused by the limitations of the algorithms.

- Know the master basic numerical techniques, emphasizing its practical applications (convergence rates, computational work effort, accuracy and stability) and limits of their appropriate use.

- Identify and classify the numerical problem to be solved and choose the most appropriate numerical method for its solution based on characteristics of the problem.

- Provide a base set of numerical skills required for the proper functioning of other course curriculum units.

- To be capable of communicating and presenting knowledge clearly and unambiguously.

Work Placement(s)

No

Syllabus

Chapter 1 - Introduction to error analysis (Taylor Series, Truncation error, Round-off error, Floating point number systems). 

Chapter 2 -  Root finding of non-linear equations (Bisection method, Newton's method, Secant method)

Chapter 3 - Interpolation: Polynomial interpolation. Cubic splines.

Chapter 4 - Curve fitting: Least squares. Linear regression. Two parameter models. Linearization.   Linear models of n parameters.

Chapter 5 - Numerical integration. Trapezoidal and Simpson Rules. Gaussian Quadrature Formulas.

Chapter 6 - Equations systems:  Systems of linear equations (direct methods and iterative methods: Jacobi and Gauss-Seidel methods). Introduction to systems of non-linear equations.

Chapter 7 - Ordinary Differential Equations (ODEs). Numerical solutions of ODEs- Initial value problems (Taylor Series methods, Runge-Kutta methods).

Chapter 8 - Introduction to Partial Differential Equations (PDEs). Finite difference method for parabolic problems.

Head Lecturer(s)

José Manuel de Eça Guimarães de Abreu

Bibliography

Abreu, J.M, Antunes do Carmo, J. S.  (2010) - Métodos Numéricos em Engenharia, DEC-FCTUC.

Chapra, S.C., Canale, R.P. (2010) - Numerical methods for engineers. McGraw-Hill Int. Eds., 6nd Edition.

Conte, S.D., de Boor, C. (1981) - Elementary Numerical Analysis: An Algorithmic Approach. McGraw-Hill Int. Eds., 3rd Edition.

Curtis, F.G., Wheatley, P.O. (1994) - Applied numerical analysis. Addison Wesley, 5th Edition.

Pina, H.L.G. (1995) - Métodos numéricos. McGraw-Hill.