Numerical Methods

Recommended Prerequisites

Mathematical Analysis I, Mathematical Analysis II, Linear Algebra and Analytical Geometry, Programming and Computer Science.     

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 classes have three components:

• Resolution of important examples to improve the understanding of the theoretical lectures;

• 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; 

• Implement and test numerical methods using Matlab.

Learning Outcomes

The primary objective of the course is to develop the basic understanding of the construction of numerical algorithms and skills to implement these algorithms to solve engineering problems on a computer.

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

• Understand discretization processes and their impact on solving engineering problems.

•  Understand the capabilities and the limitations of computer arithmetic.

•  Know basic numerical techniques, emphasizing practical application 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

• Have a solid base of computational skills and be able to write their own codes using Matlab.

• Utilizing with critical capacity more sophisticated numerical methods provided as built-in Matlab functions.     

Work Placement(s)

No

Syllabus

Chapter 1 – Introduction and errors in numerical analysis. Taylor’s formula and Taylor series. 

Chapter 2 - Numerical linear algebra. Systems of linear equations: direct methods (Cholesky and QR factorization) and iterative methods (stationay: Jacobi, Gauss-Seidel and SOR; non-stationary: steepest descent method). Eigenvalues and eigenvectors (power and inverse power methods).

Chapter 3 -  Root finding of non-linear equations. Newton's and fixed point method. Systems of non-linear equations

Chapter 4 - Interpolation: Polynomial interpolation (Lagrange and Hermite). Cubic splines.

Chapter 5 - Curve fitting: Two parameter linear and non-nonlinear models. Linearization. Linear models of n parameters.

Chapter 6 - Numerical integration. Newton-Cotes and Gaussian Quadrature Formulas. Multiple integrals.

Chapter 7 - Ordinary Differential Equations (ODEs). Initial value problems (Taylor Series methods, Runge-Kutta methods; Multistep methods).      

Head Lecturer(s)

José Manuel de Eça Guimarães de Abreu

Assessment Methods

Assessment
Mini Tests: 25.0%
Resolution Problems: 25.0%
Frequency: 50.0%

Bibliography

[1] Abreu, J.M., Carmo, J.S.A. (2015) - Métodos Numéricos em Engenharia, DEC-FCTUC.

[2] Chapra, S.C. (2017) - Applied Numerical Methods with MATLAB for Engineers and Scientists. McGraw-Hill, 4th Ed.

[3] Chapra, S.C., Canale, R.P. (2015) - Numerical methods for engineers. McGraw-Hill, 7th Ed.

[4] Conte, S.D., de Boor, C. (1981) - Elementary Numerical Analysis: An Algorithmic Approach. McGraw-Hill, 3rd Ed.

[5] Correia, A.A. (2018) - Informática. Volume II - Introdução à programação em MatLab. DEC-FCTUC.

[6] Heath, M.T. (2018) - Scientific Computing. An Introductory Survey. Siam, 2nd Ed.

[7] Khoury,R., Harder,D.W. (2016) - Numerical Methods and Modelling for Engineering. Springer.

[8] Lindfield, G.R., Penny J.E.T. (2012) -  Numerical Methods, Using MATLAB, Elsevier, 3rd Ed.

[9] Moler, C.B. (2008) - Numerical Computing with MATLAB. Siam.

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

[11] Quarteroni, A.,  Saleri, F. (2007) - Cálculo Científico Com MATLAB e Octave. Springer.