Elements of Numerical Analysis
Cod: 21035
Department: DCET
ECTS: 6
Scientific area: Mathematics
Total working hours: 156
Total contact time: 26
In concrete applications, quite often one cannot explicitly compute a solution of a problem under consideration. The methods and tools available, either limited or insufficient, are the mathematical reasons for the difficulties and failures pointed out. As examples, one has compute the roots of a function. When such situations occur, sometimes one may apply numerical methods in order to derive an approximation to the solution.  Some numerical methods for the most common cases, as for instance the examples described above, are studied in this curricular unit.
1. Error Theory
2. Approximation of functions
3. Numerical methods for Linear and Nonlinear Equations
Application of numerical methods to compute the roots of a function, to solve an integral or a system of linear equations.

• Know the classic and more common methods for the numerical solution of central problems in Numerical Analysis, as approximating the rootn of nonlinear equations, the iterative solution of linear systems and the approximation of functions;
• For each problem, know the numerical and computational advantages and drawbacks of the several numerical method, as well as to perform error analysis and determine its bounds.
• Analyze the suitability of different numerical methods to each specific example.
• Apply the theorical acquired knowledge to problem solution in different levels of difficulty,
1. Introduction to Octave
2. Introduction to numerical analysis: errors, conditioning and stability
3. Approximation of Functions: Interpolation by polynomials and regression.
4. Nonlinear equations: bisection method, fixed point, Newton's method, secant method.
5. Iterative solutions of systems of linear equations.

P. Serranho, Matemática Aplicada e Análise Numérica - uma introdução com Octave, Sebenta Online, 2017
Valença, M. R., Análise Numérica. Universidade Aberta, 1996.
A. Quarteroni & F. Saleri, Cálculo Científico com Matlab e Octave, Springer, 2006
E-learning

Continuous assessment is privileged: 2 digital written documents (e-folios) during the semester (40%) and a final digital test, Global e-folio (e-folio G) at the end of the semester (60%). In due time, students can alternatively choose to perform one final exam (100%).

Students are recommended to have previously knowledge in the Linear Algebra I and the Elements of Infinitesimal Analysis II curricular units.