Computational Mathematics

Code	8552
Year	2
Semester	S2
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	The student should already be able to: - calculate derivatives of an actual function; - operate with matrices; - use the concept of linear combination in polynomial vector spaces; - efficiently use the calculator as an auxiliary calculation tool.
Mode of delivery	Classroom
Work placements	N/A.
Learning outcomes	The general objective of this course is the study of efficient and stable numerical methods for solving certain mathematical problems. The study of each numerical method includes the analytic deduction of the formulae used, the description in algorithmic language and the presentation of techniques to estimate the solution error. This objective is realized by the transmission of the following competences: a) Analyse errors and determine error propagation; b) Calculate roots and extreme values of a non-linear function; c) Solve systems of linear and non-linear equations; d) Interpolate and approximate random data sets by polynomial functions; e) Differentiate and integrate functions numerically; f) Solve differential equations and systems of differential equations numerically. At the end the student should be able to: Identify possible methods to solve mathematical model of an engineering problem, choose the most appropriate one, implement it in MATLAB and criticize the results.
Syllabus	1. Introduction to numerical computing: Floating point computing; Approximation of real functions; Conditioning of a problem e stability of a numerical method. Numerical differentiation. 2. Systems of linear equations: Direct methods and numerical instability; Jacobi and Gauss-Seidel iterative methods. 3. Nonlinear equations: bisection method, fixed point method and Newton's method. 4. Polynomial interpolation: Lagrange and Newton formulas and interpolation by segmented polynomials. 5. Numerical integration: Newton-Cotes and Gauss rules. 6. Numerical methods for differential equations and systems of differential equations: Methods based on the Taylor series and Runge-Kutta methods; consistency, stability and convergence.
Main Bibliography	• R.I. Burden & J.D. Faires , " Numerical Analysis 7e", PWSKent, Boston, 2001. • H. Pina, "Métodos Numéricos", Mc GrawHill, Alfragide, 1995. • M.R. Valença , "Métodos Numéricos", INIC, Braga, 1988. • A. Stanoyevitch, “Introduction to MATLAB with Numerical Preliminaries", John Wiley & Sons, 2005. • A. Quarteroni e F. Saleri, “Cálculo científico com MATLAB e Octave”, Springer-Verlag, 2007. • J.C. Butcher , "The Numerical Analysis of Ordinary Differential Equations", John Wiley & Sons, Auckland, 1987. • E. Hairer , S.P. Nørsett & G. Wanner , " Solving Ordinary Differential Equations I ", Springer Series in Comput. Mathematics, Vol. 8, Springer-Verlag, Heidelberg, 1987. • E. Hairer & G. Wanner , " Solving Ordinary Differential Equations II ", Springer Series in Comput. Mathematics, Vol. 8, Springer-Verlag, Heidelberg, 1987
Teaching Methodologies and Assessment Criteria	The teaching methodology will be based on a process of learning less dependent on the teacher and more focused on self-study. Thus, the program content will be presented in video form or supporting text, which should be studied preferably outside of the classes. The classes will be reserved for the discussion of concepts and ideas and resolution of practical problems, under tutorial teaching orientation. Evaluation is performed in two phases: - Continuous evaluation: theoretical and practical tests throughout the semester and realization and discussion of works with implementation of procedures and functions in MATLAB to solve practical problems. - Final exam (with theoretical and practical part) for admitted students. Students with a final classification below 6 values and attendance below 50% are not admitted to the exam.
Language	Portuguese. Tutorial support is available in English.