Base Knowledge
Solving equations and inequations involving real functions of one variable.
Calculation of derivatives of real functions of one variable (knowledge acquired in Mathematical Analysis I)
Matrix calculus: matrix operations and calculation of the inverse of a matrix (knowledge acquired in Mathematical Analysis I)
Teaching Methodologies
Classes are taught on a theoretical-practical basis.
The expository methodology is used to present the concepts of the curricular unit, supported by exercises to apply the concepts transmitted. Some of these exercises are based on real-life practical problems.
Learning Results
Goals:
Provide students with some existing numerical methods and techniques for solving concrete problems, which occur in the most diverse areas, and which are not always solved in a direct (analytical) way.
Introduce the discussion of the numerical results obtained.
Skills:
Develop the ability to select the methods that best adapt to the resolution of various problems by studying their efficiency, applicability and stability.
Interpret the numerical results obtained.
Program
Chapter 1 – Theory of errors
Basic definitions of error theory
Chapter 2 – Nonlinear equations
2.1 Introduction
2.2 Root location
2.3 Bisection method
2.4 Newton’s method
2.5 Secant method
2.6 Application to the determination of extrema
Chapter 3 – Systems of linear equations
3.1 Introduction
3.2 Jacobi method
3.3 Gauss-Seidel method
Chapter 4 – Polynomial interpolation
4.1 Introduction
4.2 Lagrange interpolation
4.3 Newton’s divided differences interpolation
4.4 Method of least squares
Chapter 5 – Numerical differentiation and integration
5.1 Introduction
5.2 Numerical differentiation
5.3 Numerical integration: trapezoidal rule and Simpson’s rule
Chapter 6 – Ordinary differential problems
6.1 Introduction: initial condition problems
6.2 Euler’s methods
6.3 Runge-Kutta’s methods
Curricular Unit Teachers
Internship(s)
NAO
Bibliography
Fundamental bibliography:
Neves, Cidália, Support texts and pratical exercices available at NONIO, Author’s Edition.
Complementary bibliography:
Araújo, A., Sebenta de Análise Numérica, FCTUC, Coimbra, 2002.
Atkinson, K. E., An Introduction to Numerical Analysis, John Wiley and sons, New York, 1989.
Burden, R. I. e Faires, J. D., Numerical Analysis, PWS-Kent, Boston, 1988.
Lopes, N.D., Santos, F.C. & Duarte, J., Fundamentos de Análise Numérica – com Phyton 3 e R, Edições Sílabo, 2019.
Pina, Heitor, Métodos Numéricos, McGraw-Hill, Alfragide, 2010.
Valença, M. R., Métodos Numéricos, INIC, Braga, 1988.