Class Computational Mathematics

In many engineering applications the need arises to solve mathematical problems that either do not have
an analytical solution, or the analytical solution is too computationally expensive to determine. In those
situations, an approximate solution can often be obtained using numerical analysis algorithms.

In this curricular unit we study a set of algorithms that solve some of the most common classes of
problems, ranging from determining the zeros of non-linear equations to solving ordinary differential
equations. This knowledge plays a central role in an engineering course, given the complexity of many
real life mathematical models.

1. Floating point Arithmetics. Representation of integer and real numbers. Errors in floating point arithmetics. Error propagation.
2. Non-linear equations. Bissection method, false position and secant. Newton and fixed point methods.
3. Systems of linear equations. Iterative methods.
4. Systems of non-linear equations. Newton's method.
5. Polynomial interpolation. Polynomial forms. Interpolation formulas.
6. Numerical differentiation. First and second order derivatives.
7. Numerical integration. Simple and composite rules.
8. Ordinary differential equations. Euler's and Runge-Kutta's methods. Convergence. Errors

Using MATLAB / Octave in class to demonstrate in an interactive way some of the concepts of the curricular unit.

• Atkinson, K. - Numerical Analysis. John Wiley and Sons, New York;
• Chapra, S. and Canale, R. - Numerical methods for Engineers. McGraw-Hill Book Comp-New York
• Pina, Heitor - Métodos Numéricos. McGraw-Hill-Lisboa