Seminar First Course 2021-2022
In the Seminar, by approximating the derivatives in the well-known fourth-order Muller's method and in a sixth-order improved Muller's method by central difference quotients, we obtain new modifications of these methods free from derivatives. We prove the important fact that the methods obtained preserve their convergence orders 4 and 6, respectively, without calculating any derivatives. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical generalization of the secant method.
In this Seminar, Bairstow’s method has to face with numerical errors due to the termination criterion of Raphson-Newton iterations and to successive polynomial divisions. Here, an optimal termination criterion is proposed allowing to stop the iterations as soon as a good computed solution is obtained. Moreover, a simple formula to check the validity of a root of a polynomial is given. It is then possible to eliminate the numerical errors in Bairstow’s method. Numerical examples are presented.
Seminar Second Course 2021-2022
Game theory problems using saddle points
In optimization problems, often students find a single extremum of a function that is assumed to be a maximum or a minimum. At best, saddle points are discarded when checking second order conditions end route to maxima or minima. Game theory provides a setting where saddle points are the solution concept. We introduce the necessary game-theoretic background and explain how game-theoretic experiments of the Matching Pennies game can be used as a classroom activity to develop intuition about saddle points. Paralleling how students learn to find extrema, we first consider finding saddle points on the interior of a set and then consider saddle-like points that appear on the boundary of compact sets. We conclude with a couple of examples from the game theory literature.
In the Seminar, by approximating the derivatives in the well-known fourth-order Steffensen’s method and in a sixth-order improved Steffensen’s method by central difference quotients, we obtain new modifications of these methods free from derivatives. We prove the important fact that the methods obtained preserve their convergence orders 4 and 6, respectively, without calculating any derivatives. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical Newton’s method.
Game theory problems using dominance method
The principle of dominance states that if one strategy of a player dominates over the other strategy in all conditions, then the later, strategy can be ignored. A strategy dominates over the other only if it is preferable over the other in all conditions. The concept of dominance is especially useful for the evaluation of two-person zero-sum games where a saddle point does not exist. In case of pay-off matrices larger than 2 × 2 size, the dominance property can be used to reduce the size of the pay-off matrix by eliminating the strategies that would never be selected.