| Περίληψη: | The method of quantifier elimination constitutes an interesting rather recent computational method in computer algebra implemented in few computer algebra systems. Here we apply this method to the determination of sharp enclosures of the two real roots (when there exist such roots) of the classical parametric quadratic equation (in its complete form with three parameters) with one interval coefficient, which is here an interval parameter, whereas the remaining two coefficients are crisp (deterministic) parameters. The powerful computer algebra system Mathematica is used in all the present computations. The classical closed-form formulae for the above two roots are not required in the present quantifier-elimination-based approach: only the original quadratic equation is employed during quantifier elimination. All three cases of parametric coefficients in the quadratic equation are studied in detail and sharp enclosures of its roots are derived in parametric forms. The present results are also verified by using minimization and maximization commands directly on the closed-form formulae for these two roots. Several numerical applications now with numerical (instead of parametric) intervals for the interval coefficient are also made. The present results are seen to be in complete agreement with previous related original results by Elishakoff and Daphnis, who appropriately used classical interval analysis and based their results on the classical closed-form formulae for these roots. Finally, the enclosures of the roots derived by the present approach are always sharp without any possibility of overestimation contrary to what happens in the classical interval-analysis-based approach, where overestimation may be present in some cases.
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