Exact enclosures of roots of interval quadratic equations by Sridhara's and Fagnano's or modified Fagnano's formulas

摘要

In this study we investigate the question of accurate determination of the root enclosures of quadratic equations whose coefficients constitute interval variables. We treat several special cases where either Sridhara's classical formula or Fagnano's alternative expression provide exact interval enclosures for the roots of the quadratic equation. In the case of a single coefficient serving as an interval, it is shown that the classical interval analysis by either Sridhara's, Fagnano's or specifically modified Fagnano's formulas provide lower and upper bounds of roots. Then we follow with the case of two coefficients being intervals whereas the third coefficient is a deterministic quantity. Numerous examples are provided in which the solutions are compared with the direct numerical evaluation of the roots' enclosures. In three arising cases either Sridhara's or Fagnano's expressions suffice to obtain exact enclosures.