Improving the efficiency index in enclosing a root of an equation

Recently several algorithms have been deveh|ped which achieve high e|tMent:y index in enclosing a na~t of the equati|m f (x ) = 0 in an interval [a, b] ,ver whk:h f ( x ) is continu|ms and f (a) f (b) < 0. The highest efficiency index, 1.6686 . . . . was achieved in [4] using the inveme cuhic interl~|latitat. This paper studies the po.~sibility of impr|wing eftlciency index by using high order inverse interpolations. A cla.~s of algorithms are presented and the optimal |me |ff the clags has achieved t|*e efficim~cy index 1.7282... With a user-given accurary e and starting with the initial interval [a~, bt] = [a, hi, these algorithms g~mrantee to find in finitely many iterations an end|ruing interval [an,bn] thai ~a|ntains a ~ t of the equati|m and who~ length b n an is smaUer than e. Numerk.~d experiments indicate that the new algorithm perfi|rms very well in practice.