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This note givesan analysis of the order ofconvergence of some modified Newton methods. The modifications we are concerned with are well-known methods-a total-step method and a singlestep method-for refining all roots of an nth-degree polynomial simultaneously. It is shown that for the single-step method the R-order of convergence, used by Ortega and Rheinboldt in [6], is at least 2 + an > 3, where an > 1 is the unique positive root of the polynomial Pn(a)= an-a-2.

Two efficient algorithms for enclosing a zero of a continuous function are presented. They are similar to the recent methods, but together with quadratic interpolation they make essential use of inverse cubic interpolation as well. Since asymptotically the inverse cubic interpolation is always chosen by the algorithms, they achieve higher-efficiency indices: 1.6529… for the first algorithm, and 1.6686… for the second one. It is proved that the second algorithm is optimal in a certain family. Numerical experiments show that the two new methods compare well with recent methods, as well as with the efficient solvers of Dekker, Brent, Bus and Dekker, and Le. The second method from the present article has the best behavior of all 12 methods especially when the termination tolerance is small.

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