Convergence of a class of efficient quadrature-based predictor–corrector methods for root-finding

In this paper we analyze a class of predictor–corrector techniques for root-finding that are derived from quadrature methods. They are found to have a rate of convergence of 1+2 regardless of the degree of precision for the quadrature technique from which they are derived, provided it is at least one. By using previously-evaluated quantities in the predictor step, they require fewer functional evaluations than the standard class of techniques. This class is found to be superior to the standard class provided that the quantity of knots from the quadrature is 1⩽m⩽31⩽m⩽3, with the optimal method being that derived from the Midpoint Method.