A variable order method for solution of a nonlinear algebraic equation

In this paper a modified form of the IMM (Improved Memory Method) for the solution of a nonlinar equation which was proposed by Shacham [Chem. Engng Sci. 44, 1495 (1989)] is presented. The form of IMM presented in the reference uses continued fractions to pass an inverse interpolating polynomial through all the previously calculated points, to find a new estimate for the solution. This method was compared, and found to be superior in terms of function evaluation, to six commonly used methods. Here we present an m-point, globally convergent version of IMM. In this version only m previously calculated points are used to construct the interpolating polynomial, where m can be any number: m ≥ 2, according to the users' choice. A set of 90 test problems, many of them very difficult, were solved using 2-, 3-, 4- and 5-point IMM methods. The proposed method did not fail in any of these cases. It was found that in most cases the 3-point IMM converged must faster, or faster, than the 2-point IMM. The improvements in convergence rate when going to 4- or 5-point IMM from the 3-point one was much more moderate. In a few cases, higher order methods actually required more function evaluations to converge than lower order ones.