Efficient methods for the solution of n-dimensional systems of nonlinear equations are critical for numerous applications. For the system F(x) = 0, the well-known Newton’s method employs the derivative matrix F′(x) and under modest assumptions delivers quadratic convergence near a solution with the error squared at each iteration. Often, a significant cost is the production and use of F’, and effectively exploiting this matrix once calculated has motivated many extensions, a new family of which is developed in this paper.

In these approaches, improvements on the order of convergence are accomplished with the help of repeated use of a calculated F′. Starting with an existing third-order two-step method, which uses the same calculated F′ twice, the authors introduce parameters, determine their optimal values theoretically, and then inductively extend their approach to an m + 1 step method of order m + 2. Significantly, only one evaluation of F′ is used for each m + 1 step, and for the analysis F needs only to be third Fréchet differentiable.

A detailed efficiency study incorporating computational cost and order is provided for the proposed method and reviewed for a set of alternative methods in the literature that reuse F′ evaluations. Comparisons are neatly presented using graphs whose axes relate the key costs of computing elements of F and F′. Numerical experiments are included using extended precision (200-digit mantissa) to capture the high-order behavior for a number of test cases. The proposed method’s efficiency increases with m and is advantageous for larger systems.