A class of efficient quadrature-based predictor-corrector methods for solving nonlinear systems

We extend a class of quadrature-based predictor-corrector techniques for root-finding to multivariate systems. 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 reusing the linear system from the previous iterate, this class incorporates a significant improvement in computational time relative to the standard class through the inclusion of an LU-decomposition during the iteration. Complexity is equivalent to Newton's Method, as they only require knowledge of F(x) and F′(x).