A Second-Order Directional Correction Giving Genuine Eighth-Order Convergence for Nonlinear Systems with One Factorization per Iteration

Three-step methods that freeze the Jacobian at the current iterate and sample a second Jacobian at the Newton predictor are attractive for large systems, since one factorization serves the whole step. We show that such schemes cannot exceed order six. The obstruction is a single matrix, Rk = ½ℱ′(xk)−1ℱ″(xk)[Qkuk, · ], which the two Jacobians cannot determine: we exhibit two problems agreeing in ℱ(xk), ℱ′(xk) and ℱ′(yk) whose values of Rk differ. For one equation Rk = Qk2, so the scalar theory is unaffected. We also determine when the barrier is inactive: if a constant starting vector or a set of affine equations confines the iteration to a one-dimensional set, the restricted problem is scalar and order eight is recovered. One further Jacobian, evaluated at a fourth-order predictor where the missing direction is exposed, supplies Rk and yields a method PC8 of genuine order eight. Written so that no weight matrix is assembled, PC8 costs one LU factorization plus O(n2) arithmetic per step, against about 4.7n3 for the usual implementation. Measured orders are 7.99–8.00 for PC8 and 5.97–6.00 for five published weights on the same problems.