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

Abstract

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, R_k = ½ℱ′(x_k)⁻¹ℱ″(x_k)[Q_ku_k, · ], which the two Jacobians cannot determine: we exhibit two problems agreeing in ℱ(x_k), ℱ′(x_k) and ℱ′(y_k) whose values of R_k differ. For one equation R_k = Q_k², 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 R_k 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.