A modification of Newton's method for solving systems of $n$ nonlinear equations is presented. The new matrix-free method relies on a given decomposition of the invertible Jacobian of the residual into invertible sparse local Jacobians according to the chain rule of differentiation. It is motivated in the context of local Jacobians with bandwidth $2m+1$ for $m\ll n$. A reduction of the computational cost by $\mathcal{O}(\frac{n}{m})$ can be observed. Supporting run time measurements are presented for the tridiagonal case showing a reduction of the computational cost by $\mathcal{O}(n).$ Generalization yields the combinatorial Matrix-Free Newton Step problem. We prove NP-completeness and we present algorithmic components for building methods for the approximate solution. Inspired by adjoint Algorithmic Differentiation, the new method shares several challenges for the latter including the DAG Reversal problem. Further challenges are due to combinatorial problems in sparse linear algebra such as Bandwidth or Directed Elimination Ordering.

翻译：本文提出了一种用于求解$n$个非线性方程组的牛顿法改进方案。这种新的无矩阵方法依据微分链式法则，将残差的可逆雅可比矩阵分解为稀疏可逆的局部雅可比矩阵。该方法在带宽为$2m+1$（其中$m\ll n$）的局部雅可比矩阵背景下提出，计算开销可降低$\mathcal{O}(\frac{n}{m})$。针对三对角情况进行的运行时测量验证了计算开销降低至$\mathcal{O}(n)$的结论。算法的泛化引出了组合优化问题——无矩阵牛顿步问题。我们证明了该问题的NP完全性，并提出了用于构建近似解方法的算法组件。受伴随算法微分的启发，新方法面临与后者类似的若干挑战，包括DAG反转问题。其他挑战源于稀疏线性代数中的组合问题，如带宽优化或有向消元排序。