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 is NP-completeness and we present several algorithmic components for building methods for its 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：提出了一种修正的牛顿方法，用于求解包含n个非线性方程的系统。该新无矩阵方法依赖于根据微分链式法则将残差的可逆雅可比矩阵分解为可逆稀疏局部雅可比矩阵。其动机源于带宽为2m+1（其中m<<n）的局部雅可比矩阵背景。观察到计算成本降低了$\mathcal{O}(\frac{n}{m})$。针对三对角情况给出了支持性运行时间测量，结果显示计算成本降低了$\mathcal{O}(n)$。推广后得到组合无矩阵牛顿步骤问题。我们证明了该问题是NP完全的，并提出了若干算法组件，用于构建其近似解的方法。受伴随算法微分的启发，新方法与后者共享若干挑战，包括有向无环图反转问题。其他挑战则源于稀疏线性代数中的组合问题，例如带宽或直接消元排序。