The calculation of an n-dimensional Newton step for a vector-valued function is a fundamental computation in scientific computing, computational finance, and engineering (as well as applied mathematics). It is often the dominant cost in the overall computational problem. Many vector-valued functions, representing expensive computations, are also structured computations. In this case the calculation of the Newton step can be greatly accelerated by exploiting this structure. It is often not necessary, nor economic, to form the true Jacobian in the process of computing the Newton step; instead, a more cost-effective auxiliary Jacobian matrix is used. This auxiliary matrix can be sparse even when the true Jacobian matrix is dense; consequently, sparse matrix technology can be used, to great speed advantage, both in forming the matrix and in solving the auxiliary linear system.