This thesis investigates the use of a particular class of quasi-Newton method based on direct updating of both factors of the Jacobian matrix, first proposed by Hart. Using a common method, our research proves its superlinear convergence property. Our numerical results also suggest that it outperforms the Dennis and Marwil update. Applied to highly-structured nonlinear equations arising from the discretization of two-point boundary-value problems, our numerical results suggest that the Hart update is also (at least) competitive to the sparse Broyden (Schubert's) update. For this and other reasons we propose that the Hartupdate is also (at least) competititve to the sparse Broyden (Schubert's ) update. For this and other reasons we propose that the Hart update is made available as an option in the BVPSOL algorithm. Applied to uncontrained minimization, we propose two symmetrization schemes. We describe our numerical experience and give a framework for a host of new research directions. We worked in two computing environment : the location- andthe content-addresssable systems. For the latter, we used the Intelligent File Store (IFS). Since for the IFS our research represents a new direction, we identify the basic structure upon which optimal IFS-numerical computing algorithms should be developed. We find the sharp separation between the content-addressable memory (of the IFS) and the arithmetic unit (of the host computer) restricts the scope of its cost effectiveness. We find the Doolittle algorithm represents the optimal IFS - linear-solver algorithm upon which the Hart update could be added to implement the sparse nonlinear equation-solving algorithm. We propose a set of boundary-value problems to be used as a basis to develop a standard test set. Finally, we make this thesis a contribution to numerical computing of large scale systems, which, in the present state of world affairs, is becoming more significant.