Some theoretical properties of Feng-Schnabel algorithm for block bordered nonlinear systems

Large and sparse nonlinear systems arise in many areas of science and technology, very often as a core process for the model of a real world problem. Newton-like approaches to their solution imply the computation of a (possibly approximated) Jacobian: in the case of block bordered systems this results in a matrix with disjoint square blocks on the main diagonal, plus a final set of rows and columns. This sparsity class allows to develop multistage Newton-like methods (with inner and outer iterations) that are very suitable for a parallel implementation ou multiprocessors computers. Recently, Feng and Schnabel proposed an algorithm which is actually the state of the art in this field. In this paper we analyze in depth important theoretical properties of the steps generated by the Feng-Schnabel algorithm. Then we study a cheap modification that gives an improvement of the direction properties, allowing a global convergence result, as well as the extension of the convergence to a broader class of algorithms, in which different linesearch globalization rules can be applied