The Bordering Algorithm and Path Following Near Singular Points of Higher Nullity

We study the behavior of the bordering algorithm (a form of block elimination) for solving nonsingular linear systems with coefficient matrices in the partitioned form $\left( {\begin{array}{*{20}c} A & B \\ {C^ * } & D \\ \end{array} } \right)$ when $\dim \mathcal{N}(A) \geqq 1$. Systems with this structure naturally occur in path following procedures. We show that under appropriate assumptions, the algorithm, which is based on solving systems with coefficient matrix A, works as A varies along a path and goes through singular points. The required assumptions are justified for a large class of problems coming from discretizations of boundary value problems for differential equations.