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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.

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