Some Remarks on the Deflated Block Elimination Method

Abstract

In many algorithms for computing singular points (or solution curves near such points) it is necessary to solve linear systems of the form

$$ \text M\;\;\begin{pmatrix}\underset \sim{\text x}\\\text y \end{pmatrix} \equiv \begin{pmatrix} \text A\;\;\;\; \underset \sim {\text b}\\ \underset \sim {\text c}^\top \;\; \text d \end{pmatrix} \begin{pmatrix}\underset \sim{\text x}\\\text y \end{pmatrix}\;=\; \begin{pmatrix}\underset \sim{\text f}\\\text g \end{pmatrix}\;\;. $$

(1.1)

Here A is an N×N matrix; $\underset \sim {\text b}, \;\underset \sim {\text c}, \;\underset \sim {\text x} \;\text {and}\; \underset \sim {\text f}$ N-vectors; and d, y and g scalars. The complete (N+1)×(N+1) coefficient matrix M is assumed to be well-conditioned, but A is expected to be singular with a one-dimensional null-space, or to have one relatively small singular value, for problems of interest. A, however, is also assumed to possess useful properties which may be lost if the augmented matrix M is dealt with directly; e.g. A might have a band structure which could be destroyed by pivoting on certain elements of M. Thus, for efficiency’s sake, we only wish to solve linear systems with coefficient matrix A, but its possible ill-condition demands special care.