Conditioning analysis of modified block incomplete factorizations (Q1176529)

The paper brings new and better results in the conditioning analysis of modified block incomplete factorizations of Stieltjes matrices, whose definition is extended to cover the singular case. For a Stieltjes (\(n\times n\))-matrix \(A\) and a partition \(\pi\) of the set \(\{1,2,..,n\}\) an algorithm is presented for computing a modified incomplete factorization \(B=U^ tP^ +U\), where \(P\) is \(\pi\)-diagonal and \(U\) is \(\pi\)-upper triangular. The main theorem gives the conditions under which \(N(B)\subseteq N(A)\) and an upper bound for the eigenvalues of \(B^ +A\). This result is then applied to a system matrix \(A\) associated with a two- dimensional elliptic partial differential equation of the form \(- \nabla[a\nabla u]+cu=f\) with mixed boundary conditions, and it is shown that the spectral condition number is bounded by \(N\), the number of gridpoints in the \(y\)-direction.