On computing INV block preconditionings for the conjugate gradient method (Q1088378)

The INV(k) and MINV(k) block preconditionings for the conjugate gradient method require generation of selected elements of the inverses of symmetric matrices of bandwidth \(2k+1\). Generalizing the previously described \(k=1\) (tridiagonal) case to \(k=2\), explicit expressions for the inverse elements of a symmetric pentadiagonal matrix in terms of Green's matrix of rank two are given. The expressions are found to be seriously ill-conditioned; hence alternative computational algorithms for the inverse elements must be used. Behavior of the \(k=1\) and \(k=2\) preconditionings are compared for some discretized elliptic partial differential equation test problems in two dimensions.