Convergence and Instability in PCG

Bordered almost block diagonal systems arise from discretizing a linearized rst-order system of n ordinary diierential equations in a two-point boundary value problem with non-separated boundary conditions. The discretization may use spline collocation, nite diierences, or multiple shooting. After internal condensation, if necessary, the bordered almost block diagonal system reduces to a standard nite diierence structure, which can be solved using a preconditioned conjugate gradient method based on a simple matrix splitting technique. This preconditioned conjugate gradient method is \guaranteed" to converge in at most 2n + 1 iterations. We exhibit a signiicant collection of two-point boundary value problems for which this preconditioned conjugate gradient method is unstable and hence convergence is not achieved. Abstract Bordered almost block diagonal systems arise from discretizing a linearized rst-order system of n ordinary diierential equations in a two-point boundary value problem with non-separated boundary conditions. The discretization may use spline collocation, nite diierences, or multiple shooting. After internal condensation, if necessary, the bordered almost block diagonal system reduces to a standard nite diierence structure, which can be solved using a preconditioned conjugate gradient method based on a simple matrix splitting technique. This preconditioned conjugate gradient method is \guaranteed" to converge in at most 2n + 1 iterations. We exhibit a signiicant collection of two-point boundary value problems for which this preconditioned conjugate gradient method is unstable and hence convergence is not achieved.