Estimating conditioning of BVPs for ODEs

An alternative to control of the global error of a numerical solution to a boundary value problem (BVP) for ordinary differential equations (ODEs) is control of its residual, the amount by which it fails to satisfy the ODEs and boundary conditions. Among the methods used by codes that control residuals are collocation, Runge-Kutta methods with continuous extensions, and shooting. Specific codes that concern us are bvp4c of the Matlab problem solving environment and the FORTRAN code MIRKDC for general scientific computation. The residual of a numerical solution is related to its global error by a conditioning constant. In this paper, we investigate a conditioning constant appropriate for BVP solvers that control residuals and show how to estimate it numerically at a modest cost. that control residuals can compute pseudosolutions, numerical solutions to BVPs that do not have solutions. That is, a “well-behaved” approximate solution is computed for an ill-posed mathematical problem. The estimate of conditioning is used to improve the robustness of bvp4c and MIRKDC and in particular, help users identify when a pseudosolution may have been computed.