Automatic coordinate transformation for two-point boundary value problems

When one is attempting to build a general boundary value problem solver which uses Green's functions, it is often desirable to automatically transform inhomogeneous boundary value problems into homogeneous boundary value problems If the sum of the boundary matrices is singular, a change of coordinates is required which can transform the inhomogeneous boundary value problem into a homogeneous boundary value problem. This change of coordinates is an invertible C¹ path in matrix space with endpoints that can multiply the boundary matrices to produce a nonsingular sum. Here we propose a simple automatic coordinate transformation that performs this task. We prove that this algorithm constructs a coordinate transformation whenever such a transformation exists, and provide numerical examples illustrating the practicability of the method.