The Close Relationships between Methods for Solving Two-Point Boundary Value Problems

The well-conditioning of a boundary value problem is shown to be related to bounding two quantities, one involving the boundary conditions and the other involving the Green’s function. In the case of a solution dichotomy, they are related to known stability results. These results easily explain why difficulties arise using superposition and reduced superposition. It is shown how multiple shooting overcomes these difficulties by relating its matrix conditioning to the underlying boundary value problem. Some factorization methods for the multiple shooting method are considered. These show the close relationships between multiple shooting, the stabilized march, and invariant imbedding, and one factorization leads to an efficient new way to implement each of these three methods. Many of the results apply for collocation and finite difference methods, too. Finally, it is shown, both theroetically and computationally, when matrix compactification can lead to difficulty.

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