Using the Sherman-Morrison-Woodbury Formula to Solve the System of Linear Equations from the Standard Multiple Shooting Method for a Linear Two Point Boundary-Value Problem is a Bad Idea
This paper provides a negative result for a specific numerical method in solving linear boundary-value problems, but it is incremental as it only confirms known issues.
The authors show that using the Sherman-Morrison-Woodbury formula to solve the linear system from multiple shooting for linear BVPs has the same stability issues as the Condensing method and is slower, concluding it is a bad idea.
We use the standard multiple shooting method to solve a linear two point boundary-value problem. To ensure that the solution obtained by combining the partial solutions is continuous and satisfies the boundary conditions, we have to solve a system of linear equations. Our idea is to first solve a bidiagonal system related to the original system of linear equations, and then update it with the Sherman-Morrison-Woodbury formula. We study the feasibility, the numerical stability and the running time of this method. The results are: The method described above has the same stability problems like the well known Condensing method. The running time analysis shows that the new method is slower than the Condensing method. Therefore we recommend not to use the method described in this article.