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

TL;DR

This paper evaluates a method that uses the Sherman-Morrison-Woodbury formula to solve linear systems from the multiple shooting method for boundary-value problems, concluding it is less stable and slower than existing methods.

Contribution

The paper analyzes the stability and efficiency of applying the Sherman-Morrison-Woodbury formula in this context and finds it inferior to traditional condensing methods.

Findings

The method shares stability issues with the condensing method.

It is slower than the condensing method.

The method is not recommended for practical use.

Abstract

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.