Relationship between the Compound-Matrix and Riccati Methods for Solving Boundary-Value Problems.

摘要

Numerically difficult boundary-value problems for linear ODE systems can be solved with the compound-matrix and Riccati methods. A drawback with the compound-matrix method is that the dimensions of the ODE systems for the compound quantities can become large. Quadratic relations for these quantities are derived and used to reduce the systems dimensions. The Riccati method appears, and this method is viewed as a dimension-reduction technique in the context of the compound-matrix method. Certain singularity problems with the Riccati method are naturally handled by inclusion of an additional forward-sweep equation to allow reconstruction of all compound-vector elements. In particular, argument-variation techniques for determination of eigenvalues become applicable. For applications to underwater acoustics, the compound-matrix method is ideally suited to handle wave propagation through a layered solid isotropic bottom.