REMOVAL OF THE SINGULARITIES FROM THE RICCATI METHOD

When the Riccati method is used to solve a difficult linear homogeneous two-point boundary value problem it is frequently necessary to switch between the Riccati matrix and its inverse matrix when a singularity of either is approached. This switching causes a loss of numerical accuracy partly because it is not easy to decide exactly when to switch and it is generally rather a nuisance, especially if it is desired to calculate the eigenfunction as well as the eigenvalue. Herein we point out that these singularities may be removed by considering the differential equations for the numerators and the denominators separately of the elements of the Riccati matrix and its inverse. It transpires that this reformulation of the Riccati method is just the compound matrix method advocated by Gilbert and Backus and rediscovered and used by Ng and Reid. We give a brief discussion of some features of the compound matrix method and we explain why it enables the standard shooting method to be used. © 1979.