Continuous Orthonormalization Algorithms for Boundary Value Problems

Introduction. The di#culties with the simple shooting method for the solution of ordinary di#erential boundary value problems are well known, and there have been many suggested modifications to overcome them. Notable among them are the discrete orthonormalization process of Godunov and Conte (1966), the multiple shooting method (Keller(1968) and Osborne(1969)) and the Riccati transformation method (Scott(1975)). Although suggested by Abramov(1962) and discussed by Bakhvalov(1973) the continuous orthonormalization method has recieved much less attention. Essentially the same idea seems to have been independently suggested by Drury (1980) and used by Davey(1983) and Meyer(1986). Recently Van Loon (1985) and (1987) has arrived at a closely related method from a rather di#erent point of view. He concentrates on a change of variable in the di#erential equation using an orthogonal matrix rather than on orthogonalizing sets of solutions. Here it is intended to explore the relationships be