Cyclic reduction, dichotomy, and the estimation of differential equations

In the estimation of ordinary differential equations given observed data it is necessary to parametrize the solutions in a computationally tractable way in order to make comparisons with the given data. Typically, this is done by adjoining initial or boundary conditions, and this requires additional information on the solution structure in order to do this in a manner that leads to stable computation of the comparison solutions. Here, an alternative approach is described in which cyclic reduction is used to reduce the estimation problem to an optimization problem subject to a fixed number of equality constraints. If orthogonal transformations are used in the cyclic reduction process then it appears that stable computations are possible without the need for the structural information needed to devise the stable imbeddings. The aim of this paper is to provide evidence in support of this claim. In particular, it is shown that the cyclic reduction process is linked to a new family of representation of the solutions of the ODE system. Some properties of members of this family are described. These provide insight into the particular advantages of the orthogonal reduction form of cyclic reduction.