  A matrix framework is presented for the solution of ODEs, including initial-,
boundary and inner-value problems. The framework enables the solution of the
ODEs for arbitrary nodes. There are four key issues involved in the formulation
of the framework: the use of a Lanczos process with complete
reorthogonalization for the synthesis of discrete orthonormal polynomials (DOP)
orthogonal over arbitrary nodes within the unit circle on the complex plane; a
consistent definition of a local differentiating matrix which implements a
uniform degree of approximation over the complete support --- this is
particularly important for initial and boundary value problems; a method of
computing a set of constraints as a constraining matrix and a method to
generate orthonormal admissible functions from the constraints and a DOP
matrix; the formulation of the solution to the ODEs as a least squares problem.
The computation of the solution is a direct matrix method. The worst case
maximum number of computations required to obtain the solution is known
a-priori. This makes the method, by definition, suitable for real-time
applications.
  The functionality of the framework is demonstrated using a selection of
initial value problems, Sturm-Liouville problems and a classical Engineering
boundary value problem. The framework is, however, generally formulated and is
applicable to countless differential equation problems.
