  We consider the problem of finding optimally stable polynomial approximations
to the exponential for application to one-step integration of initial value
ordinary and partial differential equations. The objective is to find the
largest stable step size and corresponding method for a given problem when the
spectrum of the initial value problem is known. The problem is expressed in
terms of a general least deviation feasibility problem. Its solution is
obtained by a new fast, accurate, and robust algorithm based on convex
optimization techniques. Global convergence of the algorithm is proven in the
case that the order of approximation is one and in the case that the spectrum
encloses a starlike region. Examples demonstrate the effectiveness of the
proposed algorithm even when these conditions are not satisfied.
