  Local fixpoint iteration describes a technique that restricts fixpoint
iteration in function spaces to needed arguments only. It has been studied well
for first-order functions in abstract interpretation and also in model
checking. Here we consider the problem for least and greatest fixpoints of
arbitrary type order. We define an abstract algebra of simply typed
higher-order functions with fixpoints that can express fixpoint evaluation
problems as they occur routinely in various applications, including program
verification. We present an algorithm that realises local fixpoint iteration
for such higher-order fixpoints, prove its correctness and study its
optimisation potential in the context of several applications.
