  We extend the framework for complexity of operators in analysis devised by
Kawamura and Cook (2012) to allow for the treatment of a wider class of
representations. The main novelty is to endow represented spaces of interest
with an additional function on names, called a parameter, which measures the
complexity of a given name. This parameter generalises the size function which
is usually used in second-order complexity theory and therefore also central to
the framework of Kawamura and Cook. The complexity of an algorithm is measured
in terms of its running time as a second-order function in the parameter, as
well as in terms of how much it increases the complexity of a given name, as
measured by the parameters on the input and output side.
  As an application we develop a rigorous computational complexity theory for
interval computation. In the framework of Kawamura and Cook the representation
of real numbers based on nested interval enclosures does not yield a reasonable
complexity theory. In our new framework this representation is polytime
equivalent to the usual Cauchy representation based on dyadic rational
approximation. By contrast, the representation of continuous real functions
based on interval enclosures is strictly smaller in the polytime reducibility
lattice than the usual representation, which encodes a modulus of continuity.
Furthermore, the function space representation based on interval enclosures is
optimal in the sense that it contains the minimal amount of information amongst
those representations which render evaluation polytime computable.
