The objective of this study is to present a shape-preserving non-linear
subdivision scheme
generalizing the exponential B-spline of degree 3,
which is a piecewise exponential polynomial with the same support
as the cubic B-spline.
The subdivision of the exponential B-spline
has a crucial limitation in that it can reproduce at most two
exponential polynomials, yielding the approximation order {\em two}.
Also, finding a best-fitting shape parameter in the exponential
B-spline is a challenging and important problem.
In this regard, we present a method for selecting an optimal shape parameter
and then formulate it in the construction of new refinement rules.
As a result, the new scheme provides an improved approximation order
{\em four} while maintaining the same $C^2$ smoothness
as the (exponential) B-spline of degree 3.
Moreover, we show that the proposed method preserves geometrically
important characteristics such as
monotonicity and convexity, under some suitable conditions.
Some numerical examples are provided to demonstrate the ability of the
new subdivision scheme.