Order stars are a beautiful idea of complex analysis that resolved several open conjectures when they were introduced in 1978 by Wanner, Hairer, and Norsett [1,2,3]. Chebfun is not really a very good tool for illustrating them, since there are poles involved that must be smashed away, but let us give it a go.

Let $R(z)$ be a function of the complex variable $z$. The order star of $R$ is the region bounded by the curve(s) in the plane satisfying the condition

$$ | e^{-z} R(z)| = 1 . $$

For example, here is a function handle for the type $(2,3)$ Pade approximant of $e^z$:

c = 1./factorial(0:18);
r = padeapprox(c,2,3);

We can use this mollifying function to turn poles into constants while preserving the absolute value $f=1$:

smash = @(f) tanh(abs(f).^2)/tanh(1);

Now we can plot the order star like this:

d = 6*[-1 1 -1 1];
f = chebfun2(@(z) smash(r(z).*exp(-z)),d);
star = roots(f-1);
axis(d), axis square

Such figures reveal important properties of the function $R$. For example, the meeting of 12 sectors at the origin reflects the 6th-order agreement of the Pade approximant with $e^z$,

$$ e^z - R(z) = O(z^6). $$


  1. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, 2nd revised ed., Springer, 1996.

  2. A. Iserles and S. P. Norsett, Order Stars, Chapman and Hall, 1991.

  3. G. Wanner, E. Hairer, and S. P. Norsett, Order stars and stability theorems, BIT, 18 (1978), 475-489.