A well-known problem in approximation theory is, how well can $e^x$ be approximated in the infinity norm on the infinite interval $(-\infty,0]$ by rational functions of type $(n,n)$? To three places, the first few approximation errors are these:

  • Type $(0,0)$: error = $0.500$

  • Type $(1,1)$: error = $0.0668$

  • Type $(2,2)$: error = $0.00736$

  • Type $(3,3)$: error = $0.000799$

  • Type $(4,4)$: error = $0.0000865$

  • Type $(5,5)$: error = $0.00000934$

  • Type $(6,6)$: error = $0.000001008$

  • Type $(7,7)$: error = $0.0000001087$

  • Type $(8,8)$: error = $0.00000001172$

As $n$ increases to infinity, it is known that the asymptotic behavior is

$$ \mathrm{error} \sim 2 C^{-n-1/2}, $$

where $C$ is a number known as Halphen's constant with the following approximate numerical value:

halphen_const = 9.289025491920818918755449435951
halphen_const =
   9.289025491920819

This result comes from a sequence of contributions between 1969 and 2002 by, among others, Cody, Meinardus and Varga; Newman; Trefethen and Gutknecht; Carpenter, Ruttan and Varga; Magnus; Gonchar and Rakhmanov; and Aptekarev. For a discussion, see Chapter 25 of [5].

Here is a plot showing that the asymptotic behavior matches the actual errors very closely even for small $n$:

n = 0:10;
err = [.5 .0668 7.36e-3 7.99e-4 8.65e-5 9.35e-6 ...
       1.01e-6 1.09e-7 1.17e-8 1.26e-9 1.36e-10];
model = 2*halphen_const.^(-n-.5);
hold off, semilogy(n,model,'-b')
hold on, semilogy(n,err,'.k','markersize',14), grid on
xlabel n, ylabel error

One way to characterize Halphen's constant mathematically is that it is the inverse of the unique positive value of $s$ where the function

$$ \sum_{k=1}^\infty \frac{k s^n}{1-(-s)^n} $$

takes the value $1/8$. This is an easy computation for Chebfun:

s = chebfun('s',[1/12,1/6]);
f = 0*s; k = 0; normsk = 999;
while normsk > 1e-16
  k = k+1;
  sk = s^k;
  f = f + k*sk/(1-(-1)^k*sk);
  normsk = norm(sk,inf);
end
hold off, plot(1/s,f), grid on
h = 1/roots(f-1/8);
hold on, plot(h,1/8,'.r','markersize',24)
title('Halphen''s constant')
text(h,.135,sprintf('%16.13f',h))

Halphen's constant appears more generally than in approximation of $e^x$. Stahl and Schmelzer generalized it to a number of perturbed exponential functions and Nakatsukasa and Trefethen showed that it also governs the accuracy of rational approximations of $x^n$ on $[-1,1]$ [3]. The latter effect is explored in the Chebfun example "Rational approximation of monomials".

References

  1. A. J. Carpenter, A. Ruttan, and R. S. Varga, Extended numerical computations on the "1/9" conjecture in rational approximation theory, in P. Graves-Morris, E. B. Saff, and R. S. Varga, eds., Rational Approximation and Interpolation, Lecture Notes in Mathematics 1105, Springer, 1984.

  2. A. A. Gonchar and E. A. Rakhmanov, Equilibrium distributions and degree of rational approximation of analytic functions, Math. USSR Sbornik, 62 (1989), 305-348.

  3. Y. Nakatsukasa and L. N. Trefethen, Rational approximation of $x^n$, Proc. AMS, 146 (2018), 5219-5224.

  4. H. Stahl and T. Schmelzer, An extension of the '1/9' problem, J. Comp. Appl. Math., 233 (2009), 821-834.

  5. L. N. Trefethen, Approximation Theory and Approximation Practice, Extended Edition, SIAM, 2019.