Halphen's constant for approximation of exp(x)
Nick Trefethen, May 2011
(Chebfun example approx/Halphen.m)
A well-known problem in approximation theory is, how well can exp(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
error ~ 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.
Here is a plot showing that the asymptotic behavior matches the actual errors very closely even for small n:
LW = 'linewidth'; MS = 'markersize'; FS = 'fontsize'; 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',LW,1.2) hold on, semilogy(n,err,'.k',MS,18), grid on xlabel('n',FS,14), ylabel('error',FS,14)
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 from k=1 to infty of ks^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,LW,1.2), grid on h = 1/roots(f-1/8); hold on, plot(h,1/8,'.r',MS,20) title('Halphen''s constant',FS,14) text(h,.15,sprintf('%16.13f',h),FS,14)
 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 Aprpoximation and Interpolation, Lecture Notes in Mathematics 1005, Springer, 1984.
 A. A. Gonchar and E. A. Rakhmanov, Equilibrium distributions and degree of rational approximation of analytic functions, Math. USSR Sbornik 62 (1989), 305-348.
 L. N. Trefethen, Approximation Theory and Approximation Practice, draft available at http://www.maths.ox.ac.uk/chebfun/ATAP (chapter 24).