# 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)
``` References:

 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).