Re: "Solve[x==Erf[x], x]"
- To: mathgroup at smc.vnet.net
- Subject: [mg16612] Re: "Solve[x==Erf[x], x]"
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Fri, 19 Mar 1999 12:53:45 -0500
- Organization: University of Western Australia
- References: <7cd7ff$o7s@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Matthias Tomann wrote:
> Can you tell me how to get around this problem an solve equations like
> x = 1 - Erf [x]
Several other respondents have indicated using FindRoot and Series
methods. I'll just add that solving the more general problem with 1
replaced by a parameter, a,
In[1]:= sol[a_] := Module[{x},x /. FindRoot[x - a + Erf[x], {x, a}]]
allows you to plot the dependence of the solution on a:
In[2]:= Plot[{a - 1, a + 1, sol[a]}, {a, -7, 7},
AspectRatio -> Automatic, PlotStyle ->
{Hue[0], Hue[1/3], Hue[2/3]}];
As you will see, the behaviour of the solution is quite simple. This is
because Erf[x] approximates a step function rather well:
In[3]:= Plot[Erf[x], {x, -5, 5}];
Expanding f[x] = x - a + Erf[x] in series about x = a,
In[4]:= x - a + Erf[x] + O[x, a]^3
Out[4]=
2
2 2 a (-a + x)
Erf[a] + (1 + ------------) (-a + x) - ------------- +
2 2
a a
E Sqrt[Pi] E Sqrt[Pi]
3
O[-a + x]
and then computing the inverse series,
In[5]:= InverseSeries[%]
Out[5]=
x - Erf[a]
a + ---------------- +
2
1 + ------------
2
a
E Sqrt[Pi]
2
2 a (x - Erf[a]) 3
-------------------------------------- + O[x - Erf[a]]
2
2 2 a
(1 + ------------) (2 + E Sqrt[Pi])
2
a
E Sqrt[Pi]
In[6]:= Normal[%] /. x -> 0
Out[6]=
2
Erf[a] 2 a Erf[a]
a - ---------------- + --------------------------------------
2 2
1 + ------------ 2 2 a
2 (1 + ------------) (2 + E Sqrt[Pi])
a 2
E Sqrt[Pi] a
E Sqrt[Pi]
gives a very good approximation for a wide range of values, e.g.,
In[7]:= % /. a -> 3.
Out[7]= 2.00058
which should be compared with
In[8]:= sol[3]
Out[8]= 2.00458
Cheers,
Paul
____________________________________________________________________
Paul Abbott Phone: +61-8-9380-2734
Department of Physics Fax: +61-8-9380-1014
The University of Western Australia
Nedlands WA 6907 mailto:paul at physics.uwa.edu.au
AUSTRALIA http://www.physics.uwa.edu.au/~paul
God IS a weakly left-handed dice player
____________________________________________________________________