       Re: Need zeros of Hermite type 'e' Polynomials

• To: mathgroup at smc.vnet.net
• Subject: [mg5889] Re: [mg5159] Need zeros of Hermite type 'e' Polynomials
• From: Roland Franzius <Roland.Franzius at physik.uni-osnabrueck.de>
• Date: Sun, 2 Feb 1997 01:32:53 -0500
• Sender: owner-wri-mathgroup at wolfram.com

```[mg5159] Need zeros of Hermite type 'e' polynomials

Dear Mike, I don=B4t know if you have got an answer already. You don=B4t =
say if you are=20
satisfied with numerical results. (Zeros of polynomials of degree > 4 can=
not be given=20
in closed form except in special cases). The solution in Mathematica look=
s something=20
like this:

Use the recurrence formula for Hn: H(n+1,x) =3D 2x H(n,x) - 2 H(n-1,x) to=
define
the standard hermite polynomials

In:=3D

Clear[h0,h1,h,x,n]
h0[x_]:=3D1
h1[x_]:=3D2x
h[n_/;n>1,x_]:=3Dh[n,x]=3D 2 x h[n-1,x] -2 h[n-2,x]
h[1,x_]:=3Dh1[x]
h[0,x_]:=3Dh0[x]

In:=3D
h6[x_]=3Dh[6,x]//Expand
Out:=3D
-8 + 96*x^2 - 160*x^4 + 64*x^6
In:=3D
h8[x_]=3Dh[8,x]//Expand

Out:=3D
16 - 320*x^2 + 960*x^4 - 896*x^6 + 256*x^8

(* Find starting points for approximation of zeros *)
In:=3D
Plot[h6[x],{x,0,1.5}]

Ou:=3D
Graphics["<<>>"]

(* read them off the plot *)

In:=3D
roots6=3D{{x,0.3},{x,0.9},{x,1.2}};

In:=3D
FR6=3DFindRoot[h6[x],#]&;

In:=3D
Map[FR6,roots6]

Out:=3D
{{x -> 0.3146921227129476},=20
{x -> 0.881747733789935},=20
{x -> 1.274162392263535}}

In:=3D
Plot[h8[x],{x,0,1.5}]

Out:=3D
Graphics["<<>>"]

In:=3D
roots8=3D{{x,0.2},{x,0.7},{x,1.1},{x,1.3}};

In:=3D
FR8=3DFindRoot[h8[x],#]&;

In:=3D
Map[FR8,roots8]

Out:=3D
{{x -> 0.2455756079379457},=20
{x -> 0.7071067811865478},=20
{x -> 1.083350440839404},=20
{x -> 1.328926048777349}}

(*the roots are symmetric wrt. to x=3D0 and the same for Hn and He *)

Useful theorems: Hn has n real distinct roots. Roots of Hn+1 are located
between those of Hn plus 2 roots of absolute value greater than those of =
Hn.
=04
Hope, it will help=20
Roland

Roland Franzius
Dept. of Physics
Univ. of Osnabrueck, Germany
Roland.Franzius at physik.uni-osnabrueck.de

```

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