Re: Numerical Integration of Large Expression
- To: mathgroup at smc.vnet.net
- Subject: [mg43331] Re: Numerical Integration of Large Expression
- From: bobhanlon at aol.com (Bob Hanlon)
- Date: Mon, 25 Aug 2003 04:10:42 -0400 (EDT)
- References: <bi9vc0$cil$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
If I understood what you are asking, this can be integrated symbolically. For
example,
Simplify[Integrate[(HermiteH[9, x]*Exp[-a*x])^2, x]]
-((1/a^19)*((256*(512*a^18*x^18 + 4608*a^17*x^17 - 2304*a^16*(8*a^2 - 17)*x^16
-
18432*a^15*(8*a^2 - 17)*x^15 + 13824*a^14*(19*a^4 - 80*a^2 + 170)*x^14 +
96768*a^13*(19*a^4 - 80*a^2 + 170)*x^13 - 16128*a^12*(118*a^6 - 741*a^4 +
3120*a^2 - 6630)*
x^12 - 96768*a^11*(118*a^6 - 741*a^4 + 3120*a^2 - 6630)*x^11 +
12096*a^10*(623*a^8 - 5192*a^6 + 32604*a^4 - 137280*a^2 + 291720)*x^10 +
60480*a^9*(623*a^8 - 5192*a^6 + 32604*a^4 - 137280*a^2 + 291720)*x^9 -
272160*a^8*(60*a^10 - 623*a^8 + 5192*a^6 - 32604*a^4 + 137280*a^2 -
291720)*x^8 -
1088640*a^7*(60*a^10 - 623*a^8 + 5192*a^6 - 32604*a^4 + 137280*a^2 -
291720)*x^7 +
635040*a^6*(29*a^12 - 360*a^10 + 3738*a^8 - 31152*a^6 + 195624*a^4 -
823680*a^2 + 1750320)*
x^6 + 1905120*a^5*(29*a^12 - 360*a^10 + 3738*a^8 - 31152*a^6 +
195624*a^4 - 823680*a^2 +
1750320)*x^5 - 4762800*a^4*(2*a^14 - 29*a^12 + 360*a^10 - 3738*a^8 +
31152*a^6 -
195624*a^4 + 823680*a^2 - 1750320)*x^4 - 9525600*a^3*(2*a^14 - 29*a^12
+ 360*a^10 -
3738*a^8 + 31152*a^6 - 195624*a^4 + 823680*a^2 - 1750320)*x^3 +
1786050*a^2*(a^16 - 16*a^14 + 232*a^12 - 2880*a^10 + 29904*a^8 -
249216*a^6 + 1564992*a^4 -
6589440*a^2 + 14002560)*x^2 + 1786050*a*(a^16 - 16*a^14 + 232*a^12 -
2880*a^10 + 29904*a^8 -
249216*a^6 + 1564992*a^4 - 6589440*a^2 + 14002560)*x +
893025*(a^16 - 16*a^14 + 232*a^12 - 2880*a^10 + 29904*a^8 - 249216*a^6 +
1564992*a^4 -
6589440*a^2 + 14002560)))/E^(2*a*x)))
Larger orders merely cause longer output, but can be handled. The
simplification step takes longer. However, you can store the results (function
definition) and need only do each integration once. Subsequent evaluations of
the stored result will be very fast.
Bob Hanlon
In article <bi9vc0$cil$1 at smc.vnet.net>, reallymadsquid at hotmail.com (Musaddiq
Awan) wrote:
<< I am trying to use Rayleigh-Schrodinger Perturbation theory to modify
the harmonic oscillator. The wavefunctions are a product of the
Hermite polynomials and an exponential function. I can integrate the
2nd order approximation up to n = 8. When I try to integrate for n = 9
the computer does not return a result. The Hermite Polynomial at n = 9
is as follows
30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9.
I am integrating this times an exponential the whole quantity squared.
Is there any suggestion on how to integrate this efficiently with a
computer. I was initially planning to go up to n = 50, and still hope
to achieve that possibility. Any help would be greatly appreciated.