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Re: Help Improving this integral calculation / solution

• To: mathgroup at smc.vnet.net
• Subject: [mg108142] Re: Help Improving this integral calculation / solution
• From: DOD <dcodea at gmail.com>
• Date: Tue, 9 Mar 2010 06:22:18 -0500 (EST)
• References: <201003070903.EAA00857@smc.vnet.net> <hn2m9q$41l$1@smc.vnet.net>

Thanks for all the tips and explanations.  As shown, these methods
work.  I actually finally got decent results  by doing the integration/
root solving simultaneously:

---
result=NMinimize[{ (1/gavg Integrate[nk k pg[k], {k, 1, \[Infinity]},
Assumptions -> 0 < t < 1] - t)^2,0<t<1} , t];
answer = t/.result[[2]]
---

And this found the t I wanted.  This was after haphazard reformulation
of the problem into various forms amenable to various built-in
functions  Thanks for these methods and insight into what the problem
was; I need to do this calculation for various parameters (d,v,a), so
any increase in speed is key.




On Mar 8, 5:16 am, DrMajorBob <btre... at austin.rr.com> wrote:
> Try this:
>
> {d, v, a} = Rationalize at {.7, .05, .1};
> nk = (a + (1 - a) k v t)/(d + a + (1 - a) k v t);
>
> pg[k_] = Exp[1 - k];
> gavg = Integrate[k pg[k], {k, 1, \[Infinity]}]
>
> 2
>
> result = 1/gavg Integrate[nk k pg[k], {k, 1, \[Infinity]},
>     Assumptions -> 0 < t < 1]
>
> (9 t (-70 + 9 t) -
>   11200 E^(1 + 160/(9 t)) ExpIntegralEi[-1 - 160/(9 t)])/(81 t^2)
>
> test = FullSimplify[result]
>
> 1 - 70/(9 t) - (
>   11200 E^(1 + 160/(9 t)) ExpIntegralEi[-1 - 160/(9 t)])/(81 t^2)
>
> FindRoot[test == t, {t, .1}]
>
> {t -> 0.142045}
>
> Bobby
>
>
>
>
>
> On Sun, 07 Mar 2010 03:03:39 -0600, DOD <dco... at gmail.com> wrote:
> > I have an integral I need to calculate, with one variable left
> > symbolic, and then use the result to find a numerical solution to an
> > equation, and the result of the integration is not giving me what I
> > need:
> > ------
> > d = .7;
> > v = .05;
> > a = .1;
> > nk = (a + (1 - a) k v t)/(d + a + (1 - a) k v t);
>
> > pg[k_] = Exp[1-k];
> > gavg = Integrate[k pg[k], {k, 1, \[Infinity]}]; (* This is just 2 *)
> > result =   1/gavg Integrate[nk k pg[k], {k, 1, \[Infinity]},
> > Assumptions -> 0 < t < 1];
> > test = FullSimplify[result]
>
> > FindRoot[test == t, {t, .1}]
>
> > -----
> > This code always gives up and stays at the initial guess.  So, I look
> > at the result of the integration, "result" (or it's simplified
> > version, test2) and calculated it for various values of t, and it is
> > always zero.
> > ----
> > test/.t->{.1,.2,.3}
> > ----
> > Output:{3.36999*10^66, 0., 0.}
> > -----
>
> > So that's a problem.  If I set t=.3, say, and the beginning, and
> > calculate the integral, I get 0.160047, which is clearly not zero.
> > -----
> > d = .7;
> > v = .05;
> > a = .1;
> > t = .3;
> > nk = (a + (1 - a) k v t)/(d + a + (1 - a) k v t);
>
> > pg[k_] = Exp[1 - k];
> > gavg =Integrate[k pg[k], {k, 1, \[Infinity]}];(*This is just 2*)
>
> > result = 1/gavg Integrate[nk k pg[k], {k, 1, \[Infinity]}]
> > ----
> > Output=0.160047
> > -----
>
> >  So there is a problem in the Integrate step.  So I want to find a
> > point where the output of that integral (result), as a function of t,
> > satisfies result=t.  Is there anyway to do this using built-in
> > functions?
>
> --
> DrMajor... at yahoo.com

• References:
  • Help Improving this integral calculation / solution
    • From: DOD <dcodea@gmail.com>