Re: Re: Write an expression in a specific form
- To: mathgroup at smc.vnet.net
- Subject: [mg108110] Re: [mg108025] Re: Write an expression in a specific form
- From: dh <dh at metrohm.com>
- Date: Mon, 8 Mar 2010 06:16:35 -0500 (EST)
- References: <hmo293$qbh$1@smc.vnet.net> <201003050935.EAA29559@smc.vnet.net> <op.u83gafrbtgfoz2@bobbys-imac.local>
Hi Bob,
you were not supposed to integrate the product of two Gaussian. But the
product of two Gaussian is another Gaussian whose parameters are asked.
Daniel
On 05.03.2010 13:14, DrMajorBob wrote:
> I missed where c4 is defined in that... but starting out, we know it has
> to be equal to
>
> G[m_,s_,x_]:=(1/(s*Sqrt[2*Pi]))*Exp[-(x-m)^2/(2*s^2)];
> Integrate[G[m1,s1,x]
> G[m2,s2,x],{x,-Infinity,Infinity},Assumptions->{s1>0,s2>0}]
>
> E^(-((m1-m2)^2/(2 (s1^2+s2^2))))/(Sqrt[2 \[Pi]] Sqrt[s1^2+s2^2])
>
> No?
>
> I think c4 was supposed to be c3, and sure enough, retrieving the RHS of
> the rule for c3, I have:
>
> (E^(-((m1-m2)^2/(2 (s1^2+s2^2)))) Sqrt[(s1^2 s2^2)/(s1^2+s2^2)])/(Sqrt[2
> \[Pi]] s1 s2)//PowerExpand
>
> E^(-((m1-m2)^2/(2 (s1^2+s2^2))))/(Sqrt[2 \[Pi]] Sqrt[s1^2+s2^2])
>
> Bobby
>
> On Fri, 05 Mar 2010 03:35:13 -0600, dh <dh at metrohm.com> wrote:
>
>> Hi Ares,
>> Mathematica really seems to have difficulties (that is I do not have
>> the nerves
>> to wait so long) with a calculation that can done by hand. It look like
>> it needs some human help.
>> We may get an equation for the exponents with only 2 unknowns.
>> We denot by m1,s1 and m2,s2 the parameters of the given Gaussian. m3,s3
>> and c3 denote the searched Gaussian: c4 G[m3,s3]. tmp is an intermediate
>> result we are not interested in:
>> sol1 = Reduce[{ForAll[
>> x, (x - m1)^2/(2 s1^2) + (x - m2)^2/(2 s2^2) ==
>> tmp + (x - m3)^2/(2 s3^2)], s1 > 0, s2 > 0, s3 > 0}, {m3, s3, tmp},
>> Reals]
>> this gives us m3,s3 and an intermediate result tmp. With this we may
>> solve the equation for the pre factor. Toward this aim we change sol1 to
>> rules and use it in the equation:
>> G[m1, s1, x] G[m2, s2, x] == c3 G[m3, s3, x] /. sol1 /. sol1
>> Note that in sol1 tmp is given in term of m3,s3. We must therefore apply
>> /.sol1 twice.
>> I also delete some superfluous info from the result. Here is the whole
>> code:
>> ================
>> G[m_, s_, x_] := (1/(s*Sqrt[2*Pi]))*Exp[-(x - m)^2/(2*s^2)];
>> sol1 = Reduce[{ForAll[
>> x, -((x - m1)^2/(2 s1^2)) - (x - m2)^2/(2 s2^2) ==
>> tmp - (x - m3)^2/(2 s3^2)], s1 > 0, s2 > 0, s3 > 0}, {m3, s3,
>> tmp}, Reals];
>> sol1 = Drop[sol1, 2] // ToRules
>> sol2 = Reduce[{G[m1, s1, x] G[m2, s2, x] == c3 G[m3, s3, x] /.
>> sol1 /. sol1, s1 > 0, s2 > 0, s3 > 0, c3 > 0}, {c3},
>> Reals][[4]] // ToRules
>> ===============
>>
>> Finally we may test if the calculation is correct:
>> G[m1, s1, x] G[m2, s2, x] == c4 G[m3, s3, x] /. sol2 /. sol1 /.
>> sol1 // Simplify
>>
>> Daniel
>>
>> On 04.03.2010 11:33, Ares Lagae wrote:
>>> Hi all,
>>>
>>> I am a beginner in Mathematica, and I have the following "problem":
>>> How can
>>> I write an expression in a specific form?
>>>
>>> For example:
>>>
>>> - Define a Gaussian:
>>>
>>> G[m_, s_, x_] := (1/(s*Sqrt[2*Pi]))*Exp[-(x - m)^2/(2*s^2)];
>>>
>>> - Product of two Gaussians:
>>>
>>> G[m1, s1, x] * G[m2, s2, x]
>>>
>>> - How can I get Mathematica to write the result in terms of c * G[m_,
>>> s_,
>>> x_]? I.e., get the values for c, m and s.
>>>
>>> Thanks,
>>>
>>> Ares Lagae
>>>
>>>
>>
>>
>
>
--
Daniel Huber
Metrohm AG
International Headquarters
Oberdorfstr. 68, CH-9101 Herisau / Switzerland
Phone +41 71 353 8606, Fax +41 71 353 89 01
Mail <mailto:dh at metrohm.com>
Web <http://www.metrohm.com
- References:
- Re: Write an expression in a specific form
- From: dh <dh@metrohm.com>
- Re: Write an expression in a specific form