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MathGroup Archive 2005

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Re: Why this function does not return a single value

  • To: mathgroup at smc.vnet.net
  • Subject: [mg60195] Re: Why this function does not return a single value
  • From: Marek <marekbr at gmail.com>
  • Date: Wed, 7 Sep 2005 04:03:53 -0400 (EDT)
  • References: <dfiv9c$p99$1@smc.vnet.net> <dfj9h7$rga$1@smc.vnet.net>
  • Reply-to: marekbr at gmail.com
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Paul

Thanks for reply. I do not understand Your comments fully yet but let me
read up on stuff. For now I have only one thing to say. Those
multiplication signs appear out of nowhere when I convert the call with
function definition from Input Form to Standard Form. I can remove the m
while in Standard form but as soon as I execute this cell (or convert it to
Standard Form) they reappear.
Is that the way it is suppose to be?

Marek

Paul Abbott wrote:

> In article <dfiv9c$p99$1 at smc.vnet.net>,
>  Marek Bromberek <marek at chopin.physics.mun.ca> wrote:
> 
>> I was wondering if somebody could tell me what am I doing wrong here.
> 
> A number of things:
> 
> [1] The syntax is incorrect. You cannot use (a[1]*_)?NumericQ. This
> means a[1] times a blank, which is not what you intend.
> 
> If you enter
> 
>   (x_)?NumericQ // FullForm
> 
> and
> 
>    (a[1]*_)?NumericQ // FullForm
> 
> you will see the difference.
> 
> [2] A much better syntax is to define a test for numerical lists, say
> 
>  num[c_List] := And @@ (NumericQ /@ c)
> 
> and then use it in your definition, say
> 
>  VoigtSum[x_?NumericQ, a_List?num, b_List?num, ...] := ...
> 
>> I am constructing a function (for fitting purposes) which is a sum of 15
>> Voigt functions + one Gaussian peak and and exponential background.
>> However when I want to check if that function works and try to evaluate
>> it it does not return a value.
> 
> [3] There is no need to restrict attention to 15 Voight functions or to
> test the length of each list. Now your arguments are lists, instead
> write your definition so that it works for _any_ number of Voight
> functions (mapping the function over the lists of arguments), and use
> the last two elements of the lists a, b, and \[Delta]G to give the
> Gaussian peak and exponential background, i.e.,
> 
>   a[[-2]] Exp[(-Log[2]) ((x - b[[-2]])^2/\[Delta]G[[-2]]^2)] +
> 
>   a[[-1]] Exp[-x/b[[-1]]]
> 
> [4] You have
> 
>> Sum[a[i]*((2.*Log[2.]*\[Delta]L[i])/(N[Pi^(3/2)]*\[Delta]G[i]))*
>>      NIntegrate[Exp[-t^2]/((Sqrt[Log[2.]]*(\[Delta]L[i]/\[Delta]G[i]))^2
>>      +
>>         (Sqrt[4.*Log[2.]]*((x - b[i])/\[Delta]G[i]) - t)^2), {t,
>>         -Infinity,
>> Infinity}], {i, 1, 15}] +
>>    a[16]*Exp[(-Log[2.])*((x - b[16])^2/\[Delta]G[16]^2)] +
>> a[17]*Exp[-x/b[17]]
> 
> There is no need -- and it is usually a bad idea -- to numericalize
> constants. Since you are computing a numerical integral, all exact
> numerical values will be coerced to numerical ones (of the same
> precision). Try entering
> 
>   2. Pi
> 
> to see what I mean.
> 
> [5] Although Mathmatica cannot compute your integral directly,
> 
>   Integrate[Exp[-t^2]/((Sqrt[Log[2.]] (\[Delta]L[i]/\[Delta]G[i]))^2 +
>     (Sqrt[4.*Log[2.]]*((x - b[i])/\[Delta]G[i]) - t)^2),
>      {t, -Infinity, Infinity}]
> 
> or the equivalent integral
> 
>   Integrate[Exp[-t^2]/(a^2 + (b - t)^2), {t, -Infinity, Infinity}]
> 
> it is possible to compute this in closed form (at least for a and b
> real).  I get
> 
>  int[a_, b_] = -((1/(2 a)) (I (((-I) Pi Erf[a + I b] -
>    Log[1/(I a - b)] + Log[1/(b - I a)])/E^(b - I a)^2 -
>    (I Pi Erf[a - I b] - Log[-(1/(I a + b))] + Log[1/(I a + b)])/
>      E^(I a + b)^2)))
> 
> Although this involves I, numerical evaluation for real a and b (to any
> desired precision, followed by Chop, yields the same answer as
> 
>  nint[a_, b_, opts___] := NIntegrate[Exp[-t^2]/(a^2 + (b - t)^2),
>    {t, -Infinity, Infinity}, opts]
> 
> For example,
> 
>   nint[2,3]
>   0.14563
> 
>   int[2.,3.]//Chop
>   0.14563
> 
>   nint[2,3,WorkingPrecision->30]
>   0.14562973135699681308
> 
>   int[2`30,3`30]//Chop
>   0.1456297313569968130774404707
> 
> Cheers,
> Paul
> 
> _______________________________________________________________________
> Paul Abbott                                      Phone:  61 8 6488 2734
> School of Physics, M013                            Fax: +61 8 6488 1014
> The University of Western Australia         (CRICOS Provider No 00126G)
> AUSTRALIA                               http://physics.uwa.edu.au/~paul


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