how to use Findminimum with a function defined numerically through a parameter

• To: mathgroup at smc.vnet.net
• Subject: [mg15607] how to use Findminimum with a function defined numerically through a parameter
• From: "pascal vallotton" <Pascal.Vallotton at epfl.ch>
• Date: Thu, 28 Jan 1999 04:23:35 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Dear members
I have defined the following function which describes the decay of the
fluorescence after a light pulse in a sample containg a dye molecule
which transmits its excitation to neighbouring fluorophores, the
unknown concentration of which is Conc

d[t_] := Exp[(-t/tau)-eps Conc (t/tau)^(1/3) ]

In this equation, tau is the average lifetime of the donor molecule in
the absence of the neighbouring acceptor molecules.

With my fluorimeter, I measure the fourier transform ( the phase lag
introduced by the interaction of the exciting light with the sample as
well as its demodulation) of the latter function at several
frequencies experimentally chosen.

Therefore, d(t) is linked to the phases phi[] and modulation  values
mod[] by the following equations.

num[[i]]=NIntegrate[ d[t] Sin[2 Pi freqs[[i]] 1000000
t],{t,0,0.0000001}]; dum[[i]]=NIntegrate[ d[t] Cos[2 Pi
freqs[[i]]1000000 t],{t,0,0.0000001}];
norm[[i]]=NIntegrate[Exp[-t/tau],{t,0,0.00001}];

phi[[i]]=ArcTan[num[[i]]/dum[[i]]]*360/(2 Pi);
mod[[i]]=((num[[i]]/norm[[i]])^2+(dum[[i]]/norm[[i]])^2)^0.5,

the point is that these funtions are numerically computed and they
depend parametrically on the concentration  conc through d(t).

Now, to find the real concentration, I need to find it so that it best
fits to the observed results for the phase and demodulation. The
chisquare function is thus defined:

chi[[k]]=Sum[(1/iphases[[i]])^2 (phi[[i]]-phases[[i]])^2,{i,numboffreq}]

where iphases[i ] describes the experimentally measured standard
deviation in the phase value, phi[i] are for each frequency the
calculated value using the transform and assuming a trial concentration
and phases[i] contains the measured values.

Now my trouble is that the function FindMinimum[chi] refuses to give an
answer. I suspect that this is because it ¨fails¨ to consider chi as a
normal function of the concentration that  needs to be computed
numerically for each value.

The current version of the program uses a loop to increase the
concentration, then fits the points by a polynomial and finds its
minimum. This will get highly time consuming when the system gets more
complex (several lifetimes) and there must be a better way???

I have attached the complete version of the program as well as well as
the experimental files that are to be analyed. Thanks for any help

pascal vallotton
lcppm chem dpt
epfl
lausanne
switzerland

filename="transfer.nb"

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Notebook[{
Cell[BoxData[{
\( (*\
this\ program\ serves\ the\ aalysis\ of\ time\ resolved\ decay\ =
in\
vesicles\ *) \n
\n (*\ enter\ here\ the\ relevant\ data\ *) \n (*\
file\ containing\ time\ resolved\ data\ in\ ascii\ code\ produced\
= by\
iss\ *) \nusefile = "\<C:\Temp\Trans.txt\>";
\n (*\ nombre\ de\ fr\[EAcute]quences\ mesur\[EAcute]es\ *) \n
numboffreq = 7; \
\n (*\ forster\ radius\ for\ the\ donor\ acceptor\ pair\ in\ meters\
= *) \n
ro = 0.0000000059\ ;
\n (*\ lifetime\ of\ the\ donor\ in\ the\ absence\ of\ acceptor\ *)
= \n
tau = 0.000000006; \n
\n (*\ initial\ concentration\ in\ natural\ units\ *) \nConc = 0;
\n (*\ Wolbers\ constant\ *) \neps\  = \ 4.25409\ ; \n
\n (*\ initialisation\ des\ variables\ par\ lecture\ du\ fichier\ *)
= \n\
s1\  = \ OpenRead[usefile]; \ \nReadList[s1, \ String, \ 2]; \ \n
S1\  = \ ReadList[s1, \ \ Number, \ \ RecordLists\  -> \ True]; \n
freqs\  = \ Table[i, \ {i, \ numboffreq}]; \ \n
phases\  = \ Table[i, \ {i, \ numboffreq}]; \ \n
moduls\  = \ Table[i, \ {i, \ numboffreq}]; \ \n
iphases\  = \ Table[i, \ {i, \ numboffreq}]; \ \n
imoduls\  = \ Table[i, \ {i, \ numboffreq}]; \ \n
Do[freqs[\([i]\)]\  = \ S1[\([i, 1]\)], \ {i, \ numboffreq}]\),
\(Do[phases[\([i]\)]\  = \ S1[\([i, 2]\)], \ {i, \ numboffreq}]\), =

\(Do[moduls[\([i]\)]\  = \ S1[\([i, 3]\)], \ {i, \ numboffreq}]\), =

\(Do[iphases[\([i]\)]\  = \ S1[\([i, 4]\)], \ {i, \ numboffreq}]\ =
\),
\(Do[imoduls[\([i]\)]\  = \ S1[\([i, 5]\)], \ {i, \ numboffreq}]\n
\n (*\ fonction\ d\[EAcute]crivant\ la\ d\[EAcute]croissance\ de\ =
la\
fluorescence\ en\ pr\[EAcute]sence\ des\ accepteurs\ *) \),
\(d[t_]\  := \
Exp[\((\(-t\)/tau)\) - eps\ Conc\ \((t/tau)\)^\((1/3)\)\ ]\ =
\n\n\)}],
"Input"],

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num\  = \ Table[i, \ {i, \ numboffreq}]; \ \n
dum\  = \ Table[i, \ {i, \ numboffreq}]; \ \n
norm\  = \ Table[i, \ {i, \ numboffreq}]; \ \n
phi\  = \ Table[i, \ {i, \ numboffreq}]; \ \n
mod\  = \ Table[i, \ {i, \ numboffreq}]; \nchi = \ Table[i, \ = {i,
\ 50}];
\)\)], "Input"],

Cell[BoxData[{
\(\n\nConc = 0\),
\(Do[Conc = Conc + \ 0.1;
Do[\n (*\ calcul\ des\ transformations\ de\ fourier\ *) \n
num[\([i]\)] =
NIntegrate[\
d[t]\ Sin[2\ Pi\ freqs[\([i]\)]\ 1000000\ t], {t, 0, =
0.0000001}];
\ndum[\([i]\)] =
NIntegrate[\
d[t]\ Cos[2\ Pi\ freqs[\([i]\)] 1000000\ t], {t, 0, =
0.0000001}];
\nnorm[\([i]\)] = NIntegrate[Exp[\(-t\)/tau], {t, 0, =
0.00001}];
\n (*\ calcul\ de\ la\ phase\ et\ de\ la\ modulations\
calcul\[EAcute]e\ *) \n
phi[\([i]\)] = ArcTan[num[\([i]\)]/dum[\([i]\)]]*360/\((2\ =
Pi)\); \n
mod[\([i]\)] =
\((\((num[\([i]\)]/norm[\([i]\)])\)^2 +
\((dum[\([i]\)]/norm[\([i]\)])\)^2)\)^0.5, {i, =
numboffreq}];
\n (*\ calcul\ du\ chisquare\ pour\ le\ guess\ =
pr\[EAcute]c\[EAcute]dent
\ *) \t\nchi[\([k]\)] =
Sum[\((1/iphases[\([i]\)])\)^2\
\((phi[\([i]\)] - phases[\([i]\)])\)^2, {i, numboffreq}], =
{k, 10}]
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filename="Trans.txt"

* trans.txt
*DATA: FREQ      PHASE      MOD      DELTAP      DELTAM
5.0000000     7.450    0.981      0.021      0.001
29.1700000    29.410    0.734      0.021      0.001
53.3300000    37.740    0.587      0.031      0.001
77.5000000    42.720    0.499      0.056      0.002
101.6700000    46.670    0.440      0.050      0.002
125.8300000    49.590    0.390      0.053      0.002
150.0000000    52.060    0.358      0.036      0.001

```

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