Perplexed by the behavior of NonlinearFit in Mathematica ver 4.2 vs 5.1

• To: mathgroup at smc.vnet.net
• Subject: [mg54224] Perplexed by the behavior of NonlinearFit in Mathematica ver 4.2 vs 5.1
• From: ellisg at rogers.com (G Ellis)
• Date: Sun, 13 Feb 2005 22:17:12 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Please note that my questions are at the end of this Post.  What
follows is some necessary background:

I include below a NonlinearFit function ("nlr[t]"), which I created in
ver 4.2 of Mathematica.  It is VERY simple.  I plotted the function
and the graph was what I expected. However, I never looked at
tabulated results of the function and therefore I did not notice a
peculiar issue with my syntax.    It was only when I ran the code in
ver 5.1 that I saw the odd output.  (The code below includes
tabulations that I created after I noticed the different behavior
between ver 4.2 and ver 5.)

In ver 4.2 Mathematica does not "complain" about the function "nlr[t]"
in the following.  It appears to plot the result correctly; but it
returns different "Table" results without and with "Evaluate".  I
believe the correct results are those returned with "Evaluate" or the
use of the "ReplaceAll" function.

In[1]:=
<< "Statistics`"
<< "Graphics`"
In[3]:=
\$Version
Out[3]=
"4.2 for Microsoft Windows \
(June 5, 2002)"
In[4]:=
pop = {1650, 1750, 1860,
2070, 2300, 2560, 3040,
3710, 4450, 5280, 6080};
nlr[t_] := NonlinearFit[pop,
E^(theta*t)*k, {t},
{theta, k}]
t1 = Table[nlr[t], {t, 0, 3}]
t1 = Table[Evaluate[nlr[t]],
{t, 0, 3}]
t2 = Table[nlr[t] /. t -> i,
{i, 0, 3}]
Out[6]=
{537.6363636363639,
3158.987930628625,
1531.0837992793613,
1782.0476907177406}
Out[7]=
{1130.2076401682348,
1315.4628872314652,
1531.0837992793613,
1782.0476907177406}
Out[8]=
{1130.2076401682348,
1315.4628872314652,
1531.0837992793613,
1782.0476907177406}

p1 = Plot[nlr[t], {t, 0, 11}, PlotStyle -> IndianRed];
p2 = Plot[nlr[t] /. t -> i, {i, 0, 11},
PlotStyle -> EmeraldGreen];

In ver 5.1, Mathematica will not let me evaluate "nlr[t]" directly.
(This is good, I think).  It forces the use of "Evaluate".

\$Version
"5.1 for Microsoft Windows \
(October 25, 2004)"
pop = {1650, 1750, 1860,
2070, 2300, 2560, 3040,
3710, 4450, 5280, 6080};
nlr[t_] := NonlinearFit[pop,
E^(theta*t)*k, {t},
{theta, k}]
Table[nlr[t], {t, 0, 3}]
Table[Evaluate[nlr[t]],
{t, 0, 3}]
Table[nlr[t] /. t -> i,
{i, 0, 3}]
\!\(\*
RowBox[{\(General::"ivar"\),
":", "\<\"\\!\\(0\\) is not a valid variable.
\\!\\(\\*ButtonBox[\\\"More\
ButtonFrame->None, \
ButtonData:>\\\"General::ivar\\\"]\\)\"\>"}]\)
NonlinearFit::fitfail:The fitting algorithm failed.
\!\(\*
RowBox[{\(General::"ivar"\),
":", "\<\"\\!\\(1\\) is not a valid variable.
\\!\\(\\*ButtonBox[\\\"More\
ButtonFrame->None, \
ButtonData:>\\\"General::ivar\\\"]\\)\"\>"}]\)
NonlinearFit::fitfail:The fitting algorithm failed.
\!\(\*
RowBox[{\(General::"ivar"\),
":", "\<\"\\!\\(2\\) is not a valid variable.
\\!\\(\\*ButtonBox[\\\"More\
ButtonFrame->None, \
ButtonData:>\\\"General::ivar\\\"]\\)\"\>"}]\)
\!\(\*
RowBox[{\(General::"stop"\),
":", "\<\"Further output of \\!\\(General :: \\\"ivar\\\"\\) will
be \
suppressed during this calculation. \
\\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\",
\\\", ButtonFrame->None,
ButtonData:>\\\"General::stop\\\"]\\)\"\>"}]\)
NonlinearFit::fitfail:The fitting algorithm failed.
\!\(\*
RowBox[{\(General::"stop"\),
":", "\<\"Further output of \\!\\(NonlinearFit ::
\\\"fitfail\\\"\\) will \
be suppressed during this calculation.
\\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\
ButtonData:>\\\"General::stop\\\"]\\)\"\>"}]\)
{NonlinearFit[{1650, 1750,
1860, 2070, 2300, 2560,
3040, 3710, 4450, 5280,
6080}, k, {0},
{theta, k}], NonlinearFit[
{1650, 1750, 1860, 2070,
2300, 2560, 3040, 3710,
4450, 5280, 6080},
E^theta*k, {1},
{theta, k}], NonlinearFit[
{1650, 1750, 1860, 2070,
2300, 2560, 3040, 3710,
4450, 5280, 6080},
E^(2*theta)*k, {2},
{theta, k}], NonlinearFit[
{1650, 1750, 1860, 2070,
2300, 2560, 3040, 3710,
4450, 5280, 6080},
E^(3*theta)*k, {3},
{theta, k}]}
{1130.2091592378029,
1315.4644493997696,
1531.0853778619419,
1782.049249124357}
{1130.2091592378029,
1315.4644493997696,
1531.0853778619419,
1782.049249124357}

So my questions are:

1)Why does "nlr[t]" appear to work in ver 4.2; but return apparently
invalid results for zero and one, but correct results for higher
values?

2) Why, in ver 4.2, does "Plot" return the (apparently) correct values
with "nlr[t]" without the need for "Evaluate"; whereas "Table" does
not?

3)  What changed between versions 4.2 and 5?

```

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