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Re: FindFit

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92752] Re: [mg92733] FindFit
  • From: danl at wolfram.com
  • Date: Sun, 12 Oct 2008 04:33:05 -0400 (EDT)
  • References: <200810100835.EAA07169@smc.vnet.net>

> Dear Mathematica Gurus,
> Who have idea which procedure I can use inspite FindFit in following:
> In[1]: Table[Simplify[Expand[(1/2) ((1 + Sqrt[13])^(2^n) + (1 -
> Sqrt[13])^(2^n))]], {n, 1, 6}]
> Out[1]: {14, 248, 102272, 20489142272, 839425017825619607552,
> 1409268686920894404615927074915795024740352}
> In[2]: FindFit[{14, 248, 102272, 20489142272, 839425017825619607552,
>   1409268686920894404615927074915795024740352}, (1/
>     2) ((a + Sqrt[b])^(2^x) + (a - Sqrt[b])^(2^x)), {a, b}, x,
>  WorkingPrecision -> 100]
> Out[2]:{a -> 2.43785286368448657933626328600,
>  b -> 4.95343824602875750402007143975}
>
> Should be: {a -> 1.00000000000000000000,
>  b -> 13.000000000000000000}
>
> Who have idea how find (true) a,b when I have sequence
> Best wishes
> Artur

There might be other methods that give better results but regardless, I
would recommend rescaling. As it is, you will get results weighted much
more heavily towards matching the larger terms. To avoid this problem a
reasonable thing would be to match the logs instead.

ftab = Table[
  Simplify[Expand[(1/
       2) ((1 + Sqrt[13])^(2^n) + (1 - Sqrt[13])^(2^n))]], {n, 1, 6}];

In[38]:= fit =
 FindFit[Log[ftab],
  Log[(1/2) ((a + Sqrt[b])^(2^x) + (a - Sqrt[b])^(2^x))], {a, b}, x,
  WorkingPrecision -> 100]

Out[38]= {a -> \
-0.9999999999999999999999999999999999999999999999999999999999999999999\
9949814992518344284926007654753938,
 b -> 13.0000000000000000000000000000000000000000000000000000000000000\
0000000296782411420663054652545138944}

In general one might need to use Abs[model function] when logarithms are
involved, so as to avoid wandering off the real axis (and thus annoying
the various FindFit deities).

Daniel Lichtblau
Wolfram Research




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