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Re: Re: Bug ??????
*To*: mathgroup at smc.vnet.net
*Subject*: [mg105385] Re: [mg105346] Re: Bug ??????
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Mon, 30 Nov 2009 06:12:30 -0500 (EST)
*References*: <200911291008.FAA16112@smc.vnet.net>
On 29 Nov 2009, at 19:08, Bill Rowe wrote:
> On 11/28/09 at 1:07 AM, wkfkh056 at yahoo.co.jp (ynb) wrote:
>
>> F[x_]:=34880228747203264624081936 - 464212176939061350196344960*x^2
>> + 4201844995162976506469882880*x^4 -
>> 36736184611200699915890392480*x^6 +
>> 245136733977616412716801297320*x^8 -
>> 1144143594851571569661248433072*x^10 +
>> 3682862525053500791559515638600*x^12 -
>> 8693355704402316431096075720520*x^14 +
>> 16394872503384952006491292949865*x^16 -
>> 26387316917169915527289585290460*x^18 + ...
>
> <rest of definition snipped>
>
>> (* Bug ?; F[Sqrt[Sqrt[2] + 3^(1/3)] + 1/Sqrt[3^(1/3) + 5^(1/5)]]
>> //N
>> =3.828176627860558*^38<---Bug ? *)
>
>> (* =0? *)
>
> This is not a bug. It is inherent when converting things to
> machine precision.
>
> Your definition of F is an alternating sum of large powers of x.
> The function N converts each subterm to machine precision then
> computes the sum. Loss of precision is inevitable when such a
> large dynamic range exists in the terms to be summed as is the
> case here.
>
> Note, this is not a Mathematica issue. Rather it is inherent to
> floating point arithmetic used on any computer system.
>
> To get a simple numerical answer to this problem, there are
> several possible approaches. The simplest I can think of would
> be either
>
>
> =46[Sqrt[Sqrt[2] + 3^(1/3)] + 1/Sqrt[3^(1/3) + =
5^(1/5)]]//Simplify//N
>
> or
>
> N[F[Sqrt[Sqrt[2] + 3^(1/3)] + 1/Sqrt[3^(1/3) + 5^(1/5)]], 100]
>
> Note neither of these as written above is guaranteed to work.
> Usage of Simplify will work *if* the structure of the terms to
> be added result in cancellations giving a much simpler prior to
> be converted to machine precision.
>
> The second approach will work if the second argument to N is
> made sufficiently large.
>
>
It should work equally well even if the second argument is small. This =
is the whole point of significance arithmetic. Compare the answers:
In[14]:= N[F[Sqrt[Sqrt[2] + 3^(1/3)] + 1/Sqrt[3^(1/3) + 5^(1/5)]], 3]
Out[14]= 1.83*10^10
In[15]:= N[F[Sqrt[Sqrt[2] + 3^(1/3)] + 1/Sqrt[3^(1/3) + 5^(1/5)]], 30]
Out[15]= 1.83396597760000000000000000000*10^10
Both of these "work" equally well with respect to the specified accuracy.
Andrzej Kozlowski
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