Re: Integrating x^b*Log[x]^m gives wrong result?

• To: mathgroup at smc.vnet.net
• Subject: [mg85448] Re: [mg85434] Integrating x^b*Log[x]^m gives wrong result?
• From: Kees van Schaik <keesvanschaik at gmail.com>
• Date: Mon, 11 Feb 2008 06:04:54 -0500 (EST)
• References: <20080210084609.26Y7U.24492.root@eastrmwml03.mgt.cox.net>

```Thanks to everybody for the quick and clear answers! Apparently I'm
still not aware enough of limitations of machine size precision
arithmetic as I never guessed this could be the reason for the result
actually getting a wrong sign and be so completely wrong, not just a
relatively small error margin away from the 'exact' answer...

Thanks again, Kees

Bob Hanlon wrote:
> It is a precision issue
>
> Clear[x, m];
>
> f1[m_] := Integrate[x^(-7/2)*Log[x]^m, {x, 5, 10}];
>
> f2[m_] := NIntegrate[x^(-3.5)*Log[x]^m, {x, 5, 10}];
>
> f3[m_] := Integrate[x^(-3.5)*Log[x]^m, {x, 5, 10}];
>
> f4[m_] := Integrate[x^(-3.5`25)*Log[x]^m, {x, 5, 10}];
>
> v = {25, 40};
>
> Machine precision:
>
> N[f1 /@ v]
>
> {403155.55804573477, -4.183061773182904*^16}
>
> Extended precision:
>
> N[f1 /@ v, 20]
>
> {
>    403156.30667862031262341904898345961`19.99999999999\
>   9996,
>    6.3361619293717813936101205761`20.000000000000007*^\
>   10}
>
> Machine precision:
>
> f1 /@ N[v]
>
> {403156.25, 0.}
>
> Extended precision:
>
> f1 /@ N[v, 23]
>
> {403156.306678620312623419`13.159208013405054,
>    6.3361619293717814121`1.6036615637884584*^10}
>
> Numerical integration:
>
> f2 /@ v
>
> {403156.306678621, 6.3361619293717926*^10}
>
> Machine precision:
>
> f3 /@ v
>
> {401853.125, -5.48409663189581*^23}
>
> Extended precision:
>
> f4 /@ v
>
> {403156.3066786203126239621`14.540465311148813,
>    6.335827093390950705`2.984895910513547*^10}
>
>
> Bob Hanlon
>
> ---- KvS <keesvanschaik at gmail.com> wrote:
>
>> Dear all,
>>
>> I'm running into the following problems with symbolic vs. numerical
>> integration of the function x^(-3.5)*Log[x]^m:
>>
>> In[564]:=
>> ClearAll["Global`*"];
>> f1[m_]:=N[Integrate[x^(-3.5)*Log[x]^m,{x,5,10}]];
>> f2[m_]:=NIntegrate[x^(-3.5)*Log[x]^m,{x,5,10}];
>> Map[f1,{5,10,25,40}]
>> Map[f2,{5,10,25,40}]
>>
>> Out[567]= {0.145434,4.62609,401145.,-9.30763*10^23}
>> Out[568]= {0.145434,4.62609,403156.,6.33616*10^10}
>>
>> Of course the symbolic integration is wrong here since it shouldn't
>> yield a negative number. If the recursive formula resulting from
>> partial integration is used, things seem to go wrong as well:
>>
>> In[572]:=
>> f[m_]:=(-1/2.5)*(10^(-2.5)*Log[10]^m-5^(-2.5)*Log[5]^m)+(m/
>> 2.5)*f[m-1];
>> f[0]=(-1/2.5)*(10^(-2.5)-5^(-2.5));
>> Map[f,{5,10,25,40}]
>>
>> Out[574]= {0.145434,4.62609,403156.,-2.54037*10^16}
>>
>> So the result for m=25 still coincides with the one from NIntegrate,
>> while Integrate already gives something different; for m=40 the result
>> is different from both NIntegrate and Integrate (and wrong as it is
>> negative). If one changes the negative power of x to a positive one,
>> things seem ok btw.
>>
>> Any clues what might be going on here?
>>
>>
>> In[533]:= \$Version
>> Out[533]= 6.0 for Microsoft Windows (32-bit) (April 27, 2007)
>>
>>
>
>
>

```

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