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

On 10 Feb 2008, at 11:19, KvS 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?
> Thanks in advance, Kees
> In[533]:= $Version
> Out[533]= 6.0 for Microsoft Windows (32-bit) (April 27, 2007)

The only thing that is wrong is that you are using machine precision  
arithmetic with Integrate and run into numerical instability. Observe:

a = Integrate[x^(-7/2)*Log[x]^40, {x, 5, 10}];

with ten digits of precision:

N[a, 10]

with machine precision:


The moral is obvious. Its risky to use machine precision unless you  
are able to convince yourself that the formula you are evaluating is  
not numerically ill conditioned.

Andrzej Kozlowski

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