Re: Numeric integration

In article <[EMAIL PROTECTED]>,
C W  <[EMAIL PROTECTED]> wrote:
>I am looking for a way to integrate this with at least 50 digits after
>decimal :

[Snip integral of the form 
   Int( P(x)*cos(x*exp(x))*exp(-24*x)*(1+x)^(-48),  x = 0 .. x0 );
 where  P  is an integer polynomial 24! x^24 + ... + 5^48  and  x0=9.488...]

>What method or software should I use ???

Well, before trying numerical analysis you can use some simple calculus.

1. Integrate only to about x0=5.5 because the integrand is smaller than
   about 10^(-52) there. This means you only go through a couple hundred
   periods of the trig function.

2. Rewrite  P  as a polynomial in  1+x (still integer coefficients, degree 24)
   and cancel some (1+x)'s. This both lowers the "48" and eliminates P.

3. Writing  C(k,m) = exp(-k*x) * (1+x)^(-m) * cos(x*exp(x))  and likewise
   for  S(k,m)  (using sine) we find that
      C'(k,m) = -k*C(k,m) - m*C(k,m+1) + S(k-1,m-1)
   and similarly for  S'(k,m).  Singling out the middle term on the right
   side, this means that for m>0 we can express an integral of a  C(k,m+1)
   in terms of integrals of  C(k,m)  and  S(k-1,m-1). Iterating this
   idea, we see there is no need to compute any integrals except those
   of  C(k,0), C(k,1), S(k,0), and S(k,1). (Note that we start with
   integrals of  C(k,m)  with  k=24 and m <= 48=2k; the reduction formulas
   will preserve the inequality  m <= 2k  and in particular we need not
   compute such integrals with negative k.)

So the problem reduces to computing (to some 80 digits of accuracy!)
integrals of the forms

    int( exp(-k*x) * cos(x*exp(x)), x=0..x0)
    int( exp(-k*x) /(1+x) * cos(x*exp(x)), x=0..x0)

(and similarly for sines) for  0 <= k <= 24,  where  x0  is around  5.5 .
I don't know what's recommended when facing both rapid oscillations
and fast decay in the integrand.

It may be productive to continue the reduction in (3) to allow negative
m  as well (in an effort to reduce  k  so that the integrand does not
undergo such dramatic decay).

dave