|Re: HP48 integration error?|
Message #6 Posted by Les Wright on 17 Mar 2007, 6:31 a.m.,
in response to message #5 by k
Yup, I have an idea.
The integration algorithm makes an estimation at a few points, then doubles the number of points and makes another estimation, refines this by something called Richardson extrapolation, etc., and continues until two or three subsequent estimations are equal to each according to the tolerance implied by the setting for FIX, SCI, or ENG.
When estimating this integral, the value of the integrand is < 1e-499 for the huge majority of the range 0 to 1000. Basically, this is zero as far as the calculator is concerned. The value of integrand only becomes nonzero, as far as the calculator is concerned, somewhere between x = 33 and x = 34. That part of the range of integration is stuck way on the left end, and it is unlikely that in a few sequential estimates the algorithm even samples much from that end. This means that you are going to get two or three consecutive estimates that are effectively zero, since the integral estimates are basically weighted averages of the integrand at the sample points. The weighted average of a bunch of zeros is always zero. This happens a a few times in a row (I think at least three) and tells the calculator convergence has been reached. That happens really quickly, right at the beginning, so false convergence is reached quickly.
The stuff in the 15C manuals is not conveniently copied in this forum, as there is pages of stuff. But, in a general sense, if you want to basic idea of what the 48G does in integration, google Romberg Method or Romberg algorithm. The math is not very hard, even though the notation can be confusing--subscripts to keep track of and that sort of thing. If you can appreciate graphically this sampling issue, you may grasp why the HP48G goes to 0 when you set up the integral with the 0-1000 limits.
Edited: 17 Mar 2007, 8:32 a.m.