Re: [WP34s] Romberg Integration Message #8 Posted by Les Wright on 30 Nov 2011, 9:11 p.m., in response to message #7 by Kiyoshi Akima
The integration scheme doesn't like that function at all. Gives 1.28 and 1.625 respectively. Not even close. According to Mathematica, the correct results are 3.08 and 3.125, which is easily seen by inspection (6*1/2 + .4*.4/2 in the first case, and you get the analogy in the second case).
The integral is periodic with discontinuities. The IG Romberg scheme doesn't even seem to get past the first estimate before exiting.
The August 1980 Kahan paper and the advanced integration chapters in the 42s, 15C, 34C, and 35s do at least touch on the "know thy function" theme. In this case, what it easily computed by visual inspection and by hand boggles the calculators. I am sensing that the underlying trapezoidal rule is not fooled by the initial nonlinear distribution of sample points, and the routine gets too successive identical results off the bat and exits. I should note that the algorithm gives exact results quickly within the continuous segments, but not across the discontinuities.
Les
