Problem with integral on WP 34s

[Nigel (UK)]

(09-15-2021 02:14 AM)lrdheat Wrote:  Hi again, Nigel,

If an integral is taking a fair amount of time to run, and I am satisfied that the estimate on screen is a good one, is there a way to retrieve the latest estimate? When I stop with r/s or exit, the estimate must reside somewhere!

For the double-exponential routine, if you interrupt by pressing <- (backspace) the most recent estimate is left in the x-register. There doesn't seem to be anything similar in the Romberg routine. These routines use local registers and flags, so their internal workings are only accessible if the routine allows them to be.

(09-15-2021 02:21 AM)lrdheat Wrote:  Unrelated, but interesting…integral of x*e^-x from 0 to infinity should be 1. When integrating from, say 0 to 1,000, it converges on 1. If I integrate from 0 to 1,000,000, it converges on 0, misses the important part of the curve close to 0. How, then, does the WP 34S on DM 42 recognize the important part of the curve when integrating from 0 to infinity (it converges on 1!)?

When both limits are infinite, the integration routine does a change of variable from \(x\) to \(t\), where
\[x=\sinh\left((\pi/2)\sinh(t)\right)\]
(or something similar). When \(t=1\), \(x=3.1\); when \(t=5\), \(x=2.1\times10^{50}\)! So it's only necessary to integrate over a small range of \(t\) to reach essentially infinite values of \(x\). This means that the important region for \(x\) will automatically be found, so long as it's concentrated somewhere near \(x=0\) as it is in this case.

The more I've read about this subject, the more I am impressed by how good the double-exponential integration method is, and how well it is implemented on the WP-34S. I won't be using Romberg myself - the cases for which the double exponential method fails always seem to be those where the function being integrated or its derivatives are discontinuous (i.e., the function is non-analytic) somewhere in the range of integration. I'm happy to spot these points myself and to work around them.

Nigel (UK)