Improper Integrals with the HP-15C LE & CE

[Valentin Albillo]

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Hi, all,

Some comments on some messages posted by you. Lessee ...

(11-14-2023 07:49 PM)Commie Wrote:  I just want to point out that the integral 1/sqrt(x) =2.sqrt(x). It's easy to see that 2(sqrt(1)-sqrt(0))=2 exactly with a bit of thought before reaching for the hp15c ce.

It seems you aren't getting the point. Of course this integral is trivial to symbolically integrate, even by hand, and thus immediately get the result but that's not the idea, no one's interested in the result in and of itself.

The reason I posted this particular improper integral and its real interest is to ascertain how accurate and fast the various integration methods available in/for HP calcs are when numerically dealing with this seemingly simple integral, in particular Namir's modification of a so-so implementation of 16-point Gaussian quadrature written by someone else in the very distant past, which proves to be unable to deal with this integral.

(11-17-2023 01:07 PM)rprosperi Wrote:  If you really need this result, pick a better device to calculate it for you...[...]

No one "really needs" this result, what people might be interested in is how various integration methods either built-in or user-programmed behave in their HP calculators.

As for picking a better device, it's often the case (and here in particular) that picking a better integration procedure will get the result fast and accurately on the HP device, say the HP-15C CE or the HP-71B, for instance.

(11-17-2023 02:58 PM)J-F Garnier Wrote:  

(11-17-2023 01:07 PM)rprosperi Wrote:  If you really need this result, pick a better device to calculate it for you [...]

In that case, which tool (in the HP calculator class) may numerically evaluate the integral with more than the 6 decimal places that the 15c CE can (slowly) provide ?

Well, I posted this example of a seemingly simple, unassuming improper integral taken from the test suite I created full of hard tests for my ancient integration program which I wrote for the HP-71B several decades ago. It would compute this integral and many others to high accuracy in a matter of seconds. So this is one answer to your question.

(11-17-2023 03:27 PM)rprosperi Wrote:  Any super computer should do nicely... you'll note I did not say "HP device"... [...] I'll make the obvious guesses that the DM42, DM32, WP34 or C47 can, but no doubt these also would take longer than most reasonable people would wait.

No need and No. As I said above, your "lowly" HP calc can compute that particular integral fast and accurately by simply using a suitably better integration procedure than the ancient built-in Romberg's method. I remember that my HP-71B BASIC program mentioned above could compute normal integrals as fast as the assembler INTEGRAL keyword and difficult integrals many times faster/more accurately, up to 100x. So a fast device surely helps but a better method helps the most.

(11-17-2023 04:38 PM)J-F Garnier Wrote:  I don't know for the WPxx or Cxx, but Free42 (and likely DM42/32) can't, despite its much higher power and 34-digit accuracy. Any other challenger?

Yes, a physical WP43 calculator using its internal integration procedure takes about 24" with ACC = 1E-22 to return I2 = 2 (displayed, actually 1.999 999 999 999 999 793 311 720 040 405 29, i.e. 17 correct places save 2 ulp.

Regards.
V.