[VA] SRC #008 - 2021 is here !

[Valentin Albillo]

            
Hi, all:

Thanks for your interest in my SRC #008, some comments to your posts and some answers to your questions:

Nihotte(lma) Wrote:I have tested several configurations but it seems to me that 673x3 + 2 gives the best result With a result of 2021 and 2,5329955 x 10^321 ... I'm coming back on the second point. I didn't want to stay on a half-failure... and for the whole integral it gives : [...] 2.467401102

You gave the first correct solution to Case 1 but forgot to also give the exact 322-digit result as requested. Not bad ! And your value for the integral using the HP-15C is correct as well, congratulations.

J-F Garnier Wrote:Using your great "Identifying Constants" program, I found the symbolic value, a nice surprise, that lead me to another conclusion about the expression to integrate :-)

Hehe, you got it, both the symbolic value (Pi²/4) and the fact that the integral's value doesn't depend on the constant being 2.021 and thus you could use 2 instead to speed up the numerical integration using the HP32S no less. And thanks a lot for your kind comments and the link to my Article.

robve Wrote:Thanks for posting. A good motivation to write some code on HP Prime. You did not mention that RPN is required...

Indeed, the Prime is an HP calculator alright. And your fine solution is the very first one to give the full 322-digit correct value for Case 1.

Albert Chan Wrote:(many, many things spread over many, many posts)

Thanks for your abundant, relentless and interesting approaches both numerical and theoretical, Albert, and most especially for using HP calculators to obtain them, as I requested. At least you did for your very first posts ...

StephenG1CMZ Wrote:But if that "." is meant to be a thousands separator rather than decimal, representing the year 2021 rather than the year 2 , I get undef ...

As you may have seen, the dot may be taken as the decimal separator or the thousands separator and it willl make no difference to the result   

ijabbott Wrote:Some clues: Splitting a 1 [...]

Your detailed explanation mimics my original solution and is quite didactic for people who would like to understand the logic behind the result, well done.

Maximilian Hohmann Wrote:This may be due to the fact that most older HP calcs (which are the ones I - as well as many others - prefer) can't handle the 3-digit exponents of puzzle #1 :-)

Well, that shouldn't be a problem, many of my former Challenges and Short & Sweet Math Challenges did require multiprecision values. Remember for instance my "Sweet & Short Math Challenges #15: April 1st Spring Special", where I posted a 9-line program for the HP-71B which computed and printed in full the exact solution to the "Take 5" sub-challenge, namely 2⁶⁵⁵³⁶-3, which is a 19,729-digit integer. There were RPL programs doing likewise, too.

Gerson W. Barbosa Wrote:I chose #2 only and I chose this one not because it’s hard but because it’s easy. :-) [...] Not nearly as beautiful as Valentín’s integral, but at least easier to key in.

Hehe, Gerson, you're playing tricks here, your (very nice, by the way) expression evaluates to 2.4674011002718... while the correct value is Pi²/4 =
2.4674011002723... (Very) close but no cigar !

Gene Wrote:Couple of solutions: [...] Using Egyptian fractions. HP-41C program in the Test Stat rom.

Very correct, if different from my original solution. And a very useful reference, thanks.

Albert Chan Wrote:Can you explain why ?

J-F Garnier Wrote:Yes, please Valentin; I was expecting to get an explanation or at least some indications.

EdS2 Wrote:I second these calls for an explanation of the inner workings of this ingenious and unexpected integral

Thanks for your interest and well, my goal with these SRC's and Challenges is to show some interesting HP-calc programming techniques (mostly for the classic models), as well as some interesting, little-known and even unexpected math topics, hopefully for the enjoyment of (at least some) MoHPC forum's readers. Also, if I can provide technical details or proofs which aren't too lengthy and/or complicated, I'll usually do it.

For instance, this is not the first time I post a definite integral allegedly dependent on some parameter but actually being independent of it, such as the two integrals featured in my Short & Sweet Math Challenge #18: April 1st, 2007 Spring Special, namely:

            / Inf
            |
       I1 = | 1/((1+x²)*(1+x^4.012007)) .dx
            |
            / 0
and
            / Pi/2
            |
       I2 = | 1/(1+Tan(x)^4.012007) .dx
            |
            / 0

which actually are one and the same upon a simple change of variables and their value is Pi/4. The proof of their being independent of the parameter (the 4.012007 value) is short enough that I did post my own proof of the fact in Message #54 in that thread.

However, for the present integral the proof is not that easy or short, actually it's quite long and would require a lot of time and cumbersome math "typesetting" to post it here, which regrettably is beyond the scope of these SRC's. I'll give however a link to "some indications" for a very similar integral, which you can find here.

The proof for the similar integral I posted (I don't have a link to the proof, sorry) essentially goes along the same lines. Also, if you think that an integral with a parameter whose value is independent of it is something quite unexpected, wait till you see the next integral I'll post in a future SRC, it's 10x as much !

Best regards to all and thanks for participating.
V.