[VA] SRC#005 April, 1st Mean Minichallenge

04132019, 11:33 AM
Post: #21




RE: [VA] SRC#005 April, 1st Mean Minichallenge
Hello Valentin,
Thanks for your kind and encouraging words. I will give Tier 2 of S&SMC#24 a try. (04132019 12:08 AM)Valentin Albillo Wrote: The trick of computing the Geometric Mean by adding logarithms and then a final exponential is perfectly Ok for strictly positive datasets, as in this case, and the precision lost amounts to but a few ulps in the worst case, so it's perfectly acceptable as well. You're right it works only for positive data sets. I didn't notice this. Best regards Bernd 

04152019, 09:29 PM
(This post was last modified: 04162019 06:54 AM by Mike (Austria).)
Post: #22




RE: [VA] SRC#005 April, 1st Mean Minichallenge
Thanks Valentin for this inspiring minichallenge. I particularly enjoy math riddles when they involve littleknown facts about wellknown mathematics. This one definitely fits this description particularly well  the fact that the various means are discrete members of a continuous family had escaped my attention so far (for decades  embarrassingly enough :) ).
Since there have been no solutions for RPL, let me share my take on this. It should work for any version of RPL starting with the HP48G(X). 1. RPL function GM for the Generalized Mean function \(G_p(x_1,\ldots,x_n)=(\frac{1}{n}\sum_{i=1}^{n}x_i^p)^{1/p}\). Code: « Exponent \(p = k2\) Usage: put list {\(x_i\)} and exponent \(p\) on stack, invoke GM. I like the terseness of this function. 2. Test: four standard means Code: { 4 1 20.19 } 'X' STO @ save list 3. Solving for \(k=p+2\) such that \(G_p(x_i)=\pi\) Auxiliary function G: for list {\(x_i\)} and exponent \(p\) in global variables X and P, return GM(x,p)\(\pi\) Rationale: the HP48 solver expects all parameters and the unknown to be solved for (\(p\) in this case) in global variables. Code: « X P GM π  » Find zero of G Code: 'G' RCL @ recall G to stack 4. Bonus: Code for Generalized fMean \(G_f(x)=f^{1}\left(\frac{1}{n}\sum_{i=1}^{n}f(x_i)\right)\) A further step in generalizing the mean is by replacing the \(p\)th power of \(x_i\) by applying an arbitrary continuous strictly monotonic function and taking its inverse after computing the sum and dividing by \(n\). Not very surprising, the geometric mean results from using \(f(x)={\rm ln}(x)\) the natural logarithm. Btw, the requirement that all \(x_i\) be positive is actually not a restriction because the geometric mean is defined for positive arguments only in the first place. Since in general the inverse of \(f\) cannot be assumed to be known analytically, we first define an auxiliary function FINV, which, given \(f, y\), and an estimate \(e\), will return an approximation of \(x=f^{1}(y)\) by numerically solving \(f(x)=y\) for the unknown \(x\). Code: « This is essentially the inverse function solver on page 254 of the HP48G Advanced User's Manual (AUM). I had initially hoped to use a compiled local variable (this is a local variable whose name starts with a backarrow ←) because the AUM promises that this can be used whenever a called function expects a global variable  but ROOT does not let me do this, thus the clumsy { tX } PURGE at the end. With this complexity hidden away, it becomes straightforward to implement the generalized \(f\)Mean function GFM: Code: « Why pick one of the original list elements to estimate the solution: we make no assumptions about the domain of \(f\). As long as the user supplies a list of valid elements, any of them is suitable as the starting point for the ROOT solver of the HP48, which is quite robust. This is cheaper than e.g. taking the arithmetic mean as an estimate or bracketing the solution by searching the minimum and maximum of {\(x_i\)}. Examples: Code: X « INV » GFM > 2.30852787043 Up to the occasional 1ULP numerical error, above results are reproduced. The last example is closely related to the LogSumExp function, which is one of the Smooth Maximum Approximations used in optimization problems. I hope all this makes sense. Setting up these functions was fun  thanks for giving me an excuse to learn and play with stuff i'd otherwise never have looked at again. Despite RPL's deficiencies (such as handling many trivial cases incorrectly), I like the expressive power and abstraction / object orientation of this language. P.S. This is my first post in the new forum. While I stumbled upon a nice description how to include math, I could not find a way to use a [pre] tag. So I had to reluctantly adorn my programs with code tags lest all formatting be lost. I'd gladly replace these by a better solution. 

04162019, 07:23 PM
Post: #23




RE: [VA] SRC#005 April, 1st Mean Minichallenge
Addendum: for stackonly purists, here comes a 63.5 bytes twoliner implementation GMS which does the same as above GM in slightly less time:
Code: « DUP { DUP2 ^ ΣLIST ROT SIZE / SWAP } 

04212019, 08:38 PM
Post: #24




RE: [VA] SRC#005 April, 1st Mean Minichallenge
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Hi, Mike: (04152019 09:29 PM)Mike (Austria) Wrote: Thanks Valentin for this inspiring minichallenge. I particularly enjoy math riddles when they involve littleknown facts about wellknown mathematics. This one definitely fits this description particularly well  the fact that the various means are discrete members of a continuous family had escaped my attention so far (for decades  embarrassingly enough :) ). You're welcome, thanks to you for your interest and awesome posts on this minichallenge. And I agree with you, I'm also enthralled when I find an interesting piece of mathematics that happens to be new to me, so my goal has always been to post the kind of articles and challenges that I myself would like to read and solve. Quote:Since there have been no solutions for RPL, let me share my take on this. It should work for any version of RPL starting with the HP48G(X). Great ! I always welcome RPL solutions, they tend to be a little on the cryptic side but yours qualifies as an exception, pretty clear code made even clearer by your generous assortment of comments. Quote:I like the terseness of this function. Terse it is, and the results are of course correct. Quote:A further step in generalizing the mean is by replacing the \(p\)th power of \(x_i\) by applying an arbitrary continuous strictly monotonic function and taking its inverse after computing the sum and dividing by \(n\). Yes, it's an interesting way of generalizing it further. Nice addition to the subject matter. Quote:Why pick one of the original list elements to estimate the solution: we make no assumptions about the domain of \(f\). [...] This is cheaper than e.g. taking the arithmetic mean as an estimate or bracketing the solution by searching the minimum and maximum of {\(x_i\)}. Cheaper or not, I would've used the minimum/maximum as they're immediately available without search using the available matrix functions and they certainly bracket the solution perfectly. Not to say that your method isn't perfectly adequate, of course. Quote:I hope all this makes sense. Setting up these functions was fun  thanks for giving me an excuse to learn and play with stuff i'd otherwise never have looked at again. Despite RPL's deficiencies (such as handling many trivial cases incorrectly), I like the expressive power and abstraction / object orientation of this language. I'm glad you enjoyed it all and thanks again for your extensive, very interesting post (from which I also learned interesting things) and for your time, much appreciated. Oh, and welcome to the new forum, for a very first post you did really great ! ... Best regards. V. . 

04222019, 02:02 PM
Post: #25




RE: [VA] SRC#005 April, 1st Mean Minichallenge
(04162019 07:23 PM)Mike (Austria) Wrote: Addendum: for stackonly purists, here comes a 63.5 bytes twoliner implementation GMS which does the same as above GM in slightly less time: I kinda miss to understand the "twoliner" thing. 

04242019, 06:54 PM
Post: #26




RE: [VA] SRC#005 April, 1st Mean Minichallenge
(04222019 02:02 PM)RMollov Wrote:(04162019 07:23 PM)Mike (Austria) Wrote: Addendum: for stackonly purists, here comes a 63.5 bytes twoliner implementation GMS which does the same as above GM in slightly less time: I was just trying to be silly by applying a meaningless metric to a piece of freefrom code. No harm meant! 

04242019, 08:54 PM
Post: #27




RE: [VA] SRC#005 April, 1st Mean Minichallenge
Hi Valentin,
(04212019 08:38 PM)Valentin Albillo Wrote: You're welcome, thanks to you for your interest and awesome posts on this minichallenge. And I agree with you, I'm also enthralled when I find an interesting piece of mathematics that happens to be new to me, so my goal has always been to post the kind of articles and challenges that I myself would like to read and solve. Wow, thanks for your appreciation! Quote:Quote:Why pick one of the original list elements to estimate the solution: we make no assumptions about the domain of \(f\). [...] This is cheaper than e.g. taking the arithmetic mean as an estimate or bracketing the solution by searching the minimum and maximum of {\(x_i\)}. OK. Replacing in GFM the Code: x 1 GET Code: x « MIN » STREAM BTW, I unsystematically benchmarked both versions (GFM = original version, GFBM = bracketing version), using Emu48 on Android (authentic speed) to compute the generalized \(exp\) mean of a 100 element uniform random list 'V100': Code: 1 RDZ @ seed RAND Code: 0 'C' STO On the other hand, timing the original EXP function using an anonymous timing routine from hpcalc, Code: V100 Best wishes, Mike 

04252019, 12:05 PM
Post: #28




RE: [VA] SRC#005 April, 1st Mean Minichallenge
(04242019 06:54 PM)Mike (Austria) Wrote: I was just trying to be silly by applying a meaningless metric to a piece of freefrom code. No harm meant!Thanks Mike, I really got puzzled by the thing, interesting link  now it makes sense (in a way) Cheers, 

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