# Thread: Specialized calculation / approximation tools - is there a desire for them on here?

1. Hi all! I'll try and be brief yet concise (the former won't happen, and the jury's out on the latter), but please bear with me... Not to mention wish me luck, there's a dear. Thanks! I'm currently learning C-programming, and as such, am in the process of designing a number of 'portable', mathematical, approximation tools (simple C-programs anyone could download and run on their compooter). In short, simple programs for, say, approximating the value of certain higher functions, like the Gamma function, Clausen function, Polygamma function, Barnes' G-function, and the How-much-is-that-doggy-in-the-window function (Hmm!). On that note, here's a few random Q's:

- Since a lot of folk on here likely don't have fancy tools like Mathematica - myself included - would/could there (a) be a place to upload a few such simple tools on here, for others to use, and (b) would there be a demand for such things to begin with?

Well actually, that's it. That's all my Q's spent, and all in one go, no less. I feel quite poor now. Intellectually skint.

One other thing... (yay! I found my mental wallet again!):

Depending on the type and complexity of the functions involved - and I'd have to have a basic grasp of the function and branch of mathematics in question - I'd be up for doing a few bespoke programs, as per forum members (potential) requests.

To give just one (obscure) example, I'm currently quite interested in limits of infinite square roots, possibly alternating, so, to sate my own particular mathematical fetishes, I'm currently working on programs to approximate the following functions:

$\displaystyle \mathscr{P}(x) = \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \cdots } } } } }$

$\displaystyle \mathscr{Q}(x,y) = \sqrt{x+ \sqrt{y+ \sqrt{x+ \sqrt{y+ \sqrt{x+ \cdots } } } } }$

$\displaystyle \mathscr{R}(x,y) = \sqrt{x+ \sqrt{y- \sqrt{x+ \sqrt{y- \sqrt{x+ \cdots } } } } }$

I'll also be doing quite a few programs for varying-degree precision calculations of infinite series and infinite products. Such programs, once a template is made, could easily be adapted for things like, say, the Riemann Zeta function $\displaystyle \zeta(x)$, to name but one example. If the place to upload them is there, and the demand is there too, I'd be more than happy to take requests, and build small, portable, bespoke programs (as and when time permits).

Up to you guys and gals...

All the best!! Gethin  Reply With Quote

2.

3. I say go for it.

Incidentally, with the first one, you can do this:

\begin{align*}
y&= \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \cdots } } } } } \\
y^2&=x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \cdots } } } } \\
y^2 - x&= \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \cdots } } } } \\
y^2-x&=y \\
y^2 - y - x&=0 \\
y&=\frac{1\pm \sqrt{1+4x}}{2},
\end{align*}
for $1+4x\ge 0$.  Reply With Quote Originally Posted by Ackbach I say go for it.
OK then, will do!    Originally Posted by Ackbach Incidentally, with the first one, you can do this:

\begin{align*}
y&= \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \cdots } } } } } \\
y^2&=x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \cdots } } } } \\
y^2 - x&= \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \cdots } } } } \\
y^2-x&=y \\
y^2 - y - x&=0 \\
y&=\frac{1\pm \sqrt{1+4x}}{2},
\end{align*}
for $1+4x\ge 0$.
Yes I know, it's a beauty that one. And incidentally, it has deep ties to certain results about hypergeometric functions...   Reply With Quote

5. You can do a similar thing for the other two, but you get a quartic. Mathematica spits out the answer pretty fast, but it's quite complicated.  Reply With Quote Originally Posted by Ackbach You can do a similar thing for the other two, but you get a quartic. Mathematica spits out the answer pretty fast, but it's quite complicated.
Aha! Thanks for that, Ackbach!! Alas, I don't have Mathematica, so that's why I was thinking of creating such tools, for those poor souls like myself without them.

More generally - just on this one particular type of functions - the examples given above were of the more basic kind. The general case I'm examining is

$\displaystyle \Omega_{\infty}(s,x) = \Bigg[ \Bigg[ \Bigg[ \Bigg[ \cdots \, + x \Bigg]^s \, + x \Bigg]^s \, + x \Bigg]^s \, + x \Bigg]^s$  Reply With Quote

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