A brief diversion into genius ..

[lihan161051]

A lot has been written about Babbage's Difference Engine, and the method of divided differences it exploited to do its job, but having been exposed to the genius of it in ways I finally understand, I want to recognize it for the extraordinary clever idea it was

Comments

[notthebuddha]

And it only adds

If you go on to study abstract algebra, number theory, and/or digital logic, it is revealed that this is how everything works. That is, all the operations on rational numbers are built from addition, and a couple of extras of things included in addition like the axiom of choice.

As for polynomial approximations, that is right on track; there's a proof that you can approximate any function in L2 with three nested sets of polynomials of sufficient order... and guess who's almost done with a GPU accelerated application of it?

[notthebuddha]

BTW, L2 means any function whose square can be integrated, which is most functions that you'd want to work with anyway.

http://mathworld.wolfram.com/L2-Function.html

[lihan161051]

Yep, I can vouch for digital logic at least. (I'm pretty sure he knew about tens-complement addition for negative terms of the polynomials, because polynomials with only positive coefficients are of very limited usefulness.) And the method of finite differences is clearly a discrete example of differential calculus in general, and from there, it's obvious why this process would have worked very elegantly. I mainly posted this because now that I've gotten a detailed enough description of how the Difference Engine was actually supposed to work, I'm realizing just how far ahead of its time it really was, specialized though it might have been. :)

It's not much of a stretch to imagine a proof of that assertion, given the known examples of Taylor series approximations of trig and exponent functions and the ability to approximate many of them quite closely over at least some ranges with partial Taylor series. I'm interested in that GPU accelerated application when it reaches a stage where you can disclose it ..

[notthebuddha]

Thanks for the interest, it looks like we'll be releasing an open Beta around August. Do you have recent vintage Nvidia GeForce on any of your machines?