I've been researching continued fractions, aka fractions whose denominators are fractions whose denominators are fractions whose denominators are.... You get the picture. I'm slowly making progress with my programming language, and I want it to have a good math library, so, like I said, I've been researching these funny things called continued fractions, reading up on some freakin' genius by the name of Gosper who's probably long dead by now. Based on his method of converting continued fractions into matrix multiplications (using something he calls "homographic functions" but which Mr. Google calls "Mobius Transforms"), I've managed to actually come up with something useful. I highly doubt it's original (it's simple enough to be a math student's homework problem), but I did arrive at it on my own, so I'm kinda proud of it.
Below is my lunchtime derivation of a square-root formula, done on a napkin no less! This will give you the square-root of a (positive) number to any degree of precision you like: just stop multiplying matrices when you get tired or when your answer's good enough.
15 comments:
*shudder* Looking at that brings back some bad memories...
...however, kudos to you for working through it and coming up with the formula. I have long learned that it takes a very special mind to be able to comprehend such math. Way to go!!
Me...I'll stick to accounting, thanks! :-)
Ouch. You hurt my head.
Are these transforms related in any way to Fourier transforms?
Also - could one use this to rebuild corrupted data? I'm thinking that if one had only a fraction of the data and you make assumptions of its value you could solve for the unknown value in the corrupted data and attempt to repair it.
Ack, I've lost what little matrix math I learnt. But it *feels* right... :-)
Alex, no and nope. Actually, they may be related to Fourier transforms at some deep level, but that would be vastly above my head.
Thanks for the clarification.
No problem; one day you're going to have to beat my brains through FFTs again (offline); I never use them so I keep forgetting how they work.
::sigh::
I like math.
I just learned it so badly... and never got to this stuff... so I break down right at the very end.
It's like missing the punch line to a joke... :(
you won't get the last step without reading a BUNCH more of it: just assume that the matrix transform is valid and pay attention to the x= part.
Besides, they never told you UD people that you can't get math holistically. :-)
So, you're playing with refraction indices? Unless I'm mistaken this looks like an impossible RTM analysis. So where are you going to get a lens with a focal length of -1? ;)
Negative indices of refraction have been created, but not in "light" wavelengths. Nah, I'm looking at it for the math library that my programming language will have. I want to be able to do trigonometry with unlimited numerical precision.
What's an RTM analysis, root-mean-squared?
Sorry, that's ray transfer matrix. Ray transfer matrix analysis is used in optics to describe whether and how light is bent on a variety of optical surfaces. Your matrix expression appears to describe a lens with a refraction index of -1.
Negative index materials themselves were hypothesized in the late 1960s by Victor Veselago, and I remember seeing a couple articles on current research into negative-index materials in a back issue of PhysicsWorld.
I've read a fair amount about how to raytrace lenses and the like, but only for paraxial rays. I quit delving deeper when I realized how expensive it would be to get anything custom-manufactured.
Thinking about what you said....
These matrices don't encode transforms of 2-D position against an [x y] argument (can't type vertical matrices in this box); they represent the coefficients of (ax+b)/(cx+d).
Ah, OK. I see that now, especially after I've been away from it for a couple days, and had something that closely resembles sleep.
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