[Edu-sig] "The study of fixed points has been at the foundation of algorithms"

[Arthur]

A bit of a windy road:

starting, as usual, with the personal frame of reference....

PyGeo's current implementation supports the exploration of the geometry 
of complex numbers, and therefore speaks Mobius transformations.

http://pygeo.sourceforge.net

now has a  pretty picture of a simple recursive transformation of 4 
circles on the unit sphere (...thanks to __iter__   the ability to 
recursively transform any arbitrary set of geometric objects is now 
built-in to PyGeo).

My current exploration (current as in today) is finding the mechanism to 
build a Mobius transformation that would be based (in part) on it's 
(pickable and movable) fixed points  - of which a Mobius transformation 
has 2, which may coincide, or be located inconveniently - e.g. at  infinity.

Which has me stepping into the math of the fixed points of a function - 
it being trivial to find the fixed points, given the Mobius 
transformation matrix, but less trivial (from where I am sitting at the 
moment) to build the transformation from fixed point information. So  I 
am struggling and researching some.

In the course of which I come across this  definition of "Fixed Point" 
in a programming glossary, @

http://carbon.cudenver.edu/~hgreenbe/glossary/second.php?page=F.html

Of a function, f:X-->X, f(x)=x. Of a point-to-set map, F:X-->2^X, x is 
in F(x). The study of fixed points has been at the foundation of 
algorithms 
<http://carbon.cudenver.edu/%7Ehgreenbe/glossary/second.php?page=A.html#Algorithm>.

Having discussed here my growing interest in some study of algorithmics, 
but not getting there yet, but  pursuing something that as far as I am 
aware is unconnected to such study, and then finding this statement 
indicating there is more of a connection - perhaps - than I had 
understood, is to me interesting.

I have thought of "fixed point"  (in programming) as connected 
to/opposed to "floating point", not as something directly connected to 
the concept of "f(x)=x"

The statement above seems to be telling me otherwise.

Guess I am fishing for some exposition on the statement that the

"The study of fixed points has been at the foundation of algorithms"

Art