Lets say you needed to calculate the successive values of some function. No=
t
any specific value, but the values for successive integer inputs. E.g. not
f(x), but first f(0) then f(1), f(2), f(3) and so on.

Charles Babbage's difference engine design used Newton's method of divided
differences to calculate 8th order polynomials via finite differences. I
thought that this would allow him to effectively calculate just about any
useful function with high enough precision to print a sellable table... but
perhaps not?

I'm trying to figure out how he would have setup the difference engine for
an "interesting" function and see if that method can be adapted to a PIC or
other small uC.

Specifically, I wanted to calculate Yx^(-0.5) for fixed values of Y and x=
=3D1
to whatever as x is incremented by 1 each calculation... For example,
1132x^(-.5), or 432x^(-.5)

I tried in an excel spreadsheet, to just set up a series of columns, where
each cell in the column is calculated from the difference of the cell to th=
e
left, and the cell one to the left and one down. But the values never zero
out as you add columns. The first cell in each column always has (an ever
smaller) value, but they don't get small enough, quickly enough to make it
practical to calculate that by finite differences.

That method seems to work for the data from e.g. 1322x^0.5, y =3D -0.0014x4=
 +
0.1987x3 - 11.134x2 + 394x + 1151.9 is a very close fit and so just 4
columns gives a good approximation. I could store 4 32 bit numbers in the
PIC, and do a few subtractions at each step along the way, and have a reall=
y
good output.

So I tried getting excel to approximate the function 1132x^(-.5) as a poly,
but even with an order of 6, the poly is a poor fit. Excel won't do 8th
order, but I doubt even that would have been good enough.

Is it that there are certain functions which can not be practically
approximated in this way? E.g. most polynomials are fine, but negative powe=
r
functions don't work well at all?=20

If so, did Babbage not plan on publishing a table of negative square roots?
And was that perhaps one of the things that drove him to design the
analytical engine?

Or is there some neat mathematical manipulation of "to the power of negativ=
e
one half" that makes it easy to fit to a polynomial?

--
James Newton
1-970-462-7764=20

P.S. Hi Scott!

--=20
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