Correction:  it should be 64 + 4/129, which comes out to 4099.969954,=20
still close enough.

Kerry


Kerry Wentworth wrote:
> OK, here's some wild speculation.
>
> Assuming he'd want to live to collect the first royalty check, he=20
> probably wouldn't be doing a lot of floating point math.
> Ada Lovelace (daughter of Lord Byron and Baroness Wentworth) no doubt=20
> knew that squares were the sum of successive odd numbers, i.e 4 =3D 1+3, =
9=20
> =3D 1+3+5, 16 =3D 1+3+5+7, with the number of odds added together equalin=
g=20
> the square root.  A fair approximation of a square root calculated in=20
> this manner would be the number of odds added + (current number - last=20
> perfect square) / next odd number.
> For example, the (approximate) square root of 4100 would be 64 (square=20
> root of 4096, which is 1+3+5...+127) + 4 / 127.  My calculator shows=20
> that to actually be the square root of 4100.032488, which is close enough=
..
> So, by tracking the square root as you go instead of calculating it each=
=20
> time can greatly speed up the calculations.  With a little cleverness=20
> applied to scaling and rounding, I'm sure this is the method she would=20
> have used.
>
> Kerry
>
>
> James Newton wrote:
>  =20
>> Lets say you needed to calculate the successive values of some function.=
 Not
>> any specific value, but the values for successive integer inputs. E.g. n=
ot
>> 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 divid=
ed
>> differences to calculate 8th order polynomials via finite differences. I
>> thought that this would allow him to effectively calculate just about an=
y
>> 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 f=
or
>> 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, whe=
re
>> each cell in the column is calculated from the difference of the cell to=
 the
>> left, and the cell one to the left and one down. But the values never ze=
ro
>> out as you add columns. The first cell in each column always has (an eve=
r
>> 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 -0.001=
4x4 +
>> 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 th=
e
>> PIC, and do a few subtractions at each step along the way, and have a re=
ally
>> good output.
>>
>> So I tried getting excel to approximate the function 1132x^(-.5) as a po=
ly,
>> 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 p=
ower
>> functions don't work well at all?=20
>>
>> If so, did Babbage not plan on publishing a table of negative square roo=
ts?
>> 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 nega=
tive
>> one half" that makes it easy to fit to a polynomial?
>>
>> --
>> James Newton
>> 1-970-462-7764=20
>>
>> P.S. Hi Scott!
>>
>>  =20
>>    =20
>
>
>
>  =20



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