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Difference engine
A Difference engine is a mechanical special-purpose computer designed to
tabulate polynomial functions. Since logarithmic and trigonometric functions
can be approximated by polynomials, such a machine is more general than it
appears at first.
History
The first of these devices was conceived in 1786 by J. H. Mueller. It was
never built.
Difference engines were forgotten and then rediscovered in 1822 by Charles
Babbage. This machine used the decimal numbers system and was powered by
cranking a handle. The British government first financed the project but
then later cut off support. Babbage went on to design his much more general
analytical engine but later returned and produced an improved design (his
"Difference Engine No. 2") between 1847 and 1849. Inspired by Babbage's
difference engine plans, George Scheutz built several difference engines
from 1855 onwards; one was sold to the British government in 1859.
Based on Babbage's original plans, the London Science Museum constructed a
working Difference Engine No. 2 from 1989 to 1991. In 2000, the printer
originally designed by Babbage was also completed. Both worked flawlessly.
The difference engine and printer were constructed to tolerances achievable
with 19th century technology, resolving a long-standing debate whether
Babbage's design would actually have worked.
Method of Differences
The principle of a difference engine is Newton's method of differences. It
may be illustrated with a small example. Consider the quadratic polynomial
p(x) = 2x2 - 3x + 2
and suppose we want to tabulate the values p(0), p(0.1), p(0.2), p(0.3),
p(0.4) etc. The table below is constructed as follows: the first column
contains the values of the polynomial, the second column contains the
differences of the two left neighbors in the first column, and the third
column contains the differences of the two neighbors in the second column:
p(0)=2.0
2.0-1.72=0.28
p(0.1)=1.72 0.28-0.24=0.04
1.72-1.48=0.24
p(0.2)=1.48 0.24-0.20=0.04
1.48-1.28=0.20
p(0.3)=1.28 0.20-0.16=0.04
1.28-1.12=0.16
p(0.4)=1.12
Notice how the values in the third row are constant. This is no coincidence.
In fact, if you start with any polynomial of degree n, the column number n +
1 will always be constant. This crucial fact makes the method work, as we
will see next.
We constructed this table from the left to the right, but now we can
continue it from the right to the left in order to compute more values of
our polynomial.
To calculate p(0.5) we use the values from the lowest diagonal. We start
with the rightmost column value of 0.04. Then we continue the second column
by subtracting 0.04 from 0.16 to get 0.12. Next we continue the first column
by taking its previous value, 1.12 and subtracting the 0.12 from the second
column. Thus p(0.5) is 1.12-0.12 = 1.0. In order to compute p(0.6), we
iterate the same algorithm on the p(0.5) values: take 0.04 from the third
column, subtract that from the second column's value 0.12 to get 0.08, then
subtract that from the first column's value 1.0 to get 0.92, which is
p(0.6).
This process may be continued ad infinitum. The values of the polynomial are
produced without ever having to multiply. A difference engine only needs to
be able to subtract. From one loop to the next, it needs to store 2 numbers
in our case (the last elements in the first and second columns); if we
wanted to tabulate polynomials of degree n, we'd need enough storage to hold
n numbers.
Babbage's difference engine No. 2, finally built in 1991, could hold 7
numbers of 31 decimal digits each and could thus tabulate 7th degree
polynomials to that precision. The best machines from Scheutz were able to
store 4 numbers with 15 digits each.
This content from Wikipedia is licensed under the GNU Free Documentation License.
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