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2 Method of Differences 3 External links |
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.
The principle of a difference engine is Newton's method of differences. It may be illustrated with a small example. Consider the quadratic polynomial
History
Method of Differences
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.
See also: The Difference Engine