Difference engine

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The Difference Engine was designed to tabulate polynomial functions, and as both logarithmic and trigonometric functions can be approximated by polynomials, such a machine is more general than it appears at first.

Contents

[edit] History

Closeup of the London Science Museum's replica difference engine.
Closeup of the London Science Museum's replica difference engine.

J.H. Müller, an engineer in the Hessian army conceived the idea in a book published in 1786, but failed to find funding to progress this further.[1]
In 1822, Charles Babbage proposed the use of such a machine in a paper to the Royal Astronomical Society on 14 June entitled "Note on the application of machinery to the computation of very big mathematical tables." [2] This machine used the decimal number system and was powered by cranking a handle. The British government initially financed the project, but withdrew funding when Babbage repeatedly asked for more money whilst making no apparent progress on building the machine. Babbage went on to design his much more general analytical engine but later produced an improved difference engine design (his "Difference Engine No. 2") between 1847 and 1849. Inspired by Babbage's difference engine plans, Per Georg Scheutz built several difference engines from 1855 onwards; one was sold to the British government in 1859. Martin Wiberg improved Scheutz's construction but used his device only for producing and publishing printed logarithmic tables.[citation needed]

Based on Babbage's original plans, the London Science Museum constructed a working Difference Engine No. 2 from 1989 to 1991, under Doron Swade, the then Curator of Computing. This was to celebrate the 200th anniversary of Babbage's birth. In 2000, the printer which Babbage originally designed for the difference engine was also completed. The conversion of the original design drawings into drawings suitable for engineering manufacturers' use revealed some minor errors in Babbage's design, which had to be corrected. Once completed, both the engine and its printer worked flawlessly, and still do. 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. (One of the reasons formerly advanced for the non-completion of Babbage's engines had been that engineering methods were insufficiently developed in the Victorian era.) In addition to funding the construction of the output mechanism for the Science Museum's Difference Engine No. 2, Nathan Myhrvold commissioned the construction of a second complete Difference Engine No. 2, which will be on exhibit at the Computer History Museum in Mountain View, California from 10 May 2008 through April 2009. [3]

[edit] Operation

The difference engine consists of a number of columns, numbered from 1 to N. Each column is able to store one decimal number. The only operation the engine can do is add the value of a column n + 1 to column n to produce the new value of n. Column N can only store a constant, column 1 displays (and possibly prints) the value of the calculation on the current iteration.

The engine is programmed by setting initial values to the columns. Column 1 is set to the value of the polynomial at the start of computation. Column 2 is set to a value derived from the first and higher derivatives of the polynomial at the same value of X. Each of the columns from 3 to N is set to a value derived from the (n − 1) first and higher derivatives of the polynomial.

[edit] Timing

In the Babbage design, one iteration i.e. one full set of addition and carry operations happens once for four rotations of the column axis. Odd and even columns alternatively perform the addition every two rotations. The sequence of operations for column n is thus:

  1. Addition from column n + 1
  2. Carry propagation
  3. Addition to column n - 1
  4. Rest

[edit] Method of differences

The London Science Museum's replica difference engine, built from Babbage's design. The design has the same precision on all columns, but when calculating converging polynomials, the precision on the higher-order columns could be lower.
The London Science Museum's replica difference engine, built from Babbage's design. The design has the same precision on all columns, but when calculating converging polynomials, the precision on the higher-order columns could be lower.

As the differential engine cannot do multiplication, it is unable to calculate the value of a polynomial. However, if the initial value of the polynomial (and of its finite differences) is calculated by some means for some value of X, the difference engine can calculate any number of nearby values, using the method generally known as the method of finite differences.

The principle of a difference engine is Newton's method of divided 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:

polynomial differences differences
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 column are constant. This is no mere 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.

[edit] Initial values

The initial values of columns can be calculated by first manually calculating N consecutive values of the function, and by backtracking, i.e. calculating the required differences.

Col 10 gets the value of the function at the start of computation f(0). Col 20 is the difference between f(1) and f(0)...

[edit] Use of derivatives

A more general and in many cases more useful method is to calculate the initial values from the values of the derivatives of the function at the start of computation. Each value is thus represented as power series of the different derivates. The constants of the series can be calculated by first expressing a function as a Taylor series i.e. a sum of its derivatives. Setting 0 as the start of computation we get the Maclaurin series

\sum_{n=0}^{\infin} \frac{f^{(n)}(0)}{n!} (x)^{n}.

Calculating the values numerically, we get the following serial representations for the initial values of the columns:

Let f,f',f'',f''',f''''... be the values of the function and its derivatives at the start of computation

  • Col 10 = f
  • Col 20 = f' + 1 / 2f'' + 1 / 6f''' + 1 / 24f'''' + 1 / 120f''''' + ...
  • Col 30 = f'' + f''' + 14 / 24f'''' + 23 / 120f''''' + ...
  • Col 40 = f''' + 36 / 24f'''' + 171 / 120f''''' + ...
  • Col 50 = f'''' + 378 / 120f''''' + ...

[edit] References

  1. ^ Swedin, E.G. & Ferro, D.L. (2005). Computers: The Life Story of a Technology. Greenwood Press, Westport, CT. Retrieved on 2007-11-17.
  2. ^ Charles Babbage. The MacTutor History of Mathematics archive. School of Mathematics and Statistics, University of St Andrews, Scotland (1998). Retrieved on 2006-06-14.
  3. ^ Computer History Museum unboxes a Babbage difference engine. Retrieved on 2008-04-28.

[edit] Further reading

  • Swade, Doron (2002). The Difference Engine: Charles Babbage and the Quest to Build the First Computer. Penguin (reprint). ISBN 0-14-200144-9. 
  • Swade, Doron (2001). The cogwheel brain. Abacus. ISBN 0-349-11239-8. 

[edit] See also

[edit] External links