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Difference engine



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''''' + ...

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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.

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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. 

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See also

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External links




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