Editing Difference engine (section)

== 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 <math>1_0</math> gets the value of the function at the start of computation <math>f(0)</math>. Col <math>2_0</math> is the difference between  <math>f(1)</math> and <math>f(0)</math>...<ref name="Thelen">{{cite web
|url=http://ed-thelen.org/bab/bab-intro.html
|title=Babbage Difference Engine #2 – How to Initialize the Machine –
|last=Thelen |first=Ed
|date=2008 }}</ref>

If the function to be calculated is a [[polynomial function]], expressed as
: <math> f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0 \, </math>
the initial values can be calculated directly from the constant coefficients ''a''<sub>0</sub>, ''a''<sub>1</sub>,''a''<sub>2</sub>, ..., ''a<sub>n</sub>'' without calculating any data points. The initial values are thus:

* Col <math>1_0</math> = ''a''<sub>0</sub>
* Col <math>2_0</math> = ''a''<sub>1</sub> + ''a''<sub>2</sub> + ''a''<sub>3</sub> + ''a''<sub>4</sub> + ... + ''a<sub>n</sub>''
* Col <math>3_0</math> = 2''a''<sub>2</sub> + 6''a''<sub>3</sub> + 14''a''<sub>4</sub> + 30''a''<sub>5</sub> + ...
* Col <math>4_0</math> = 6''a''<sub>3</sub> + 36''a''<sub>4</sub> + 150''a''<sub>5</sub> + ...
* Col <math>5_0</math> = 24''a''<sub>4</sub> + 240''a''<sub>5</sub> + ...
* Col <math>6_0</math> = 120''a''<sub>5</sub> + ...
* <math>...</math>

=== Use of derivatives ===
Many commonly used functions are [[analytic function]]s, which can be expressed as [[power series]], for example as a [[Taylor series]]. The initial values can be calculated to any degree of accuracy; if done correctly the engine will give exact results for first N steps. After that, the engine will only give an [[approximation]] of the function.

The Taylor series expresses the function as a sum obtained from its [[derivative]]s at one point.  For many functions the higher derivatives are trivial to obtain; for instance, the [[sine]] function at 0 has values of 0 or <math>\pm1</math> for all derivatives. Setting 0 as the start of computation we get the simplified [[Maclaurin series]]
:<math>
\sum_{n=0}^{\infin} \frac{f^{(n)}(0)}{n!}\  x^{n}
</math>

The same method of calculating the initial values from the coefficients can be used as for polynomial functions. The polynomial constant coefficients will now have the value
:<math>
a_n \equiv \frac{f^{(n)}(0)}{n!}
</math>

=== Curve fitting ===
The problem with the methods described above is that errors will accumulate and the series will tend to diverge from the true function. A solution which guarantees a constant maximum error is to use [[curve fitting]]. A minimum of ''N'' values are calculated evenly spaced along the range of the desired calculations. Using a curve fitting technique like [[Gaussian reduction]] an ''N''−1th degree [[polynomial interpolation]] of the function is found.<ref name="Thelen" /> With the optimized polynomial, the initial values can be calculated as above.