(12C Platinum) Sums of Powers of N numbers

[Albert Chan]

Modified Aitken's method can interpolate slope too, then recover interpolated value.
Example, with 5-digits precision, calculate LN(12.3), with tables of LN (integer domain):

LN(12.3) is between LN(12)=2.4849, and LN(13)=2.5649
So, LN(12.3) = 2.5 (2 digits accurate), only 3 digits slope required

Code:

X   LN(X)   Slopes, interpolate for X=12.3
12  2.4849
13  2.5649  0.0800
11  2.3979  0.0870  825
14  2.6391  0.0771  820  823

Each interpolated diagonals gained 1 digits accuracy, so only 4 points needed.
Recover interpolated slope to value:

LN(12.3) = 0.0823 * (12.3-12) + 2.4849 = 2.5096 (5 digits)

Interpolations needed are reduced (only 3 interpolations for cubic fit, down 50%)
Also, with slopes interpolated to full precision, recovered result may be more accurate.