Peeking at different interpolation algorithms

[Albert Chan]

(08-02-2019 09:27 PM)Albert Chan Wrote:  

Code:

p   y(p)      Difference Table
-2  18.4708  
-1  17.8144  -0.6564
+0  17.107   -0.7074  -0.0510
+1  16.3432  -0.7638  -0.0564  -0.0054
+2  15.5154  -0.8278  -0.0640  -0.0076  -0.0022

y(p) = \(17.107 + \binom{p}{1}(-0.7638) + \binom{p}{2}(-0.0564)
+ \binom{p+1}{3}(-0.0076) + \binom{p+1}{4}(-0.0022) \)

We can build interpolation formula on the fly, for any path, without looking up.
Just think of what the interpolation X values sequence.
Then assume the list were originally sorted the same way, with numbers on the diagonal.

For Above Gauss Forward Formula, X's = 0, 1, -1, 2, -2

y(p) = 17.107 + (p-0)/1*(-0.7638 + (p-1)/2*(-0.0564 + (p+1)/3*(-0.0076 + (p-2)/4*(-.0022)))))

Had we wanted Gauss Backward Formula, X's = 0, -1, 1, -2, 2

y(p) = 17.107 + (p-0)/1*(-0.7074 + (p+1)/2*(-0.0564 + (p-1)/3*(-0.0054 + (p+2)/4*(-.0022)))))