Peeking at different interpolation algorithms

[Albert Chan]

With divided difference, there is no requirement that the table be sorted.
Just do Forward Divided Difference, with the entries closest to interested region up on top.

Example, from https://jp.mathworks.com/matlabcentral/f...on-formula
(note: the link had the entries entered wrong, below is corrected points)

Points: (-2, 18.4708), (-1, 17.8144), (0, 17.107), (1, 16.3432), (2, 15.5154), find y(0.25):

Assume calculations with 6 sig. digits, and we want best interpolated values in the middle:

Code:

p   y(p)      Divided Difference Table
+0  17.107
+1  16.3432  -0.763800
-1  17.8144  -0.735600  -0.0282000
+2  15.5154  -0.766333  -0.0307330  -0.00126650
-2  18.4708  -0.738850  -0.0274830  -0.00108333  -0.0000915850

y(p) = 17.107 + (p-0)*(-0.7638 + (p-1)*(-0.0282 + (p+1)*(-0.0012665 + (p-2)*(-0.000091585))))

→ y(0.25) ≈ 16.9216

Above formula is same (except slight rounding errors) as Gauss Forward Formula:

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