We know that the interpolation polynomial is unique and the formulae like Newton's forward formula, Newton's backward formula lagrange's formula, Stirling or bessel's formula etc. are just different forms of one and the same interpolation polynomial. Hence for interpolating at a point the results computed by the different formulas should be identical but in practice we see that above formulas are appropriate for interpolating at different zones then why does this Apparent contradiction arise? (For example, we use NFI for interpolating at the beginning of a table) Thank you.
2026-03-24 23:40:54.1774395654
A question on Interpolation..
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The Lagrangian polynomial is indeed unique and the other formulas must yield precisely the same polynomial (to numerical errors, possibly), as long as they are of the same degree. Notice anyway that Stirling and Bessel require an arithmetic progression of the independent variable.
"appropriate for interpolating for interpolating at different zones" is just a matter of operational convenience, i.e. more an algorithmic issue. But actually, the Neville scheme is more appropriate. https://en.wikipedia.org/wiki/Neville%27s_algorithm