Interpolation

[mahjk17]

Consider the following table of values for a function \(\displaystyle j_0(x)\)

\(\displaystyle \begin{array}{c|ccccc} x & \delta_0(x)
\\ \hline 0.0&1.00000\\0.1&.99833\\0.2&.99335\\0.3&.98507\\0.4&.97355\\0.5&.95885\\0.6&.94107\\0.7&.92031\\0.8&.89670\\0.9&.87036\\1.0&.84147\\1.1&.81019\\1.2&.77670\\1.3&.74120
\end{array}\)

What should be the maximum degree of polynomial interpolation used
with the table?


I know that I must use the forward difference table so that I can detect the influence of the rounding errors.

 

[HallsofIvy]

Consider the following table of values for a function \(\displaystyle j_0(x)\)

\(\displaystyle \begin{array}{c|ccccc} x & \delta_0(x)
\\ \hline 0.0&1.00000\\0.1&.99833\\0.2&.99335\\0.3&.98507\\0.4&.97355\\0.5&.95885\\0.6&.94107\\0.7&.92031\\0.8&.89670\\0.9&.87036\\1.0&.84147\\1.1&.81019\\1.2&.77670\\1.3&.74120
\end{array}\)

What should be the maximum degree of polynomial interpolation used
with the table?


I know that I must use the forward difference table so that I can detect the influence of the rounding errors.

Given n points, there exist a unique polynomial of degree n- 1 passing through those points. Here, you have 13 points.

 

[DrPhil]

What should be the maximum degree of polynomial interpolation used
with the table?


I know that I must use the forward difference table so that I can detect the influence of the rounding errors.

Although it is mathematically possible to create a polynomial of degree n-1, your observation that you have to watch out for rounding errors is well founded. Take 1st differences, then 2nd differences, etc. I did not examine the entire table, but for the part I looked at I would stop at the third difference, where the values wobble around a bit. Can you enter the table in a spreadsheet to make the calculation of the nth differences easier?

From what I saw, I would probably take the average of all the 3rd differences and use that as a constant value. What degree of polynomial would that generate?