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Newton Forward Interpolation

Newton Forward Interpolation is a numerical method used to estimate the values of a function based on a set of known values. This technique is particularly useful when the function is tabulated, and we want to predict values for points that fall within the range of the known data. The method is based on the concept of finite differences and utilizes polynomial interpolation to achieve accurate estimates.

Key Concepts:

Finite Differences: The differences between successive values in a table of data are called finite differences. These differences are used to construct the polynomial interpolation.

Newton's Forward Difference Formula: This method uses forward differences, which involve the data points arranged in increasing order, to compute the interpolated value.

Interpolation Polynomial: The goal is to find a polynomial P(x) that passes through the given data points. Newton’s Forward Interpolation constructs this polynomial based on the first n points.

Finite Difference Table

x	y	Δ	Δ²	Δ³	Δ⁴
1	10
2	26
3	58
4	112
5	194

X: r: NFIF:

\({y_{NFIF}} = {y_0} + r \times \Delta {y_0} + \frac{{r \times \left( {r - 1} \right)}}{{2!}}{\Delta ^2}{y_0} + \frac{{r \times \left( {r - 1} \right) \times \left( {r - 2} \right)}}{{3!}}{\Delta ^3}{y_0} + \frac{{r \times \left( {r - 1} \right) \times \left( {r - 2} \right) \times \left( {r - 3} \right)}}{{4!}}{\Delta ^4}{y_0}\)