Newton Interpolation

Updated 01 Oct 2026

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What it is
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  builds the interpolating polynomial using divided differences
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Formula
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  $f[x_i,x_{i+1}]=(f(x_{i+1})-f(x_i))/(x_{i+1}-x_i)$ .
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  Higher divided differences apply this difference-quotient rule recursively.
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  $P_n(x)=f[x_0]+\sum_{k=1}^{n}f[x_0,\ldots,x_k]\prod_{j=0}^{k-1}(x-x_j)$ .
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Example
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  For $(0,1),(1,3),(3,25),(4,49)$ : $P_3(x)=1+2x+3x(x-1)+\frac13x(x-1)(x-3)$ .
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Implementation
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  nested multiplication evaluates the polynomial efficiently
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  adding a point can extend the divided-difference representation without rebuilding all earlier coefficients
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  Newton-Gregory formulas specialize the construction to equally spaced nodes
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Related
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  Polynomial Interpolation

Linked references 3