Newton Interpolation Polynomial

Source inspiration: (Mathew 2000-2019).

Animations

Each animation below shows the degree-\(n\) Newton interpolating polynomial built by adding one equally-spaced node per frame using divided differences. The first case shows well-behaved convergence; the second demonstrates the Runge phenomenon for a function with a steep peak.

Julia source scripts that generated these animations are linked under each case.

Case 1 — Convergent interpolation, \(f(x) = \cos(x)\), 7 nodes on \([0,2\pi]\)

Behavior: With 7 equally-spaced nodes the degree-6 Newton polynomial closely tracks \(\cos(x)\). Adding each node reduces the error substantially, and the improvement is monotone because \(\cos(x)\) is smooth and the nodes are well-placed relative to the function’s variation.

Julia source

Newton interpolating polynomial for cos(x) on [0,2pi] growing from degree 0 to degree 6 as nodes are added one at a time; the polynomial converges to the true function

Case 2 — Runge phenomenon, \(f(x) = 1/(1+25x^2)\), 11 nodes on \([-1,1]\)

Behavior: With equally-spaced nodes the high-degree Newton polynomial oscillates wildly near the endpoints even though the function is smooth. This is the Runge phenomenon: high-degree polynomial interpolation at equally-spaced nodes can diverge, motivating the use of Chebyshev nodes or piecewise methods.

Julia source

Newton polynomial for the Runge function 1/(1+25x^2) on [-1,1] showing increasing oscillation near the endpoints as degree grows from 0 to 10; the classic Runge phenomenon

Derivation Notes (Planned)

Short derivations will be added to explain the core equations and assumptions.

References

Mathew, John H. 2000-2019. Numerical Analysis - Numerical Methods Modules. https://web.archive.org/web/20190808102217/http://mathfaculty.fullerton.edu/mathews/n2003/NumericalUndergradMod.html.