Newton Interpolation Polynomial

← Numerical Methods

Source inspiration: (Mathew 2000-2019).

Description

Newton interpolation represents the interpolating polynomial in divided-difference form,

\[ P_n(x)=a_0 + a_1(x-x_0)+a_2(x-x_0)(x-x_1)+\cdots+a_n\prod_{j=0}^{n-1}(x-x_j), \]

where \(a_k=f[x_0,\ldots,x_k]\) are divided differences. For the same nodes and data, Newton and Lagrange forms produce the same polynomial; they differ only in representation and numerical workflow.

The 29 animations below intentionally reuse the same interpolation cases as the Lagrange page, matching the legacy Mathews case ordering.

Animations

Each animation increases polynomial degree one step at a time for a fixed function and interval.

Julia source for all cases

Case 01

Newton interpolation animation for case 01 from the legacy function-and-interval list.

Case 02

Newton interpolation animation for case 02 from the legacy function-and-interval list.

Case 03

Newton interpolation animation for case 03 from the legacy function-and-interval list.

Case 04

Newton interpolation animation for case 04 from the legacy function-and-interval list.

Case 05

Newton interpolation animation for case 05 from the legacy function-and-interval list.

Case 06

Newton interpolation animation for case 06 from the legacy function-and-interval list.

Case 07

Newton interpolation animation for case 07 from the legacy function-and-interval list.

Case 08

Newton interpolation animation for case 08 from the legacy function-and-interval list.

Case 09

Newton interpolation animation for case 09 from the legacy function-and-interval list.

Case 10

Newton interpolation animation for case 10 from the legacy function-and-interval list.

Case 11

Newton interpolation animation for case 11 from the legacy function-and-interval list.

Case 12

Newton interpolation animation for case 12 from the legacy function-and-interval list.

Case 13

Newton interpolation animation for case 13 from the legacy function-and-interval list.

Case 14

Newton interpolation animation for case 14 from the legacy function-and-interval list.

Case 15

Newton interpolation animation for case 15 from the legacy function-and-interval list.

Case 16

Newton interpolation animation for case 16 from the legacy function-and-interval list.

Case 17

Newton interpolation animation for case 17 from the legacy function-and-interval list.

Case 18

Newton interpolation animation for case 18 from the legacy function-and-interval list.

Case 19

Newton interpolation animation for case 19 from the legacy function-and-interval list.

Case 20

Newton interpolation animation for case 20 from the legacy function-and-interval list.

Case 21

Newton interpolation animation for case 21 from the legacy function-and-interval list.

Case 22

Newton interpolation animation for case 22 from the legacy function-and-interval list.

Case 23

Newton interpolation animation for case 23 from the legacy function-and-interval list.

Case 24

Newton interpolation animation for case 24 from the legacy function-and-interval list.

Case 25

Newton interpolation animation for case 25 from the legacy function-and-interval list.

Case 26

Newton interpolation animation for case 26 from the legacy function-and-interval list.

Case 27

Newton interpolation animation for case 27 from the legacy function-and-interval list.

Case 28

Newton interpolation animation for case 28 from the legacy function-and-interval list.

Case 29

Newton interpolation animation for case 29 from the legacy function-and-interval list.

Derivation Notes (Planned)

Short derivations will be added for divided differences, nested evaluation, and incremental updates.

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.