What it is
The Newton interpolation polynomial service helps you interpolate or extrapolate the value of a function at an arbitrary point using a table of known values, automatically choosing the forward or backward Newton interpolation formula.
If the interpolation nodes are equally spaced and ordered by value, so that xi+1 − xi = h = const, i.e. xi = x0 + i·h, i = 0, 1, …, n, the interpolation polynomial can be written in Newton form. Exactly one polynomial of degree at most n passes through n + 1 points — the same one the Lagrange polynomial gives, but Newton's form is more convenient for tables with a constant step.
Finite differences
The formulas use finite differences: Δyi = yi+1 − yi (first), Δ2yi = Δyi+1 − Δyi (second), and so on. For y = x3 at the nodes 1, 2, 3, 4:
| x | y | Δy | Δ2y | Δ3y |
|---|---|---|---|---|
| 1 | 1 | 7 | 12 | 6 |
| 2 | 8 | 19 | 18 | |
| 3 | 27 | 37 | ||
| 4 | 64 |
The third differences are constant — the sign of a cubic polynomial.
Newton formulas
First (forward) Newton interpolation formula, q = (x − x0) / h:
Pn(x) = y0 + qΔy0 + q(q−1)/2!·Δ2y0 + … + q(q−1)…(q−n+1)/n!·Δny0
Second (backward) Newton interpolation formula, q = (x − xn) / h:
Pn(x) = yn + qΔyn−1 + q(q+1)/2!·Δ2yn−2 + … + q(q+1)…(q+n−1)/n!·Δny0
Newton interpolation polynomials are convenient when the interpolation point is near the beginning (forward Newton formula) or the end (backward Newton formula) of the table: the first, most significant terms then come from the nearest nodes and the rounding error is smaller.
Example
- Forward formula, x = 1.5: h = 1, q = (1.5 − 1)/1 = 0.5. P = 1 + 0.5·7 + 0.5·(−0.5)/2·12 + 0.5·(−0.5)·(−1.5)/6·6 = 1 + 3.5 − 1.5 + 0.375 = 3.375 = 1.53.
- Backward formula, x = 3.5: q = (3.5 − 4)/1 = −0.5. P = 64 + (−0.5)·37 + (−0.5)·0.5/2·18 + (−0.5)·0.5·1.5/6·6 = 64 − 18.5 − 2.25 − 0.375 = 42.875 = 3.53.
Since the function is a cubic and there are four nodes, the result is exact.
Good to know
- The step h must be constant and non-zero; for an uneven grid use the Lagrange polynomial or divided differences.
- Adding a new node does not require recomputing the whole polynomial — one more term is appended. This is the main practical advantage of Newton's form.
- With many nodes a high-degree polynomial can oscillate strongly between them (Runge's phenomenon); in practice 3–6 nearest nodes are used.
- Outside the table (extrapolation) accuracy drops quickly.
- You can also use the services Linear interpolation and extrapolation, Bilinear interpolation and Lagrange polynomial.
Where it is used
- Function tables and handbooks — values between rows of tables with a constant step (logarithms, trigonometry, material properties by temperature).
- Numerical methods — numerical differentiation and integration (Simpson's and Adams formulas) are built on Newton's formulas.
- Measurement processing — instrument readings taken at equal time intervals: the value at an intermediate moment.
- Astronomy and navigation — interpolating ephemerides (positions of planets and satellites) tabulated at equal intervals.
- Economics — filling gaps in time series with a constant step (months, quarters).
- Studying — checking numerical-methods homework.
How to use
- Enter x0 and x1 — they define the step h; the remaining nodes x2, x3, … are filled automatically.
- Enter f(xi) at each node. The "+" and "−" buttons add and remove points.
- Put the point you need into xN and press "Calculate" (or enable auto-calculation with the box in the button).
- The calculator shows h, q, the result f(xN) and which formula — forward or backward — was used.