Newton's Interpolation Calculator

Newton's interpolation is a numerical method for estimating unknown values between known data points. It constructs a polynomial (Newton's interpolating polynomial) that passes exactly through all the given (x, y) data points using a system of finite differences or divided differences. The resulting polynomial can then be evaluated at any intermediate x value.

Newton's Forward Difference formula is most accurate when interpolating near the beginning of the data table, while the Backward Difference formula is preferred when interpolating near the end of the table. Both require equally spaced x values. The 'Auto Select' option in this calculator automatically picks the better method based on where your target x falls relative to the dataset.

Newton's Divided Differences are a generalization of finite differences that work with unequally spaced x values. Instead of computing Δ (delta) differences with a fixed step h, divided differences are computed as recursive quotients: f[x0,x1] = (f(x1)−f(x0))/(x1−x0). This method is more flexible than the forward/backward formulas and is related to Lagrange interpolation.

The s value (sometimes called p or u) is a normalized parameter defined as s = (x − x₀) / h for the forward formula, where x is the interpolation point, x₀ is the first (or last) tabular value, and h is the step size. It represents how many steps away from the reference point your target x lies, and it is used in the Newton forward/backward polynomial formula with binomial-like coefficients.

For Newton's Forward and Backward Difference formulas, yes — the x values must be equally spaced (constant step h). If your data has unequal spacing, the calculator will automatically use Newton's Divided Difference formula, which works for any spacing. Always double-check that your data is sorted in ascending order of x.

You need at least 2 data points to perform interpolation. In practice, 4–6 points give a good balance between accuracy and polynomial degree. More points can increase accuracy but also increase the degree of the polynomial, which may introduce oscillation (Runge's phenomenon) for large datasets.

Both Newton's and Lagrange's methods construct the same unique interpolating polynomial for a given set of points — they just use different mathematical representations. Newton's form is typically easier to update when new data points are added, while Lagrange's form is symmetric and straightforward to write down. The two polynomials, when fully simplified, produce identical results.

The difference table displays successive finite differences (Δ¹, Δ², Δ³, …) computed from your f(x) values. Each column is obtained by subtracting consecutive entries from the previous column. The first diagonal of this table contains the values used in Newton's Forward formula, while the last diagonal contains the values for the Backward formula. It gives insight into the smoothness and behaviour of your data.