Calculator Inputs
Use equally spaced x-values. Separate numbers with commas, spaces, or new lines.
Example Data Table
This built-in example comes from y = x² + x + 1, so the second forward difference is constant.
| x | y | Δy | Δ²y |
|---|---|---|---|
| 0 | 1 | 2 | 2 |
| 1 | 3 | 4 | 2 |
| 2 | 7 | 6 | — |
| 3 | 13 | 8 | — |
| 4 | 21 | — | — |
Formula Used
u = (x - x0) / h f(x) ≈ y0 + uΔy0 + [u(u-1)/2!]Δ²y0 + [u(u-1)(u-2)/3!]Δ³y0 + ... Second-order form: f(x) ≈ y0 + uΔy0 + [u(u-1)/2]Δ²y0This method estimates an unknown function value from equally spaced tabulated data. The calculator creates forward differences, evaluates u, applies each interpolation term, and sums the chosen order.
How to Use This Calculator
- Enter x-values in increasing order. They must be equally spaced.
- Enter the matching y-values in the same sequence.
- Type the target x-value where you want an estimate.
- Choose the interpolation order, decimal precision, and curve density.
- Press Calculate to see the estimate, difference table, term contributions, and graph above the form.
- Use the CSV and PDF buttons to export the visible results.
FAQs
1) What does Newton forward interpolation calculate?
It estimates a missing or intermediate y-value from tabulated data with equal x-spacing. The method uses forward differences starting from the first row and builds a polynomial approximation around the beginning of the table.
2) When should I use the forward formula?
Use it when x-values are equally spaced and the target x lies near the first tabulated value. For points near the end, backward interpolation is usually a better fit.
3) What is the 2nd order Newton's forward interpolating formula?
The second-order form is y ≈ y₀ + uΔy₀ + [u(u−1)/2]Δ²y₀, where u = (x − x₀)/h. It uses the first two forward-difference levels and often gives strong accuracy for gently changing data.
4) Why must x-values be equally spaced?
The forward-difference structure depends on a constant step size h. Unequal spacing breaks the standard coefficient pattern, so the formula no longer represents the data correctly.
5) What does the value u represent?
u measures how far the target x is from the first x-value in units of the common spacing h. It controls the weight of each forward-difference term in the polynomial.
6) Does a higher interpolation order always improve accuracy?
Not always. Higher orders can fit the table more closely, but they may also amplify noise or rounding errors. Moderate orders are often more stable for real data.
7) Can this calculator extrapolate outside the table?
Yes, it can compute a value outside the listed x-range, but extrapolation is riskier. Errors typically grow faster beyond the known data than inside it.
8) What do the CSV and PDF exports include?
They export the visible result summary and the forward-difference table. This helps you save the estimate, term buildup, and tabulated differences for reports or later checking.