Calculator Inputs
Example Data Table
| x | f(x) | Meaning |
|---|---|---|
| 1 | 1 | Known point from the dataset |
| 2 | 8 | Second measured value |
| 3 | 27 | Third measured value |
| 4 | 64 | Fourth measured value |
With target x = 2.5, the calculator estimates the missing value using the Newton divided difference polynomial.
Formula Used
Newton interpolation builds a polynomial from known points using divided differences. The general form is shown below.
The first divided difference is:
Higher-order divided differences use earlier differences recursively:
How to Use This Calculator
- Enter the known x-values in order, separated by commas or spaces.
- Enter the matching function values using the same count.
- Type the target x where you want an estimate.
- Choose decimal precision for the final output tables.
- Optionally sort the points if your dataset is unsorted.
- Press the calculate button to show results above the form.
- Review the polynomial, the divided difference table, and the term contributions.
- Use the CSV or PDF buttons to save your work.
Why This Tool Is Useful
This calculator supports arbitrary spacing between x-values, so it works beyond equally spaced tables. It also reveals each coefficient, term product, and contribution, helping you audit the full interpolation process.
You can use it for classroom exercises, lab reports, engineering datasets, forecasting checkpoints, and numerical analysis practice where a transparent step-by-step polynomial estimate matters.
Frequently Asked Questions
1. What does Newton interpolation estimate?
It estimates an unknown function value using nearby known data points. The method builds a polynomial that passes through the supplied coordinates.
2. Does this method require equal spacing?
No. Newton divided difference interpolation works with unevenly spaced x-values, which makes it more flexible than some table-based interpolation formulas.
3. Why are duplicate x-values rejected?
Duplicate x-values make the divided difference denominator zero. That breaks the interpolation table and prevents a valid polynomial from being formed.
4. What is the difference between interpolation and extrapolation?
Interpolation estimates inside the known x-range. Extrapolation estimates outside that range and can become less reliable, especially with high-degree polynomials.
5. How many points should I enter?
Use at least two points. More points can improve fit for smooth data, but very high-degree polynomials may oscillate and reduce stability.
6. What does the divided difference table show?
It shows the recursive coefficients used to build the Newton polynomial. Each higher order captures how the function changes across wider intervals.
7. Can I save the computed results?
Yes. After calculation, you can export the main output as CSV or generate a clean PDF-style print file for reports.