Enter equally spaced values
Sample equally spaced dataset
| x | y |
|---|---|
| 10 | 100 |
| 20 | 121 |
| 30 | 144 |
| 40 | 169 |
| 50 | 196 |
Example query point: x = 47. This sample follows y = (x / 10 + 9)2, so the expected interpolated value is 189.61.
Newton backward interpolation equation
Step size: h = xi - xi-1
Backward parameter: p = (x - xn) / h
Main formula: f(x) = yn + pāyn + [p(p + 1) / 2!]ā2yn + [p(p + 1)(p + 2) / 3!]ā3yn + ...
Use case: This method works best when x-values are equally spaced and the query point lies near the last tabulated x-value.
Practical workflow
- Enter equally spaced x-values in ascending or mixed order.
- Enter matching y-values in the same pair positions.
- Provide the interpolation point where you need an estimate.
- Set decimal places and spacing tolerance if needed.
- Enable extrapolation only when the query lies outside the data range.
- Submit the form to view the estimate, steps, table, and chart.
- Download the results as CSV or PDF for documentation.
FAQs
1. When should I use Newton backward interpolation?
Use it when the x-values are equally spaced and the required estimate lies near the last known data point. It is especially helpful for tabulated sequences where building a backward difference structure is convenient.
2. Why must the x-values be equally spaced?
The backward interpolation formula depends on a constant step size h. If the spacing changes noticeably, the parameter p and the difference terms no longer match the derivation, so the estimate can become unreliable.
3. Can I use this calculator for extrapolation?
Yes, but you should enable extrapolation first. Results outside the data interval are more sensitive to data noise and higher-order differences, so treat those estimates as less stable than true interpolation.
4. What does the backward parameter p mean?
The value p measures how far the query point sits from the final tabulated x-value in step-size units. Smaller absolute values of p generally indicate a more suitable case for backward interpolation.
5. Why does the calculator show a forward difference table?
Backward interpolation can be implemented from the end of a standard forward difference table. The calculator extracts the last needed entries from each difference order to assemble the backward polynomial cleanly.
6. How many data points should I enter?
At least two points are required, but more points usually provide a richer polynomial model. Still, very high-degree interpolation may oscillate, especially with noisy or poorly behaved datasets.
7. Does the input need to be sorted?
No. The calculator automatically sorts x-y pairs by x-value before checking the spacing and computing the interpolation. Duplicate x-values are rejected because they break the single-valued table structure.
8. What can I export with the download buttons?
The CSV file includes the summary values and original data pairs. The PDF file contains the same summary and data table in a compact report, which is useful for study notes or project documentation.