Calculator Input
Enter equally spaced x values when you want Newton backward interpolation. The tool still builds the backward difference table for non-uniform spacing.
Example Data Table
This sample follows the function y = x3 + 2x + 1. It is useful for checking table generation and interpolation behavior.
| x | y |
|---|---|
| 0 | 1 |
| 1 | 4 |
| 2 | 13 |
| 3 | 34 |
| 4 | 73 |
Formula Used
Backward difference definition
∇yi = yi − yi−1
∇2yi = ∇yi − ∇yi−1
Higher orders continue the same subtraction pattern.
Newton backward interpolation
u = (x − xn) / h
y(x) = yn + u∇yn + u(u+1)/2! ∇2yn + u(u+1)(u+2)/3! ∇3yn + ...
This works best near the last tabulated point when x values are equally spaced.
The calculator builds the full backward difference table first. If spacing is constant and a target x is provided, it then evaluates the Newton backward series up to the selected order.
How to Use This Calculator
- Enter x values in increasing order using spaces, commas, or new lines.
- Enter matching y values in the same sequence.
- Provide a target x if you want an interpolated or extrapolated estimate.
- Optionally enter an order limit to restrict the series depth.
- Click the calculation button to generate the table and graph.
- Review the summary cards, interpolation terms, and difference table.
- Use the CSV or PDF buttons to export the results.
- Use the example button if you want instant sample data.
FAQs
1. What does a backward difference measure?
It measures how each tabulated value changes from the previous one. Higher-order backward differences track changes of changes and help build interpolation formulas.
2. When should I prefer backward interpolation?
Use it when the target x lies near the final entries of an equally spaced table. It usually performs better there than formulas centered near the beginning.
3. Must the x values be equally spaced?
Equal spacing is required for Newton backward interpolation. The table itself can still be generated for irregular spacing, but the estimate is not applied.
4. What does the order limit control?
It controls how many backward-difference terms enter the interpolation series. Lower orders are simpler, while higher orders may capture more curvature from the data.
5. Can this tool extrapolate outside the table?
Yes, if spacing is constant and a target is entered outside the range. Extrapolation is often less reliable, so interpret those estimates carefully.
6. Why are higher-order differences useful?
They reveal how the data pattern evolves. Constant third or fourth differences often indicate an underlying polynomial relationship of matching degree.
7. What does the graph show?
The graph plots the original y values and the first backward differences. When an estimate is available, the interpolated target point is also highlighted.
8. Which inputs are accepted?
You can enter integers, decimals, and scientific notation. Separate values with spaces, commas, semicolons, or new lines for faster data entry.