Calculated Results
The results appear here directly below the header and above the form after calculation.
Finite Difference Table
Plotly Graph
Enter Tabulated Values
Provide x-values and y-values in matching order. Use commas, spaces, or line breaks.
Formula Used
Forward finite differences
First difference:
Δyi = yi+1 - yi
Higher order recurrence:
Δkyi = Δk-1yi+1 - Δk-1yi
If one order becomes nearly constant, the data often comes from a polynomial of that degree.
Newton interpolation
For equal spacing h, with u = (x - x0) / h:
y(x) = y0 + uΔy0 + u(u-1)Δ²y0/2! + ...
Backward form uses p = (x - xn) / h.
This page applies forward or backward interpolation when x-values are evenly spaced.
How to Use This Calculator
1. Enter ordered data
Type x-values and y-values in the same order. Each x must match one y. At least two pairs are required.
2. Add an optional target
Enter a target x when you want an interpolated estimate. Leave it blank if you only need the finite difference table.
3. Choose display settings
Set decimal places, spacing tolerance, and chart style. Auto interpolation picks a direction based on the target position.
4. Review exports and graph
After calculation, inspect the difference table, interpolation summary, and chart. Then download CSV or PDF for reporting.
Example Data Table
This sample follows the quadratic rule y = x² + 2x + 1, so the second differences are constant.
| x | y | Δy | Δ²y |
|---|---|---|---|
| 0 | 1 | 3 | 2 |
| 1 | 4 | 5 | 2 |
| 2 | 9 | 7 | - |
| 3 | 16 | - | - |
FAQs
1. What are finite differences?
Finite differences measure how values change between consecutive entries in a table. Repeating the process builds higher-order differences, which help reveal patterns, estimate polynomial degree, and support interpolation on discrete data.
2. When are finite differences most useful?
They are useful when data is sampled at regular intervals and you need to study change without continuous formulas. Common uses include interpolation, table-based forecasting, numerical analysis, and checking whether data follows a polynomial trend.
3. Why do constant second differences matter?
Constant second differences usually indicate a quadratic relationship. In general, constant kth differences often suggest a polynomial of degree k, assuming the x-values are equally spaced and rounding noise is small.
4. Can I interpolate with uneven x-values?
This page still builds the difference table, but Newton forward and backward interpolation require nearly equal spacing. If your x-values are uneven, use the table for pattern study and switch to another interpolation method for estimates.
5. Which interpolation method should I choose?
Choose forward interpolation when the target x is closer to the beginning of the table. Choose backward interpolation when the target is closer to the end. Auto mode makes that decision for you.
6. What if my data contains rounding errors?
Small measurement or rounding errors can stop a difference order from appearing perfectly constant. The tolerance field helps the calculator judge whether spacing is effectively equal despite tiny floating-point differences.
7. What does the graph show?
The graph plots the original data points and connects them according to your chosen chart style. When interpolation runs, the estimated target point is added so you can compare it with the known table values visually.
8. What do the CSV and PDF exports include?
The CSV export includes summary details and the difference table. The PDF export includes the same key results in a report format, which is useful for homework, revision packs, and quick documentation.