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Example data table
This sample uses points from the polynomial y = x² + 1. It produces a clean second-order interpolation table.
| x | y |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 5 |
| 3 | 10 |
Formula used
Base value: f[xi] = yi
Recursive divided difference:
f[xi, ..., xi+k] = ( f[xi+1, ..., xi+k] - f[xi, ..., xi+k-1] ) / ( xi+k - xi )
Newton interpolation polynomial:
P(x) = a0 + a1(x - x0) + a2(x - x0)(x - x1) + ... + an∏j=0n-1(x - xj)
The first entry in each divided-difference column becomes a Newton coefficient. Those coefficients define the interpolation polynomial used for table generation, point estimates, and plotting.
How to use this calculator
- Enter the x values in order. Keep every x value unique.
- Enter the matching y values. Each position must correspond to the same row index.
- Optionally add a target x value to estimate an intermediate or outside point.
- Set decimal precision and graph sample count for the displayed output.
- Press Calculate divided differences to build the table, coefficients, expressions, and graph.
- Use the export buttons to download the table as CSV or create a quick PDF summary.
FAQs
1) What does a divided difference measure?
It measures how function values change across unevenly spaced x positions. Higher-order divided differences capture curvature and help construct the Newton interpolation polynomial efficiently.
2) Why must x values be unique?
Duplicate x values make the denominator zero in the recursive formula. That breaks the divided-difference table and prevents the interpolation polynomial from being defined in this form.
3) Can the calculator handle uneven spacing?
Yes. Newton divided differences are especially useful when x values are not equally spaced. That is one reason this method is popular for interpolation from measured data.
4) What is the difference between interpolation and extrapolation?
Interpolation estimates inside the x range supplied by your data. Extrapolation estimates outside that range and often carries more risk because the polynomial may change rapidly.
5) Why show both Newton and expanded polynomial forms?
The Newton form reveals the divided-difference coefficients directly. The expanded form is convenient for analysis, differentiation, reporting, and comparing with standard polynomial expressions.
6) Does the method always match every input point?
Yes, when all x values are distinct. The constructed polynomial passes exactly through every provided data point, which is why the plotted curve intersects all markers.
7) What does the derivative at the target x mean?
It gives the local slope of the interpolating polynomial at the chosen x value. That can help estimate rate of change from tabulated data.
8) When should I avoid high-degree interpolation?
Very high degrees can oscillate, especially near interval edges. For large noisy datasets, piecewise interpolation or spline methods often behave more reliably than one global polynomial.