Neville Interpolation Calculator

Calculator Inputs

x-values

Use commas, spaces, or new lines.

y-values

Each y-value matches the x-value in the same position.

Interpolation x

The calculator estimates y at this x.

Decimal places

Choose between 2 and 12 decimal places.

Graph sample points

Higher values make smoother curves.

Quick Example

Fills x = 1,2,3,4 and y = 1,8,27,64.

Graph minimum x

Leave empty to use an automatic padded minimum.

Graph maximum x

Leave empty to use an automatic padded maximum.

Clear Inputs

Example Data Table

Sample interpolation target: x = 2.5. Expected result: 15.625000.

Index	x_i	y_i	Meaning
0	1	1	Known point
1	2	8	Known point
2	3	27	Known point
3	4	64	Known point

Formula Used

Neville interpolation builds the result recursively from known points. The base term is:

P[i,i](x) = y_i

The recursive relation is:

P[i,j](x) = ((x - x_i)P[i+1,j](x) - (x - x_j)P[i,j-1](x)) / (x_j - x_i)

The final interpolated value is P[0,n-1](x), where n is the total number of data points.

How to Use This Calculator

Enter the x-values in order, separated by commas, spaces, or line breaks.
Enter the matching y-values in the same order.
Provide the target x where you need an estimated y value.
Choose decimal precision and optional graph sampling density.
Optionally set custom graph minimum and maximum x limits.
Click the calculate button to generate the result, table, and graph.
Use the CSV button for tabular export and the PDF button for report output.

Frequently Asked Questions

1. What does Neville interpolation calculate?

It estimates an unknown y-value at a chosen x using known data points. The method builds the answer recursively through a triangular interpolation table.

2. When should I use Neville interpolation instead of linear interpolation?

Use Neville interpolation when you need a smoother polynomial estimate from several known points. Linear interpolation only uses two nearby points and gives straight-line estimates.

3. Do my x-values need to be sorted?

Sorting is helpful for reading results, but the recursion can still work with unsorted distinct x-values. Matching x and y positions correctly remains essential.

4. Can this calculator extrapolate outside the data range?

Yes. If the target x falls outside the known x-range, the tool labels the result as extrapolation. Extrapolated values can be less reliable than interpolated ones.

5. Why are duplicate x-values invalid?

Duplicate x-values make the denominator x_j − x_i equal zero. That breaks the recurrence and makes the interpolation undefined.

6. How accurate is the interpolated result?

Accuracy depends on point quality, spacing, rounding, and the behavior of the underlying function. More points do not always improve real-world reliability.

7. What does the Neville table show?

It shows how the algorithm combines smaller interpolations into larger ones. The last value in the top row is the final estimated result.

8. What do the CSV and PDF exports include?

The CSV export includes the input data, final estimate, and Neville table. The PDF export captures the visible result section for reporting.