Calculator Input
Large screens use three columns, medium screens use two, and mobile uses one.
Example Data Table
This example uses the polynomial f(x) = x³ - 2x + 1. At x = 0.5, the interpolated value should be 0.125.
| x | f(x) |
|---|---|
| -2 | -3 |
| -1 | 2 |
| 0 | 1 |
| 1 | 0 |
| 2 | 5 |
Formula Used
Barycentric interpolation rewrites the Lagrange polynomial into a stable rational-looking form. It evaluates faster and more safely than expanding basis polynomials directly.
Meaning of the terms:
- xi are the interpolation nodes.
- yi are the known data values.
- wi are the barycentric weights.
- P(x) is the interpolated value at the requested point.
How to Use This Calculator
- Enter distinct x-nodes in the first input box.
- Enter the matching y-values in the second box.
- Set one evaluation point for a highlighted result.
- Optionally add several batch query points.
- Choose the graph range and number of plot samples.
- Press Calculate Interpolation.
- Review the result card, graph, weights, and batch table.
- Use the export buttons to save CSV or PDF output.
Frequently Asked Questions
1) What is barycentric interpolation?
It is a numerically stable way to evaluate the Lagrange interpolation polynomial. Instead of expanding basis terms directly, it uses precomputed weights and a compact evaluation formula.
2) Why is this form better than direct Lagrange expansion?
It usually reduces roundoff problems, avoids repeated basis recomputation, and evaluates quickly at many query points after the weights have been built once.
3) Must the x-nodes be evenly spaced?
No. The nodes only need to be distinct. Uneven spacing is common, although some node choices are more stable than others for high-degree interpolation.
4) What happens when the query point matches a node?
The interpolated value is exactly the stored y-value at that node. This rule avoids dividing by zero and preserves the original data values.
5) Why can high-degree interpolation oscillate?
Poor node placement, especially equally spaced nodes over wide intervals, can cause Runge-type oscillation. Chebyshev-like nodes often reduce that behavior.
6) What do the barycentric weights represent?
They encode the relative influence of each node in the interpolation formula. Large weight magnitudes can indicate stronger sensitivity to node placement.
7) Can I evaluate many x-values at once?
Yes. Add multiple batch query points in the input area. The calculator returns a table with interpolated values and exact-hit indicators.
8) When should I use Chebyshev nodes?
Use them when you can choose the sample locations and want better stability for higher-degree interpolation over a fixed interval.