What is the Lagrange interpolating polynomial?

Given n+1 data points, the Lagrange polynomial is the unique polynomial of degree ≤ n that passes through all of them. It is expressed as a weighted sum of basis polynomials, each equal to 1 at one data point and 0 at all others.

How many data points can I enter?

The tool supports 2 to 10 data points. More points produce a higher-degree polynomial and can introduce Runge's phenomenon (oscillation near the edges), so practical use rarely exceeds 6–8 points.

Do all x values have to be distinct?

Yes. Lagrange interpolation requires all x values to be unique. If two points share an x coordinate, the polynomial is undefined and the tool will show an error.

How accurate is the interpolation?

The polynomial passes exactly through every data point. Accuracy at other x values depends on how well the underlying function is approximated by a polynomial — it can be poor far outside the data range (extrapolation).

What is the difference from linear interpolation?

Linear interpolation uses only two points and gives a straight line. Lagrange interpolation uses all your points and produces a smooth curve of higher degree that honors every point simultaneously.

Lagrange Interpolation Calculator

Enter a set of (x, y) data points and a target x to evaluate the Lagrange interpolating polynomial instantly.

Data Points

x y

Evaluate at x

Enter data points and click Calculate

Minimum 2 points required

Summary

Lagrange interpolation constructs a unique polynomial of minimum degree that passes exactly through a given set of data points. For n+1 points it produces a degree-n polynomial built from weighted basis functions. This tool lets you enter any number of (x, y) pairs, choose an evaluation point, and instantly see the interpolated value alongside the full polynomial expression and each basis term.

How it works

Enter at least two (x, y) data points — one per row.
Add or remove rows with the + / − buttons.
Type a target x value in the "Evaluate at x" field.
Click Calculate to run the interpolation.
The tool displays the interpolated y value, the full polynomial, and the step-by-step basis polynomials.

Use cases

Estimate function values between tabulated measurements.
Reconstruct a smooth curve through sensor or experiment data.
Verify polynomial interpolation homework problems step by step.
Approximate transcendental functions with polynomial series.
Check numerical methods course exercises.
Interpolate calibration tables in engineering.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu

Lagrange Interpolation Calculator

Data Points

Polynomial Expression

Basis Polynomials

Summary

How it works

Use cases

Frequently Asked Questions

Lagrange Interpolation Calculator

Use the Lagrange Interpolation Calculator

Data Points

Polynomial Expression

Basis Polynomials

Summary

How it works

Use cases

Frequently Asked Questions