Lagrange Interpolation Calculator

A way to find the polynomial of lowest degree that passes exactly through a set of points. With n points you get a polynomial of degree n − 1 or less, and there is only one.

P(x) = Σ yᵢ Lᵢ(x), where each basis polynomial Lᵢ(x) = ∏ (x − xⱼ) / (xᵢ − xⱼ) over every other point j. Lᵢ is 1 at xᵢ and 0 at every other xⱼ.

Each Lᵢ divides by xᵢ − xⱼ, which would be zero. A function also can't pass through two different y values at the same x.

Yes. Both give the same unique polynomial. Newton's divided differences just build it in a different order, which makes adding a point cheaper.

With many equally spaced points, the polynomial can swing wildly near the ends of the interval, even for a smooth function like 1/(1 + 25x²). Chebyshev points, bunched toward the ends, fix it. Try it on the Runge & Error tab.

That's extrapolation. Polynomials often shoot off beyond the data, so trust values between your smallest and largest x much more.

The polynomial, its coefficients and P(x) are worked out with exact fractions, so nothing is rounded until it's displayed. The charts use the barycentric formula, a stable way to plot the same polynomial.