Construct the unique interpolating polynomial of degree ≤ n-1 that passes through n given data points. Evaluate at any x, visualize the polynomial curve, and get explicit coefficients.
Lagrange interpolation is a classical technique for finding a polynomial that passes exactly through a given set of points. For n distinct points (x₀, y₀), (x₁, y₁), …, (xₙ₋₁, yₙ₋₁), there exists a unique polynomial P(x) of degree at most n-1 such that P(xᵢ) = yᵢ. The Lagrange form is expressed as:
This representation is numerically stable for moderate n and provides a direct construction without solving linear systems. The method is fundamental in numerical analysis, computer-aided design, and scientific computing. Our calculator uses a robust coefficient aggregation algorithm to compute the explicit polynomial coefficients, evaluate efficiently, and visualize the curve with high precision.
Developed by Joseph-Louis Lagrange in the 18th century, the interpolation polynomial underpins many areas of mathematics including finite element methods, quadrature rules (Newton-Cotes), and function approximation. The uniqueness stems from the fact that a polynomial of degree ≤ n-1 is uniquely determined by n conditions. The error term is given by P(x) - f(x) = f⁽ⁿ⁾(ξ)/n! · ∏(x - xᵢ), which highlights the importance of point distribution — equispaced nodes can lead to severe oscillations (Runge phenomenon).
Our implementation expands each Lagrange basis polynomial ℓᵢ(x) into its standard monomial form by repeatedly multiplying linear factors (x - xⱼ). The coefficients of each basis polynomial are accumulated with weight yᵢ, and the final sum yields the coefficient array [a₀, a₁, …, aₙ₋₁] such that P(x) = a₀ + a₁·x + … + aₙ₋₁·xⁿ⁻¹. This explicit coefficient representation allows fast evaluation using Horner’s method and precise rendering of the curve. The algorithm runs in O(n³) time, suitable for up to 8 points, and includes safeguards against duplicate x-coordinates.
The Runge phenomenon demonstrates that high-degree polynomial interpolation at equally spaced points can oscillate wildly near the boundaries, especially for functions like f(x)=1/(1+25x²). Our preset "Runge" example shows this instability: even with 6 points, the interpolant deviates significantly from the true function. This highlights the need for alternative approaches such as Chebyshev nodes or spline interpolation for equispaced data. Use the interactive graph to explore how adding points affects oscillation.
Our implementation uses double-precision arithmetic and a robust polynomial multiplication algorithm. For high-degree polynomials (≥ 8), numerical errors may increase. For more than 6 points, we recommend using Chebyshev nodes or spline interpolation to mitigate Runge's phenomenon and improve stability. The tool is optimized for educational use and small-to-medium datasets (2 to 8 points). Lagrange interpolation is ideal for exact reconstruction but can be unstable for large n due to Runge's phenomenon. When working with equispaced data beyond 6 points, consider piecewise interpolation methods.
To deepen your understanding of polynomial interpolation and its variants, explore these resources:
These methods are essential for practical data fitting and are widely used in scientific computing, computer graphics, and engineering simulation.