Lagrange Interpolation Calculator

Lagrange Basis Polynomials

The foundation of Lagrange interpolation is the construction of basis polynomials L_i(x) for each data point (x_i, y_i). Each basis polynomial has the special property that L_i(x_i) = 1 and L_i(x_j) = 0 for all j ≠ i. The basis polynomial is constructed as a product of linear factors: L_i(x) = \prod_j=0, j≠ i^n (x - x_j)/(x_i - x_j) This ensures that at x = x_i, all factors in the numerator and denominator cancel to give 1, while at any other data point x = x_k, the factor (x - x_k) in the numerator makes the entire product zero. These basis polynomials form a complete set for polynomial interpolation.

L_i(x) = \prod_j=0, j≠ i^n (x - x_j)/(x_i - x_j)

The Interpolating Polynomial

Once the basis polynomials are constructed, the interpolating polynomial P(x) is formed as a linear combination of these basis functions, weighted by the corresponding y-values: P(x) = Σ_i=0^n y_i L_i(x) This formula guarantees that P(x_i) = y_i for all data points because L_j(x_i) = 0 for j ≠ i and L_i(x_i) = 1. The resulting polynomial has degree at most n for n+1 data points. The Lagrange form provides an explicit representation without requiring matrix inversion or solving systems of equations, making it particularly elegant for theoretical analysis and straightforward for computational implementation.

P(x) = Σ_i=0^n y_i L_i(x)

Uniqueness of Polynomial Interpolation

A fundamental theorem in numerical analysis states that given n+1 distinct points (x_0, y_0), (x_1, y_1), \ldots, (x_n, y_n) with distinct x-coordinates, there exists a unique polynomial P(x) of degree at most n such that P(x_i) = y_i for all i. This uniqueness is guaranteed by the Vandermonde matrix determinant being non-zero when all x_i are distinct. While different interpolation methods (Newton, Lagrange, monomial basis) may give different algebraic forms, they all represent the same unique polynomial. The Lagrange form is particularly useful because it explicitly constructs this unique polynomial without requiring preliminary calculations or matrix operations.

\det(V) = \prod_0 ≤ i < j ≤ n (x_j - x_i) ≠ 0

Error Analysis and Bounds

The interpolation error for a function f(x) when approximated by the Lagrange interpolating polynomial P(x) can be bounded using the error formula: E(x) = f(x) - P(x) = \fracf^(n+1)(\xi)(n+1)! \prod_i=0^n (x - x_i) where \xi is some point in the interval containing x and all x_i, assuming f has n+1 continuous derivatives. The product term \omega(x) = \prod_i=0^n (x - x_i) is called the nodal polynomial and represents the distance from x to the interpolation points. This error bound shows that interpolation error depends on both the smoothness of the function (through the derivative term) and the distribution of interpolation nodes. For functions with bounded derivatives, the error can be made arbitrarily small by increasing the number of points, though care must be taken to avoid Runge's phenomenon with equally-spaced points.

E(x) = \fracf^(n+1)(\xi)(n+1)! \prod_i=0^n (x - x_i)

Computational Considerations

While Lagrange interpolation provides an elegant theoretical framework, direct computation using the formula requires O(n^2) operations for each evaluation point. The nested form can reduce this to O(n) operations per evaluation after an O(n^2) preprocessing step. For multiple evaluations or when adding new data points dynamically, Newton's divided difference form may be more efficient. The Lagrange form, however, excels in clarity and is particularly useful when interpolating at only a few points or when the interpolating polynomial itself needs to be analyzed algebraically. Modern implementations often use barycentric forms of Lagrange interpolation, which provide numerical stability and efficiency comparable to Newton's method while maintaining the explicit structure of Lagrange's approach.

P(x) = Σ_i=0^n y_i \prod_j ≠ i (x - x_j)/(x_i - x_j)

Runge's Phenomenon and Node Selection

A critical limitation of polynomial interpolation is Runge's phenomenon, where interpolating polynomials with equally-spaced nodes can exhibit wild oscillations near the boundaries of the interval, even for smooth functions. For example, interpolating f(x) = 1/(1+25x^2) on [-1,1] with equally-spaced points shows increasing oscillations as the degree increases. This problem can be mitigated by using Chebyshev nodes: x_i = \cos((2i+1)/(2(n+1))π) These nodes cluster near the endpoints, reducing oscillations and providing near-optimal approximation properties. Alternatively, piecewise polynomial interpolation (splines) avoids high-degree polynomials entirely while maintaining smoothness.

x_i = \cos((2i+1)/(2(n+1))π)

Applications in Numerical Methods

Lagrange interpolation forms the theoretical foundation for numerous numerical methods. In numerical integration, Newton-Cotes formulas (including the trapezoidal rule and Simpson's rule) derive from integrating Lagrange interpolating polynomials. In differential equations, finite difference approximations and Runge-Kutta methods use polynomial interpolation concepts. The method appears in computer graphics for curve and surface fitting, in signal processing for resampling and filter design, and in scientific computing for approximating complex functions with polynomials. Understanding Lagrange interpolation provides insight into the behavior, accuracy, and limitations of these numerical techniques, making it a cornerstone of numerical analysis education and practice.

∫_a^b f(x)dx ≈ ∫_a^b P(x)dx

Worked Example

Inputs

• points: (0,1), (1,3), (2,7)
• evaluateAt: 1.5

Result: 5

Explanation

Using Lagrange interpolation through points (0,1), (1,3), (2,7), the interpolating polynomial is P(x) = 1 + 2x + x², evaluated at x=1.5 gives P(1.5) = 5

Linear Interpolation - Two Points

Inputs

• points: (1,2), (3,8)
• evaluateAt: 2

Result: 5

Explanation

With two points, Lagrange interpolation reduces to linear interpolation: P(x) = 2·(x-3)/(1-3) + 8·(x-1)/(3-1) = -x+3 + 4x-4 = 3x-1, so P(2) = 5