Lagrange Interpolation

In computational mathematics, Lagrange interpolation is a method for constructing interpolating polynomials. This approach utilizes Lagrange basis functions, with the core concept being to construct a polynomial function based on known function values (and optionally derivative values) to interpolate between given data points. The method can be widely applied in numerical analysis, numerical approximation, and data fitting applications. Therefore, in practical implementations, calling Lagrange basis functions to perform Lagrange interpolation is a commonly used technique.

From a code implementation perspective, the Lagrange interpolation algorithm typically involves calculating Lagrange basis polynomials for each data point. The key mathematical formulation uses the product formula: L_i(x) = Π (x - x_j)/(x_i - x_j) for j≠i, where each basis polynomial equals 1 at its corresponding node x_i and 0 at all other nodes. The final interpolation polynomial is constructed as a weighted sum P(x) = Σ y_i * L_i(x), where y_i are the function values at nodes x_i. Implementation commonly involves nested loops - an outer loop iterating through data points to compute basis polynomials, and an inner loop calculating the product terms while avoiding division by zero through conditional checks.