Lagrange Interpolation

Lagrange interpolation provides a direct way to find the unique polynomial of lowest degree that passes through a given set of data points. For n points, the resulting polynomial has a degree of at most n - 1.

Verify the interpolation with the module

Open Lagrange Interpolation to explore how moving points affects the global polynomial and local basis weights.

1. Select the Quadratic preset. The nodes are (-1, 1), (0, 0), and (1, 1). Set the evaluation point to 0.5. Verify the basis weights are exactly -0.1250, 0.7500, and 0.3750, summing to 1.0000, and p(0.5) = 0.2500.
2. Move the evaluation point precisely to one of the nodes, for example x = 1.00. Notice that the weight for L₂ becomes 1.0000 and all other weights become 0.0000, yielding p(1.00) = 1.0000 exactly. This demonstrates that Lagrange polynomials perfectly interpolate the given nodes.
3. Switch to the Linear preset with nodes (0, 1) and (2, 5). The resulting polynomial is p(x) = 1 + 2x. Set the evaluation point to 1 and verify p(1) = 3.0000.
4. Switch to the Constant preset. Every interpolated value across the domain is 2. Evaluating outside the node interval (e.g., x < -1 or x > 1) is correctly labeled as extrapolation.