Vandermonde Node Conditioning

A Vandermonde matrix is defined by a set of nodes or points x_0, x_1, …, x_(n-1). For a row-node configuration, the i-th row corresponds to the powers of the node x_i, such that V_(i,j) = x_i^j. For a 3 × 3 matrix: V = [[1, x_0, x_0²], [1, x_1, x_1²], [1, x_2, x_2²]]

Experiment

You can interact with the Vandermonde Node Conditioning module to see how clustering and shifting nodes affect the condition number.

1. Change the Nodes to Clustered. Notice the nodes bunch together near 0.5. The determinant decreases to 0.25, and the condition number increases to 24.
2. Change the Shift to 1 (for Separated nodes). The nodes shift to (0, 1, 3). The determinant remains 6.0, but the condition number increases to 39.
3. Select the Repeated nodes configuration. The nodes become (0, 0, 1), and the determinant drops to 0.0, making the matrix singular (rank 2) with an unavailable finite condition number.