Optimally scaled and optimally conditioned Vandermonde and Vandermonde-like matrices

Vandermonde matrices with real nodes are known to be severely ill-conditioned. We investigate numerically the extent to which the condition number of such matrices can be reduced, either by row-scaling or by optimal configurations of nodes. In the latter case we find empirically the condition of the optimally conditioned n×n Vandermonde matrix to grow exponentially at a rate slightly less than \((1+\sqrt{2})^{n}\). Much slower growth—essentially linear—is observed for optimally conditioned Vandermonde-Jacobi matrices. We also comment on the computational challenges involved in determining condition numbers of highly ill-conditioned matrices.