Optimal Node Placement for Constrained Polynomial Interpolation

In this paper, an algorithm for optimizing the Lebesgue constant of a set of nodes is demonstrated. First, the standard Lebesgue function and Lebesgue constant are used to show how the location of the nodes affect the interpolating qualities of a set of polynomials. Next, the definition of the Lebesgue function and constant is generalized to any set of polynomials, including constrained polynomials. To that end, an algorithm is developed for optimizing a set of nodes by indirectly minimizing the Lebesgue constant. Finally, the performance of the algorithm is shown on several examples, including one with a known theoretical solution and a boundary value problem. The optimal nodes obtained via the algorithm can then be used for generating admissible polynomials for hybrid machine learning which have good interpolating and approximating qualities.