A fast algorithm for computing mock-Chebyshev nodes with reduced uniform grid size

Abstract

In computational applications, it is often the case that measurements are collected at uniformly distributed locations. In such cases, using ordinary polynomial interpolation may lead to divergence due to the Runge phenomenon; furthermore, the interpolation process is known to be severely ill-conditioned. To address these challenges, an effective strategy involves selecting mock-Chebyshev points for polynomial interpolation from a dense set of uniformly spaced points, thereby replicating the favorable properties of Chebyshev nodes. Yet, few studies in the literature address the computation of these nodes. Moreover, developing an algorithm that can generate mock-Chebyshev points with $ \mathcal {O}(n) $ O(n) computational complexity is of significant practical importance. This study proposes a new version of the fast algorithm introduced in Ibrahimoglu [A fast algorithm for computing the mock-Chebyshev nodes, J. Comput. Appl. Math. 373 (2020), p. 112336.], which employs the floor function to calculate the ratio of distances between consecutive Chebyshev-Lobatto interpolation points. The resulting algorithm is shown to be fast and stable, consistently generating a distribution of nodes fulfilling the mock-Chebyshev requirements with the linear complexity $ \mathcal {O}(n) $ O(n), simultaneously reducing the cardinality of the corresponding satisfactory uniform grid. This study also introduces a theoretical lower bound on the minimum number of equispaced nodes required to satisfy the mock-Chebyshev conditions. A bivariate extension of the approach to mock-Padua points on $ [-1, 1] 2 $ [-1,1]2 is also presented and validated through numerical experiments.