Akademik Veri Yönetim Sistemi
İBRAHİMOĞLU B. A. (Yürütücü), ŞEHİTOĞULLARI D.
Yükseköğretim Kurumları Destekli Proje, 2023 - 2023
Vandermonde matrices arise frequently in computational mathematics in problems that require polynomial approximation. The polynomial interpolation problem through data given at a set of distinct interpolation points can be expressed in the monomial basis. This gives rise to a linear system of equations with a Vandermonde matrix. This Vandermonde matrix, however, brings the difficulty that this matrix including real nodes is usually ill-conditioned. This is also a difficulty even for not very high orders. The distribution of the points may cause variations in this ill-conditioning in a reasonable way. In order to treat this ill-conditioning, the generally suggested method is to deploy the highly non-uniform Chebyshev nodes, but there is still a problem on what to do when the experimental data at hand are available only at equispaced points. In this situation, polynomial interpolation with equidistant nodes becomes unreliable owing to the well-known drawback called the Runge phenomenon and the polynomial interpolation is numerically ill-conditioned. Interpolation processes on Chebyshev-Lobatto nodes offer near-optimal solutions and defeating the Runge phenomenon by using the mock-Chebyshev subset interpolation proves to be one of the best strategies. In this study, we investigate the condition number of the Vandermonde matrix numerically and find out that the condition number can be reduced by using mock-Chebyshev nodes, as in the case of using Chebyshev-Lobatto nodes. Moreover, we extend our investigation to a Vandermonde-like matrix by using a basis of Chebyshev polynomials and show numerically that, in this case, the condition number of this matrix can be reduced significantly.