Accurate computations with Gram and Wronskian matrices of geometric and Poisson bases
E. Mainar, Juan Manuel Peña, Beatriz Rubio · Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas · 2022
Abstract In this paper we deduce a bidiagonal decomposition of Gram and Wronskian matrices of geometric and Poisson bases. It is also proved that the Gram matrices of both bases are strictly totally positive, that is, all their minors are positive. The mentioned bidiagonal decompositions are used to achieve algebraic computations with high relative accuracy for Gram and Wronskian matrices of these bases. The provided numerical experiments illustrate the accuracy when computing the inverse matrix, the eigenvalues or singular values or the solutions of some linear systems, using the theoretical results.