On the accurate basis conversion of univariate polynomials

Abstract This paper summarizes the bidiagonal factorizations of change of basis matrices between polynomial bases commonly used in interpolation and computer-aided geometric design. These factorizations enable the efficient and highly accurate resolution of linear algebra problems, offering significant advantages in terms of precision. The article also highlights a variety of applications, including the accurate computation of divided differences and the efficient solution of algebraic problems involving collocation matrices or Gram matrices, underscoring the practical relevance of these factorizations in mathematical and computational contexts.