On the Total Positivity and Accurate Computations of <i>r</i>-Bell Polynomial Bases

Abstract

A new class of matrices defined in terms of r-Stirling numbers is introduced. These r-Stirling matrices are totally positive and determine the linear transformation between monomial and r-Bell polynomial bases. An efficient algorithm for the computation to high relative accuracy of the bidiagonal factorization of r-Stirling matrices is provided and used to achieve computations to high relative accuracy for the resolution of relevant algebraic problems with collocation, Wronskian, and Gramian matrices of r-Bell bases. The numerical experimentation confirms the accuracy of the proposed procedure.

Keywords

• high relative accuracy
• bidiagonal decompositions
• totally positive matrices
• <i>r</i>-Stirling numbers
• <i>r</i>-Bell polynomials