High relative accuracy through Newton bases

Bidiagonal factorizations for the change of basis matrices between monomial and Newton polynomial bases are obtained. The total positivity of these matrices is characterized in terms of the sign of the nodes of the Newton bases. It is shown that computations to high relative accuracy for algebraic problems related to these matrices can be achieved whenever the nodes have the same sign. Stirling matrices can be considered particular cases of these matrices, and then computations to high relative accuracy for collocation and Wronskian matrices of Touchard polynomial bases can be obtained. The performed numerical experimentation confirms the accurate solutions obtained when solving algebraic problems using the proposed factorizations, for instance, for the calculation of their eigenvalues, singular values, and inverses, as well as the solution of some linear systems of equations associated with these matrices.