In this note, we extend the Vandermonde with Arnoldi method recently advocated by Brubeck et al. (2021) to dealing confluent matrix. To apply process, it is critical find a Krylov subspace which generates column space of A theorem established for such subspaces any order derivatives. This enables us compute derivatives high degree polynomials precision. It also makes many applications involving...

A confluent Vandermonde-like matrix P(a0, a, an) is a generalisation of the confluent Vandermonde matrix in which the monomials are replaced by arbitrary polynomials. For the case where the polynomials satisfy a three-term recurrence relation algorithms for solving the systems Px b and Pra f in O(n2) operations are derived. Forward and backward error analyses that provide bounds for the relativ...

A confluent Vandermonde-like matrix P(a0, a, an) is a generalisation of the confluent Vandermonde matrix in which the monomials are replaced by arbitrary polynomials. For the case where the polynomials satisfy a three-term recurrence relation algorithms for solving the systems Px b and Pra f in O(n2) operations are derived. Forward and backward error analyses that provide bounds for the relativ...

An arbitrary derivative of a Vandermonde form in N variables is given as [ n 1 ⋯ ], where the ith variable differentiated − i times, ≤ 1. A simple decoding table introduced to evaluate it by inspection. The special cases 0 i+1 for < are one-to-one correspondence with ribbon Young diagrams. respective N! standard tableaux map complete graded basis space S -harmonic polynomials. mapping realiz...

Lower bounds on the condition number minκp(V ) of a Vandermonde matrix V are established in terms of the dimension n or n and the largest absolute value among all nodes that define the Vandermonde matrix. Many bounds here are asymptotically sharp, and compare favorably to those of the same kind due to Gautschi and Inglese (Numer. Math., 52 (1988), 241–250) who considered either all positive nod...