Fast Solution of Confluent Vandermonde-Like Linear Systems Using Polynomial Arithmetic

In the present paper we propose a method for the fast solution of some confluent Vandermonde-like systems by polynomial arithmetic. The results (though less general than known results of Hao Lu) can be formulated explicitly in terms of numerator polynomials. In order to avoid numerical difficulties with the canonical (monomial) basis, algorithms for multiplication and division in arbitrary orthogonal bases are developed. We discuss applications to the numerical computation of Gauss-rules from given modified moments and to the solution of Gram systems.