The accurate and efficient solution of a totally positive generalized Vandermonde linear system

Vandermonde, Cauchy, and Cauchy - Vandermonde totally positive linear systems can be solved extremely accurately in O( n(2)) time using Bjorck-Pereyra-type methods. We prove that Bjorck-Pereyra-type methods exist not only for the above linear systems but also for any totally positive linear system as long as the initial minors ( i.e., contiguous minors that include the first row or column) can be computed accurately. Using this result we design a new O( n2) Bjorck-Pereyra-type method for solving generalized Vandermonde systems of equations by using a new algorithm for computing the Schur function. We present explicit formulas for the entries of the bidiagonal decomposition, the LDU decomposition, and the inverse of a totally positive generalized Vandermonde matrix, as well as algorithms for computing these entries to high relative accuracy.