On Computations with Dense Structured Matrices

We reduce several computations with Hilbert and Vandermonde type matrices to matrix computations of the Hankel-Toeplitz type (and vice versa). This unifies various known algorithms for computations with dense structured matrices and enables us to extend any progress in computations with matrices of one class to the computations with other classes of matrices. In particular, this enables us to compute the inverses and the determinants of $n \times n$ matrices of Vandermonde and Hilbert types for the cost of $O(n{\log ^2}n)$ arithmetic operations. (Previously, such results were only known for the more narrow class of Vandermonde and generalized Hilbert matrices.)