We estimate the Boolean complexity of multiplication of structured matrices
by a vector and the solution of nonsingular linear systems of equations with
these matrices. We study four basic most popular classes, that is, Toeplitz,
Hankel, Cauchy and Van-der-monde matrices, for which the cited computational
problems are equivalent to the task of polynomial multiplication and division
and polynomial and rational multipoint evaluation and interpolation. The
Boolean cost estimates for the latter problems have been obtained by Kirrinnis
in \cite{kirrinnis-joc-1998}, except for rational interpolation, which we
supply now. All known Boolean cost estimates for these problems rely on using
Kronecker product. This implies the d-fold precision increase for the d-th
degree output, but we avoid such an increase by relying on distinct techniques
based on employing FFT. Furthermore we simplify the analysis and make it more
transparent by combining the representation of our tasks and algorithms in
terms of both structured matrices and polynomials and rational functions. This
also enables further extensions of our estimates to cover Trummer's important
problem and computations with the popular classes of structured matrices that
generalize the four cited basic matrix classes.Comment: (2014-04-10
