DFT representations of Toeplitz-plus-Hankel Bézoutians with application to fast matrix-vector multiplication
The authors discuss possibilities of fast matrix-vector multiplication by inverse Toeplitz-plus-Hankel matrices. The approach is based on the following two facts: Inverses of Toeplitz-plus-Hankel matrices are Toeplitz-plus-Hankel Bézoutians and these can be transformed into Cauchy-like matrices. Then, these Cauchy-like matrices can be represented by using discrete Fourier transforms (DFT's). The authors show how Bézoutians can be transformed into Cauchy matrices with the help of Vandermonde matrices, in particular DFT's. Using these properties, one can perform a matrix-vector multiplication by an inverse Toeplitz-plus-Hankel matrix with six DFT's and eight DFT's for preprocessing. This are fewer DFT's than in the so far best known algorithm with seven DFT's plus 10 DFT's for preprocessing [see \textit{E. Bozzo}, Linear Algebra Appl. 230, 127-150 (1995; Zbl 0839.15007)].
scientific article; zbMATH DE number 1371014