SVD of Hankel matrices in Vandermonde-Cauchy product form

Drmač, Zlatko

Electronic transactions on numerical analysis, Tome 44 (2015), pp. 593-623.

Résumé

Summary: Structured matrices of Cauchy, Vandermonde, Hankel, Toeplitz, and other types arise in a variety of applications, and their SVD decomposition provides key information, e.g., in various rational approximation tasks. In particular, Hankel matrices play an important role in the Adamyan-Arov-Krein and Carathéodory-Feyér rational approximation theories as well as in various applications in signal processing and control theory. This paper proposes new algorithms to compute the SVD of a Hankel matrix given implicitly as the product $\mathcal{V}^T D \mathcal{V}$, where $\mathcal{V}$ is a complex Vandermonde matrix and $D$ is a diagonal matrix. The key steps are the discrete Fourier transform and the computation of the SVD of $\mathcal{C}^T \widetilde{D}\mathcal{C}$, where $\mathcal{C}$ is a Cauchy matrix and $\widetilde{D}$ is diagonal. This SVD is computed by a specially tailored version of the Jacobi SVD for products of matrices. Error and perturbation analysis and numerical experiments confirm the robustness of the proposed algorithms, capable of computing to high relative accuracy all singular values in the full range of machine numbers.

Détail

Classification : 15A09, 15A12, 15A18, 15A23, 65F15, 65F22, 65F35
Keywords: Cauchy matrix, discrete Fourier transform, eigenvalues, Hankel matrix, Jacobi method, rational approximations, singular value decomposition, Toeplitz matrix, Vandermonde matrix