Zhongyun Liu

Zhongyun Liu, Siheng Chen, Weijin Xu et al.

The circulant matrices and skew-circulant matrices are two special classes of Toeplitz matrices and play vital roles in the computation of Toeplitz matrices. In this paper, we focus on real circulant and skewcirculant matrices. We first investigate their real Schur forms, which are closely related to the family of discrete cosine transform (DCT) and discrete sine transform (DST). Using those real Schur forms, we then develop some fast algorithms for computing real circulant, skew-circulant and Toeplitz matrix-real vector multiplications. Also, we develop a DCT-DST version of circulant and skew-circulant splitting (CSCS) iteration for real positive definite Toeplitz systems. Compared with the fast Fourier transform (FFT) version of CSCS iteration, the DCTDST version is more efficient and saves a half storage. Numerical experiments are presented to illustrate the effectiveness of our method.

Zhongyun Liu, Yulin Zhang, Carla Ferreira et al.

Our contribution is two-folded. First, starting from the known fact that every real skew-Hamiltonian matrix has a real Hamiltonian square root, we give a complete characterization of the square roots of a real skew-Hamiltonian matrix W. Second, we propose a structure exploiting method for computing square roots of W. Compared to the standard real Schur method, which ignores the structure, our method requires significantly less arithmetic.