  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.
