The eigenstructures of real (skew) circulant matrices with some applications
For researchers and practitioners working with Toeplitz systems, this work provides a more efficient algorithm that reduces storage requirements by half.
The paper derives real Schur forms for real circulant and skew-circulant matrices using DCT and DST, leading to faster algorithms for matrix-vector multiplications and a more efficient DCT-DST version of the CSCS iteration for real positive definite Toeplitz systems, which saves half the storage compared to the FFT version.
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.