Abstract
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.
Highlights
Recall that a matrix T =nj,−k=1 0 is said to be Toeplitz if tjk = tj−k; a matrix C =nj,−k=1 0 is said to be circulant if cjk = cj−k and c−l = cn−l for 1 ≤ l ≤ n − 1; and a matrix S =nj,−k=1 0 is said to be skew-circulant if sjk = sj−k and s−l = −sn−l for 1 ≤ l ≤ n − 1.Toeplitz matrices arise in a variety of applications in mathematics, scientific computing and engineering, for instance, signal processing, algebraic differential equation, time series and control theory, see e.g. [3] and a large literature therein
We show the calculations of Cx and Sx for any real n-vector x using discrete cosine transform (DCT) and discrete sine transform (DST)
Consider the case n = 2m (The odd case is similar to the even case), we have the following version, The DCT-DST version of circulant and skew-circulant splitting (CSCS) iteration: Given an initial guess x(0) ∈ Rn, compute x(k), for k = 0, 1, · · ·, until {x(k)} converges:
Summary
Real Schur form, real circulant matrices, real skew-circulant matrices, real Toeplitz matrices, CSCS iteration. The eigenvalues of a real skew-circulant matrix S can be arranged in the following order Theorem 2.5 (Real Schur form of real skew-circulant matrices).
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