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On Optimal Backward Perturbation Analysis for the Linear System with Skew Circulant Coefficient Matrix
We first give the style spectral decomposition of a special skew circulant matrix and then get the style decomposition of arbitrary skew circulant matrix by making use of the Kronecker products between the elements of first row in skew circulant and the special skew circulant . Besides that, we obtain the singular value of skew circulant matrix as well. Finally, we deal with the optimal backward perturbation analysis for the linear system with skew circulant coefficient matrix on the base of its style spectral decomposition.
A skew circulant matrix with the first row is a square matrix of the form denoted by .
Skew circulant matrices have important applications in various disciplines including image processing, signal processing, solving Toeplitz matrix problems, and preconditioner. The skew circulant matrices are considered as preconditioners for linear-multistep-formulae (LMF-) based ordinary differential equations (ODEs) codes; Hermitian and skew-Hermitian Toeplitz systems are considered in [1–4]. Lyness and Sörevik  employed a skew circulant matrix to construct -dimensional lattice rules. Spectral decompositions of skew circulant and skew left circulant matrices are discussed in . Akhondi and Toutounian  presented a new iteration method for the numerical solution of Hermitian positive definite Toeplitz systems of linear equations. Narasimha  believed that the linear convolution required in block filtering can be decomposed into a sum of skew-circulant convolutions and such convolutions can be realized efficiently with half-length complex transforms when the signals are real. Liu and Vaidyanathan  presented a new family of normal form state-space structures, the method used allows people to synthesize in normal form, most IIR transfer functions, and the state transition matrices involved are either circulant or skew circulant matrices. Vaidyanathan and Pal  examined a case where two arrays are generated by matrices that are adjugates of each other; in this case, it is possible to obtain a dense rectangular tiling of the 2 frequency plane from a pair of coarse 2 DFT filter banks; the special case where the adjugate pairs are generated by skew circulant matrices has some advantages, which are examined in detail. An additional convolution-multiplication property for the skew-circulant convolution operation , where is a skew-circulant matrix; besides, skew-circulant convolution is the underlying form of convolution in half of the 40 cases of symmetric convolution, and the convolution is an extension of a result Vernet's , Foltz and Welsh provided the convolution performed between and is skew-circulant rather than circulant in .
Liu and Guo  gave the optimal backward perturbation analysis for a linear system with block circulant coefficient matrix. The optimal backward perturbation bound for underdetermined systems is studied by J.-G. Sun and Z. Sun in . Some new theorems generalizing a result of Oettli and Prager are applied to the a posteriori analysis of the compatibility of a computed solution to the uncertain data of a linear system by Rigal and Gaches in .
In this paper, we first give the style spectral decomposition of a special skew circulant matrix and then get the style spectral decomposition of arbitrary skew circulant matrix by making use of Kronecker products between the elements of first row in skew circulant and the special skew circuant . Besides that, we obtain the singular value of skew circulant matrix as well. Finally, we deal with the optimal backward perturbation analysis for the linear system with skew circulant coefficient matrix on the base of its style spectral decomposition.
2. The Style Spectral Decomposition of Skew Circulant Matrix
2.1. Style Spectral Decomposition of a Special Skew Circulant Matrix
Let Some properties of this matrix are given in the following theorem.
Lemma 1. (1) The eigenvalues of matrix are
(2) If is even, the matrix has no real eigenvalue and
The basis of the associated two-dimensional invariant subspace can be taken as
(3) If is odd, the matrix has only one real eigenvalue , and the associated eigenvector is
The basis of the associated two-dimensional invariant subspace can be taken as
Specially, if is even, then and span the two-dimensional invariant subspace associated with and .
Lemma 2. (1) and are orthogonal.
(2) and are orthogonal () ().
(3) Also , ,
Let , where Then is an orthogonal matrix, and if is even, where .
When is odd, where .
In fact, (10) and (11) are the style spectral decomposition of the matrix .
2.2. The Style Spectral Decomposition of the Skew Circulant Matrix
We have where ( is even, the same case as (10)), ( is odd, the same case as (11)).
Noticing that is an orthogonal matrix, hence (13) is the style spectral decomposition of the matrix .
The following are the computation formulae of the factors in (13): Hence, when is even, where, for arbitrary , When is odd, where is defined by (18), and
Hence the style spectral decomposition of the matrix is
3. The Structured Perturbation Analysis
In this section we give the structured perturbation analysis for linear systems with skew circulant coefficient matrix.
3.1. Condition Number and Relative Error of Linear Skew Circulant Equation System
Consider the following: where is defined in (2).
From (13), we know that the style spectral decomposition of the matrix is When is even and , When is odd and , is defined in (24) () and
Lemma 3. is an invertible matrix if and only if , where
Remark 4. The singular values of matrix are .
The proof of Lemma 3 and Remark 4 is given in the following: Consequently, the spectral decomposition of the matrix (by using the complex style spectral decomposition of ) is where is a unitary matrix.
Let , be the perturbation of the coefficient matrix and vector , respectively, where Let If then Hence is an invertible matrix. Let By and , we get where Notice that is a skew circulant matrix, and . So we get Hence we have the following theorem.
Theorem 5. Let , , , , and be defined as above. If , then where
Remark 6. From (38) and (39), the condition number of the skew circulant system can be defined as . It is easily computed, as well as the bound of perturbation (38).
3.2. Optimal Backward Perturbation Bound of the Linear Skew Circulant Equation System
Let be an approximate solution to and let which is equivalent to Due to , we have
If the recycling property of is not utilized in the algorithm in forming , then can be used to estimate the backward stability for this algorithm.
Let be an approximate solution to , where is defined in (2): Then (such that is a skew circulant matrix, and ) and Since so Besides that, we can get where
Let then which is equivalent to Hence the is a convex function about , and the point of minimal value is Substituting it back into (49), we can get the following.
Theorem 7. One has
Let be the singular value decomposition of , where and are unitary (in fact, and can be real orthogonal), , and . Hence we have where , and .
Remark 8. By , we get , and hence .
Step 1. Form the block style spectral decomposition of the matrix
Step 2. Compute .
Step 3. Compute .
Step 4. Compute .
Step 5. Form .
Step 6. Compute .
The related problems of skew-circulant matrix are considered in this paper. We not only present style spectral decomposition and singular value but also study backward perturbation analysis for the linear system with skew-circulant coefficient matrix. The reason why we focus our attentions on skew-circulant is to explore the application of skew circulant in the related field in medicine. Wittsack et al. in  validated a deconvolution method originating from magnetic resonance techniques and apply it to the calculation of dynamic contrast enhanced computed tomography perfusion imaging, and the application of a block circulant matrix approach for singular value decomposition renders the analysis independent of tracer arrival time to improve the results. On the basis of existing application situation, we conjecture that SVD decomposition of skew circulant matrix will play an important role in CT-perfusion imaging of human brain.
The research was supported by the Development Project of Science & Technology of Shandong Province (Grant no. 2012GGX10115) and NSFC (Grant no. 11201212) and the AMEP of Linyi University, China.
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