On Optimal Backward Perturbation Analysis for the Linear System with Skew Circulant Coefficient Matrix

Abstract

We first give the style spectral decomposition of a special skew circulant matrix 𝐶 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 𝐶. 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.

1. Introduction

A skew circulant matrix with the first row (𝑎₁,𝑎₂,…,𝑎_𝑛) is a square matrix of the form 𝑎₁𝑎₂⋯𝑎_𝑛−1𝑎_𝑛−𝑎_𝑛𝑎₁𝑎₂⋯𝑎_𝑛−1⋮⋱⋱⋱⋮−𝑎₃⋯−𝑎_𝑛𝑎₁𝑎₂−𝑎₂−𝑎₃⋯−𝑎_𝑛𝑎_1𝑛×𝑛, (1) denoted by SCirc(𝑎₁,𝑎₂,…,𝑎_𝑛).

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 [5] employed a skew circulant matrix to construct 𝑠-dimensional lattice rules. Spectral decompositions of skew circulant and skew left circulant matrices are discussed in [6]. Akhondi and Toutounian [7] presented a new iteration method for the numerical solution of Hermitian positive definite Toeplitz systems of linear equations. Narasimha [8] 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 [9] 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 [10] 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 2𝐷 frequency plane from a pair of coarse 2𝐷 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 𝑦=ℎⓈ𝑥=𝐻_𝑠𝑥, where 𝐻_𝑠 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 [11], Foltz and Welsh provided the convolution performed between ℎ and 𝑥 is skew-circulant rather than circulant in [12].

Liu and Guo [13] 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 [14]. 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 [15].

In this paper, we first give the style spectral decomposition of a special skew circulant matrix 𝐶 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 𝐶. 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 𝐶=010⋯0001⋯0⋮⋱⋱⋱⋮0⋯001−10⋯00. (2) Some properties of this matrix are given in the following theorem.

Lemma 1. (1) The eigenvalues of matrix 𝐶 are 𝜆_𝑗=𝑒^{𝑖((2𝑗−1)/𝑛)𝜋},𝑗=1,2,…,𝑛. (3)
(2) If 𝑛 is even, the matrix 𝐶 has no real eigenvalue and 𝜆_𝑗=𝜆_{𝑛+1−𝑗}𝑛,𝑗=1,2,…,2. (4)
The basis of the associated two-dimensional invariant subspace can be taken as 𝑥_𝑗⁽¹⁾=1cos𝜃_𝑗⋮cos(𝑛−1)𝜃_𝑗,𝑥_𝑗⁽²⁾=0sin𝜃_𝑗⋮sin(𝑛−1)𝜃_𝑗,𝜃_𝑗=2𝑗−1𝑛𝑛𝜋,𝑗=1,2,…,2. (5)
(3) If 𝑛 is odd, the matrix 𝐶 has only one real eigenvalue 𝜆_(𝑛+1)/2=−1, and the associated eigenvector is 𝑥_(𝑛+1)/2=11⋮1,𝜆−1−1_𝑗=𝜆_{𝑛+1−𝑗},𝑗=1,2,…,𝑛−12. (6)
The basis of the associated two-dimensional invariant subspace can be taken as 𝑥_𝑗⁽¹⁾=1cos𝜃_𝑗⋮cos(𝑛−1)𝜃_𝑗,𝑥_𝑗⁽²⁾=0sin𝜃_𝑗⋮sin(𝑛−1)𝜃_𝑗,𝜃_𝑗=2𝑗−1𝑛𝜋,𝑗=1,2,…,𝑛−12. (7)
Specially, if 𝑛 is even, then 𝑥_𝑗⁽¹⁾ and 𝑥_𝑗⁽²⁾ span the two-dimensional invariant subspace associated with 𝜆_𝑗 and 𝜆_𝑗.

Lemma 2. (1)  𝑥_𝑗⁽¹⁾ and 𝑥_𝑗⁽²⁾ are orthogonal.
(2)  𝑥_𝑗^(𝑙) and 𝑥_𝑘^(𝑠) are orthogonal (𝑗≠𝑘) (𝑙,𝑠=1,2).
(3) Also ‖𝑥_(𝑛+1)/2‖₂=√𝑛, ‖𝑥_𝑗⁽¹⁾‖₂=‖𝑥_𝑗⁽²⁾‖₂=𝑛/2, 1𝜆_𝑗⋮𝜆_𝑗^𝑛−1₂=√𝑛. (8)

Let 𝑄=[𝑞₁,𝑞₂,…,𝑞_𝑛], where 𝑞_2𝑗−1=2𝑛𝑥_𝑗⁽¹⁾,𝑞_2𝑗=2𝑛𝑥_𝑗⁽²⁾𝑛𝑗=1,2,…,2,𝑞_𝑛=1𝑛11⋮1().−1−1𝑛isodd (9) Then 𝑄 is an orthogonal matrix, and if 𝑛 is even, 𝐶𝐶=𝑄¹𝐶₂⋱𝐶_𝑛/2𝑄^𝑇, (10) where 𝐶_𝑗=^{cos𝜃𝑗sin𝜃𝑗}_{−sin𝜃𝑗cos𝜃𝑗}(𝑗=1,2,…,𝑛/2).

When 𝑛 is odd, 𝐶𝐶=𝑄¹⋱𝐶_(𝑛−1)/2𝑄−1^𝑇, (11) where 𝐶_𝑗=^{cos𝜃𝑗sin𝜃𝑗}_{−sin𝜃𝑗cos𝜃𝑗}(𝑗=1,2,…,(𝑛−1)/2).

In fact, (10) and (11) are the style spectral decomposition of the matrix 𝐶.

2.2. The Style Spectral Decomposition of the Skew Circulant Matrix

We have 𝑎𝐴=¹𝑎₂⋯𝑎_𝑛−1𝑎_𝑛−𝑎_𝑛𝑎₁𝑎₂⋯𝑎_𝑛−1⋮⋱⋱⋱⋮−𝑎₃⋯−𝑎_𝑛𝑎₁𝑎₂−𝑎₂−𝑎₃⋯−𝑎_𝑛𝑎₁=𝑎₁⋅𝐶⁰+𝑎₂⋅𝐶¹+⋯+𝑎_𝑛⋅𝐶^𝑛−1=𝑎₁⋅𝑄𝐶⁰₀𝑄^𝑇+⋯+𝑎_𝑛⋅𝑄𝐶₀^𝑛−1𝑄^𝑇=𝑄𝑎₁⋅𝐶⁰₀𝑄^𝑇+⋯+𝑄𝑎_𝑛⋅𝐶₀^𝑛−1𝑄^𝑇=𝑄^𝑛_𝑘=1𝑎_𝑘⋅𝐶₀^𝑘−1𝑄^𝑇, (12)(13) where 𝐶₀=𝐶₁𝐶₂⋱𝐶_𝑛/2 (14) (𝑛 is even, the same case as (10)), 𝐶₀=𝐶₁⋱𝐶_(𝑛−1)/2−1 (15) (𝑛 is odd, the same case as (11)).

Noticing that 𝑄 is an orthogonal matrix, hence (13) is the style spectral decomposition of the matrix 𝐴.

The following are the computation formulae of the factors in (13): 𝐶^𝑘_𝑗=cos𝑘𝜃_𝑗sin𝑘𝜃_𝑗−sin𝑘𝜃_𝑗cos𝑘𝜃_𝑗. (16) Hence, when 𝑛 is even, ^𝑛_𝑘=1𝑎_𝑘⋅𝐶₀^𝑘−1=𝐴₁𝐴₂⋱𝐴_𝑛/2, (17) where, for arbitrary 𝑗=1,2,…,𝑛/2, 𝐴_𝑗=^𝑛_𝑘∑₌₁𝑎_𝑘cos𝑘𝜃∑_𝑗^𝑛_𝑘=1𝑎_𝑘sin𝑘𝜃_𝑗−∑^𝑛_𝑘=1𝑎_𝑘sin𝑘𝜃∑_𝑗^𝑛_𝑘=1𝑎_𝑘cos𝑘𝜃_𝑗. (18) When 𝑛 is odd, ^𝑛_𝑘=1𝑎_𝑘⋅𝐶₀^𝑘−1=𝐴₁⋱𝐴_(𝑛−1)/2𝐴_∗, (19) where 𝐴_𝑗 is defined by (18), and 𝐴_∗=(−1)^𝑘𝑛_𝑘=1𝑎_𝑘. (20)

Hence the style spectral decomposition of the matrix 𝐴 is 𝐴=𝑄^𝑛_𝑘∑₌₁𝑎_𝑘𝐶_1𝑛^𝑘−1_𝑘∑₌₁𝑎_𝑘𝐶₂^𝑘−1⋱⋱𝑄^𝑇. (21)

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: 𝐴𝑥=𝑏, (22) where 𝐴 is defined in (2).

From (13), we know that the style spectral decomposition of the matrix 𝐴 is 𝐴𝐴=𝑄¹¹⋱𝐴_𝑡𝑡𝑄^𝑇. (23) When 𝑛 is even and 𝑡=𝑛/2, 𝐴_𝑗𝑗=^𝑛_𝑘=1𝑎_𝑘cos(𝑘−1)𝜃_𝑗sin(𝑘−1)𝜃_𝑗−sin(𝑘−1)𝜃_𝑗cos(𝑘−1)𝜃_𝑗,𝜃_𝑗=2𝑗−1𝑛𝜋,𝑗=1,2,…,𝑡. (24) When 𝑛 is odd and 𝑡=(𝑛−1)/2+1, 𝐴_𝑗𝑗 is defined in (24) (𝑗=1,2,…,(𝑛−1)/2) and 𝐴_𝑡𝑡=(−1)^𝑘𝑛_𝑘=1𝑎_𝑘. (25)

Lemma 3. 𝐴 is an invertible matrix if and only if 𝑓(𝜔_𝑗)≠0(𝑗=1,2,…,𝑛), where 𝑓𝜔_𝑗=^𝑛_𝑘=1𝑎_𝑘𝜔_𝑗^𝑘−1,𝜔_𝑗=𝑒^{𝑖((2𝑗−1)/𝑛)𝜋},𝑗=1,2,…,𝑛. (26)

Let 𝜎_𝑗=𝑓𝜔_𝑗𝜎,𝑗=1,2,…,𝑛,𝒦=max_𝑗min𝜎_𝑗. (27)

Remark 4. The singular values of matrix 𝐴 are 𝜎₁,𝜎₂,…,𝜎_𝑛.

The proof of Lemma 3 and Remark 4 is given in the following: 𝐴=^𝑛_𝑘=1𝑎_𝑘𝐶^𝑘−1. (28) Consequently, the spectral decomposition of the matrix 𝐴 (by using the complex style spectral decomposition of 𝐶=𝑄₀^𝜔1⋱_𝜔𝑛𝑄^∗₀) is 𝐴=𝑄₀𝑓𝜔₁𝑓𝜔₂⋱𝑓𝜔_𝑛𝑄^∗₀, (29) where 𝑄₀ is a unitary matrix.

Let Δ𝐴, Δ𝑏 be the perturbation of the coefficient matrix 𝐴 and vector 𝑏, respectively, where Δ𝐴=𝛿𝑎₁𝛿𝑎₂⋯𝛿𝑎_𝑛−1𝛿𝑎_𝑛−𝛿𝑎_𝑛𝛿𝑎₁𝛿𝑎₂⋯𝛿𝑎_𝑛−1⋮⋱⋱⋱⋮−𝛿𝑎₃⋯−𝛿𝑎_𝑛𝛿𝑎₁𝛿𝑎₂−𝛿𝑎₂−𝛿𝑎₃⋯−𝛿𝑎_𝑛𝛿𝑎₁. (30) Let ̂𝑓𝜔𝐴=𝐴+Δ𝐴,𝑏=𝑏+𝛿𝑏,_𝑗=^𝑛_𝑘=1𝑎_𝑘+𝛿𝑎_𝑘𝜔_𝑗^𝑘−1. (31) If ^𝑛_𝑘=1𝛿𝑎_𝑘<min_{1≤𝑗≤𝑛}𝜎_𝑗, (32) then 𝑓𝜔_𝑗≥^𝑛_𝑘=1𝑎_𝑘𝜔_𝑗^𝑘−1−^𝑛_𝑘=1𝛿𝑎_𝑘𝜔_𝑗^𝑘−1≥min_{1≤𝑗≤𝑛}𝜎_𝑗−^𝑛_𝑘=1𝛿𝑎_𝑘>0. (33) Hence 𝐴 is an invertible matrix. Let 𝜎_min=min_{1≤𝑗≤𝑛}𝜎_𝑗,Δ=^𝑛_𝑘=1𝛿𝑎_𝑘. (34) By 𝐴𝑥=𝑏 and 𝐴𝑥=𝑏̂, we get 𝑥−𝑥=𝐴⁻¹𝑏̂−𝐴⁻¹𝑏=𝐴⁻¹(𝑏+𝛿𝑏)−𝐴⁻¹𝑏=𝐴⁻¹𝛿𝑏+𝐴⁻¹−𝐴⁻¹𝑏=𝐴⁻¹𝛿𝑏+𝐴⁻¹−𝐴⁻¹=𝐴𝐴𝑥₋₁𝛿𝑏+𝐴⁻¹𝐴‖‖𝐴−𝑥,𝑥−𝑥₂≤𝐴⁻¹₂‖𝛿𝑏‖₂+𝐴⁻¹₂𝐴−𝐴₂‖𝑥‖₂≤‖𝛿𝑏‖₂𝜎_min+−Δ𝐴−𝐴₂‖𝑥‖₂𝜎_min,‖‖−Δ𝑥−𝑥₂‖𝑥‖₂≤‖𝛿𝑏‖₂𝜎_min−Δ‖𝑥‖₂+𝐴−𝐴₂𝜎_min≤‖𝐴‖−Δ₂𝜎_min‖‖−Δ𝛿𝑏₂‖𝑏‖₂+𝐴−𝐴₂‖𝐴‖₂, (35) where ‖𝐴‖₂=max_{1≤𝑗≤𝑛}𝜎_𝑗. (36) Notice that 𝐴−𝐴=Δ𝐴 is a skew circulant matrix, and ‖𝐴−𝐴‖₂=|−1|‖𝐴−𝐴‖₂=‖𝐴−𝐴‖₂. So we get 𝐴−𝐴₂=max_{1≤𝑗≤𝑛}^𝑛_𝑘=1𝛿𝑎_𝑘𝜔_𝑗^𝑘−1≤^𝑛_𝑘=1𝛿𝑎_𝑘𝜔_𝑗^𝑘−1=^𝑛_𝑘=1𝛿𝑎_𝑘=Δ. (37) Hence we have the following theorem.

Theorem 5. Let 𝐴, 𝐴, 𝛿𝑏, Δ, and 𝜎_min be defined as above. If Δ<𝜎_min, then ‖𝑥−𝑥‖₂‖𝑥‖₂≤𝜎_max𝜎_min‖‖−Δ𝛿𝑏₂‖𝑏‖₂+Δ𝜎_max, (38) where 𝜎_max=‖𝐴‖₂. (39)

Remark 6. From (38) and (39), the condition number of the skew circulant system can be defined as 𝒦=max{𝜎_𝑗}/min{𝜎_𝑗}. 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 𝑥 be an approximate solution to 𝐴𝑥=𝑏 and let )∣()},𝜂(𝑥)Ω≡{(Δ𝐴,Δ𝑏𝐴+Δ𝐴𝑥=𝑏+Δ𝑏≡inf_{()Δ𝐴,Δ𝑏∈Ω},()‖[Δ𝐴,Δ𝑏]‖𝐴+Δ𝐴𝑥=𝑏+Δ𝑏 (40) which is equivalent to ()𝑥Δ𝐴,Δ𝑏−1=𝑏−𝐴𝑥. (41) Due to [15], we have 𝜂(𝑥)=‖𝑏−𝐴𝑥‖₂1+‖𝑥‖²₂‖⋅‖beingunitaryinvariantnorm. (42)

If the recycling property of 𝐴 is not utilized in the algorithm in forming 𝑥, then 𝜂(𝑥) can be used to estimate the backward stability for this algorithm.

Let 𝑥 be an approximate solution to 𝐴𝑥=𝑏, where 𝐴 is defined in (2): )∣()}𝜂(𝑥)Ω≡{(Δ𝐴,Δ𝑏𝐴+Δ𝐴𝑥=𝑏+Δ𝑏,Δ𝐴isaskewcirculantmatrix≡inf_{()Δ𝐴,Δ𝑏∈Ω}‖(Δ𝐴,Δ𝑏)‖_𝐹. (43) Then Ω≠𝜙 (such that Δ𝐴=0 is a skew circulant matrix, and Δ𝑏=𝐴𝑥−𝑏) and 𝜂²(𝑥)=inf_{()Δ𝐴,Δ𝑏∈Ω}‖Δ𝐴‖²_𝐹+‖Δ𝐴𝑥+𝐴𝑥−𝑏‖²_𝐹. (44) Since Δ𝐴=^𝑛_𝑘=1𝛿𝑎_𝑘𝐶^𝑘−1, (45) so ‖Δ𝐴‖²_𝐹=𝑛^𝑛_𝑘=1𝛿𝑎_𝑘². (46) Besides that, we can get ‖Δ𝐴𝑥+𝐴𝑥−𝑏‖²_𝐹=𝑄𝐶_∗𝑄^𝑇𝑥+𝐴𝑥−𝑏²_𝐹=^𝑛_𝑘∑₌₁𝛿𝑎_𝑘𝐶₁^𝑘−1𝑥₁⁽⁰⁾_𝑛⋮_𝑘∑₌₁𝛿𝑎_𝑘𝐶_𝑡^𝑘−1𝑥_𝑡⁽⁰⁾−𝑟₀²_𝐹=𝐺₀𝛿𝑎₁⋮𝛿𝑎_𝑛−𝑟₀²_𝐹, (47) where 𝐶_∗=^𝑛_𝑘∑₌₁𝛿𝑎_𝑘𝐶₁^𝑘−1⋱_𝑛𝑘∑₌₁𝛿𝑎_𝑘𝐶_𝑡^𝑘−1,𝑟₀=𝑄^𝑇(𝑏−𝐴𝑥),𝑄^𝑇𝑥𝑥=⋮¹⁽⁰⁾𝑥_𝑡⁽⁰⁾,𝐺₀=𝐶⁰₁𝑥₁⁽⁰⁾⋯𝐶₁^𝑛−1𝑥₁⁽⁰⁾𝐶⋮⋱⋮_0𝑡𝑥_𝑡⁽⁰⁾⋯𝐶_𝑡^𝑛−1𝑥_𝑡⁽⁰⁾. (48)

Let 𝑓𝛿𝑎₁,…,𝛿𝑎_𝑛=𝑛^𝑛_𝑘=1𝛿𝑎_𝑘²+𝐺₀𝛿𝑎₁⋮𝛿𝑎_𝑛−𝑟₀²_𝐹; (49) then 𝜕𝑓𝜕𝛿𝑎_𝑘=0 (50) which is equivalent to 2𝑛𝐼_𝑛+2𝐺^𝑇₀𝐺₀𝛿𝑎₁⋮𝛿𝑎_𝑛−2𝐺^𝑇₀𝑟₀𝜕=0.₂𝑓𝜕𝛿𝑎_𝑘²=2𝑛𝐼_𝑛+2𝐺^𝑇₀𝐺₀>0, (51) Hence the 𝑓 is a convex function about (𝛿𝑎₁,…,𝛿𝑎_𝑛), and the point of minimal value is 𝛿𝑎₁⋮𝛿𝑎_𝑛=𝑛𝐼_𝑛+𝐺^𝑇₀𝐺₀⁻¹𝐺^𝑇₀𝑟₀. (52) Substituting it back into (49), we can get the following.

Theorem 7. One has 𝜂(𝑥)²=𝑛𝑟^𝑇₀𝐺₀𝑛𝐼_𝑛+𝐺^𝑇₀𝐺₀⁻²𝐺^𝑇₀𝑟₀+𝐺₀𝑛𝐼_𝑛+𝐺^𝑇₀𝐺₀⁻¹𝐺^𝑇₀−𝐼_𝑛𝑟₀²_𝐹. (53)

Let 𝐺₀=𝑈Σ𝑉^∗ be the singular value decomposition of 𝐺₀, where 𝑈 and 𝑉 are unitary (in fact, 𝑈 and 𝑉 can be real orthogonal), Σ=diag(𝜎^′₁,…,𝜎^′_𝑛), and 𝜎^′_𝑗≥0(𝑗=1,2,…,𝑛). Hence we have 𝜂(𝑥)²=𝑛𝑟^𝑇₀𝑈Σ𝑉^𝑇𝑛𝐼_𝑛+Σ²⁻²𝑉Σ𝑈^𝑇𝑟₀+𝑈Σ𝑉^𝑇𝑛𝐼_𝑛+Σ²⁻¹𝑉Σ𝑈^𝑇−𝐼_𝑛𝑟₀²_𝐹=𝑛𝑟^𝑇₁Σ𝑛𝐼_𝑛+Σ²⁻²Σ𝑟₁+Σ𝑛𝐼_𝑛+Σ²⁻¹Σ−𝐼_𝑛𝑟₀²_𝐹=𝑛𝑟^𝑇₁Σ𝑛𝐼_𝑛+Σ²⁻²Σ𝑟₁+Σ𝑛𝐼_𝑛+Σ²⁻¹Σ−𝐼_𝑛𝑟₁²_𝐹=𝑛𝑟^𝑇₁Σ𝑛𝐼_𝑛+Σ²⁻²Σ𝑟₁+𝑛²𝑟^𝑇₁𝑛𝐼_𝑛+Σ²⁻²𝑟₁=𝑟^𝑇₁𝑑₁⋱𝑑_𝑛𝑟₁, (54) where 𝑟₁=𝑈^𝑇𝑟₀, and 𝑑_𝑗=(𝑛𝜎_𝑗^′2+𝑛²)/(𝑛+𝜎_𝑗^′2)²=𝑛/(𝑛+𝜎_𝑗^′2).

Remark 8. By 𝜎_𝑗^′2≤‖𝐺₀‖²_𝐹=𝑛‖𝑥‖²₂, we get 1+(𝜎_𝑗^′2/𝑛)≤1+‖𝑥‖²₂, and hence 1/(1+‖𝑥‖²₂)≤𝑛/(𝑛+𝜎_𝑗^′2).

Algorithm 9.   
Step  1. Form the block style spectral decomposition of the matrix 𝐶𝐶𝐶=𝑄¹⋱𝐶_𝑡𝑄^𝑇. (55)
Step  2. Compute 𝑟=𝑏−𝐴𝑥.
Step  3. Compute 𝑟₀=𝑄^𝑇𝑟.
Step  4. Compute 𝑄^𝑇𝑥=^𝑥1(0)⋮_𝑥1(0).
Step  5. Form 𝐺₀.
Step  6. Compute 𝜂²(𝑥).

4. Conclusion

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 [16] 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.

Acknowledgments

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.