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

From (13), we know that the style spectral decomposition of the matrix A is
(23)A=Q(A11⋱Att)QT.
When n is even and t=n/2,
(24)Ajj=∑k=1nak(cos(k-1)θjsin(k-1)θj-sin(k-1)θjcos(k-1)θj),θj=2j-1nπ, j=1,2,…,t.
When n is odd and t=(n-1)/2+1, Ajj is defined in (24) (j=1,2,…,(n-1)/2) and
(25)Att=(-1)k∑k=1nak.

Lemma 3.
A
is an invertible matrix if and only if f(ωj)≠0 (j=1,2,…,n), where
(26)f(ωj)=∑k=1nakωjk-1, ωj=ei((2j-1)/n)π, j=1,2,…,n.

Let
(27)σj=|f(ωj)|, j=1,2,…,n,𝒦=max{σj}min{σj}.

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

The proof of Lemma 3 and Remark 4 is given in the following:
(28)A=∑k=1nakCk-1.
Consequently, the spectral decomposition of the matrix A (by using the complex style spectral decomposition of C=Q0(ω1⋱ωn)Q0*) is
(29)A=Q0(f(ω1)f(ω2)⋱f(ωn))Q0*,
where Q0 is a unitary matrix.

Let ΔA, Δb be the perturbation of the coefficient matrix A and vector b, respectively, where
(30)ΔA=(δa1δa2⋯δan-1δan-δanδa1δa2⋯δan-1⋮⋱⋱⋱⋮-δa3⋯-δanδa1δa2-δa2-δa3⋯-δanδa1).
Let
(31)A^=A+ΔA, b^=b+δb,f^(ωj)=∑k=1n(ak+δak)ωjk-1.
If
(32)∑k=1n|δak|<min1≤j≤n{σj},
then
(33)|f^(ωj)|≥|∑k=1nakωjk-1|-∑k=1n|δak||ωj|k-1≥min1≤j≤n{σj}-∑k=1n|δak|>0.
Hence A^ is an invertible matrix. Let
(34)σmin=min1≤j≤n{σj}, Δ=∑k=1n|δak|.
By Ax=b and A^x^=b^, we get
(35)x^-x=A^-1b^-A-1b=A^-1(b+δb)-A-1b=A^-1δb+(A^-1-A-1)b=A^-1δb+(A^-1-A-1)Ax=A^-1δb+A^-1(A-A^)x,∥x^-x∥2≤∥A^-1∥2∥δb∥2+∥A^-1∥2∥A^-A∥2∥x∥2≤∥δb∥2σmin-Δ+∥A^-A∥2∥x∥2σmin-Δ,∥x^-x∥2∥x∥2≤∥δb∥2(σmin-Δ)∥x∥2+∥A^-A∥2σmin-Δ≤∥A∥2σmin-Δ(∥δb∥2∥b∥2+∥A^-A∥2∥A∥2),
where
(36)∥A∥2=max1≤j≤n{σj}.
Notice that A^-A=ΔA is a skew circulant matrix, and ∥A-A^∥2=|-1|∥A^-A∥2=∥A^-A∥2. So we get
(37)∥A^-A∥2=max1≤j≤n|∑k=1nδakωjk-1|≤∑k=1n|δak||ωj|k-1=∑k=1n|δak|=Δ.
Hence we have the following theorem.

Theorem 5.
Let A, A^, δb, Δ, and σmin be defined as above. If Δ<σmin, then
(38)∥x^-x∥2∥x∥2≤σmaxσmin-Δ(∥δb∥2∥b∥2+Δσmax),
where
(39)σmax=∥A∥2.

Remark 6.
From (38) and (39), the condition number of the skew circulant system can be defined as 𝒦=max{σj}/min{σj}. 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
(40)Ω≡{(ΔA,Δb)∣(A+ΔA)x^=b+Δb},η(x^)≡inf(ΔA,Δb)∈Ω∥[ΔA,Δb]∥,(A+ΔA)x^=b+Δb
which is equivalent to
(41)(ΔA,Δb)(x^-1)=b-Ax^.
Due to [15], we have
(42)η(x^)=∥b-Ax^∥21+∥x^∥22(∥·∥ being unitary invariant norm).

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):
(43)Ω≡{(ΔA,Δb)∣(A+ΔA)x^=b+Δb, ΔA is a skew circulant matrix}η(x^)≡inf(ΔA,Δb)∈Ω{∥(ΔA,Δb)∥F}.
Then Ω≠ϕ (such that ΔA=0 is a skew circulant matrix, and Δb=Ax^-b) and
(44)η2(x^)=inf(ΔA,Δb)∈Ω{∥ΔA∥F2+∥ΔAx^+Ax^-b∥F2}.
Since
(45)ΔA=∑k=1nδakCk-1,
so
(46)∥ΔA∥F2=n∑k=1n(δak)2.
Besides that, we can get
(47)∥ΔAx^+Ax^-b∥F2 =∥QC*QTx^+Ax^-b∥F2 =∥(∑k=1nδakC1k-1x1(0)⋮∑k=1nδakCtk-1xt(0))-r0∥F2 =∥G0(δa1⋮δan)-r0∥F2,
where
(48)C*=(∑k=1nδakC1k-1⋱∑k=1nδakCtk-1),r0=QT(b-Ax^), QTx^=(x1(0)⋮xt(0)),G0=(C10x1(0)⋯C1n-1x1(0)⋮⋱⋮Ct0xt(0)⋯Ctn-1xt(0)).

Let
(49)f(δa1,…,δan)=n∑k=1n(δak)2+∥G0(δa1⋮δan)-r0∥F2;
then
(50)∂f∂δak=0
which is equivalent to
(51)(2nIn+2G0TG0)(δa1⋮δan)-2G0Tr0=0.∂2f∂(δak)2=2nIn+2G0TG0>0,
Hence the f is a convex function about (δa1,…,δan), and the point of minimal value is
(52)(δa1⋮δan)=(nIn+G0TG0)-1G0Tr0.
Substituting it back into (49), we can get the following.

Theorem 7.
One has
(53)η(x^)2=nr0TG0(nIn+G0TG0)-2G0Tr0 +∥[G0(nIn+G0TG0)-1G0T-In]r0∥F2.

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(σ1′,…,σn′), and σj′≥0 (j=1,2,…,n). Hence we have
(54)η(x^)2=nr0TUΣVT(nIn+Σ2)-2VΣUTr0+∥[UΣVT(nIn+Σ2)-1VΣUT-In]r0∥F2=nr1TΣ(nIn+Σ2)-2Σr1+∥[Σ(nIn+Σ2)-1Σ-In]r0∥F2=nr1TΣ(nIn+Σ2)-2Σr1+∥[Σ(nIn+Σ2)-1Σ-In]r1∥F2=nr1TΣ(nIn+Σ2)-2Σr1+n2r1T(nIn+Σ2)-2r1=r1T(d1⋱dn)r1,
where r1=UTr0, and dj=(nσj′2+n2)/(n+σj′2)2=n/(n+σj′2).

Remark 8.
By σj′2≤∥G0∥F2=n∥x^∥22, we get 1+(σj′2/n)≤1+∥x^∥22, and hence 1/(1+∥x^∥22)≤n/(n+σj′2).

Algorithm 9.

Step 1. Form the block style spectral decomposition of the matrix C(55)C=Q(C1⋱Ct)QT.

Step 2. Compute r=b-Ax^.

Step 3. Compute r0=QTr.

Step 4. Compute QTx^=(x1(0)⋮x1(0)).

Step 5. Form G0.

Step 6. Compute η2(x^).