Structured backward error and structured linearizations

given byMP(z) =P_m

j=0λ^jM Ajis eitherT-symmetric,T-skew-symmetric,T-even orT-odd.

Hence replacing Aj and r by M Aj and M r, respectively, in the above results, we obtain corresponding results for the polynomial P.

Similarly, whenM is unitary andM =M^HorM =−M^H,we consider the Jordan algebra J := {A ∈ C^n×n : M⁻¹A^HM = A} and the Lie algebra L := {A ∈ C^n×n : M⁻¹A^HM =

−A} associated with the scalar product (x, y) 7→y^HM x.Now, let P(z) = P_m

j=0z^jAj be a polynomial where Aj’s are in J and/or in L. Then the polynomial MP(z) = P_m

j=0z^jM Aj

is either H-Hermitian, H-skew-Hermitian, H-even or H-odd. Hence the results obtained above extend easily to the polynomial P by replacing Aj, r by M Aj, M r, respectively. In particular, whenM :=J,whereJ :=

Ã

0 I

−I 0

!

∈C^2n×2n,the Jordan algebraJconsists of skew-Hamiltonian matrices and the Lie algebra L consists of Hamiltonian matrices. So, for example, considering the polynomial P(z) :=P_m

j=0z^jAj,whereAjs are Hamiltonian whenj is even and skew-Hamiltonian whenjis odd, we see that the polynomialJP(z) =P_m

j=0z^jJA_j is H-even. Hence extending the results obtained for H-even polynomial to the case of P,we have the following.

Theorem 4.3.13. LetSbe the space of polynomials of the formP(z) =P_m

j=0z^jAj whereAj

is Hamiltonian when j is even, andAj is skew-Hamiltonian whenj is odd. Let P∈S. Then for(λ, x)∈C×Cⁿ, setr:=−P(λ)x.Then we have

η_F^S(λ, x,P) =







√2krk²₂−|x^HJr|² kΛmk2 ≤√

2η(λ, x,P), if λ∈iR q

kbrk²₂+^2(krk_kΛ^{22−|xHJr|2)}

mk²₂ , if λ∈C\iR.

η₂^S(λ, x,P) =







η(λ, x,P), ifλ∈iR q

kbrk²₂+^krk22_kΛ^−|xHJr|2

mk²₂ , ifλ∈C\iR where br=

"

Πe(Re(Λm)^T)−(I−Πe)(Im(Λm)^T) Π_e(Im(Λ_m)^T) + (I−Π_e)(Re(Λ_m)^T)

#_†"

re(x^HJr) im(x^HJr)

# .

4.4 Structured backward error and structured lineariza-

strong linearizations of P.Generalizing the concepts of companion forms, Mackey at al. [67]

introduced two vector spaces of potential linearizations

L1(P) = {L(λ) : L(λ).(Λm−1⊗In) =v⊗P(λ), v∈C^m} L2(P) = {L(λ) : (Λ^T_m−1⊗In).L(λ) =ω^T⊗P(λ), ω∈C^m}

of dimension m(m−1)n²+m, where Λm−1 = [λ^m−1, λ^m−2, . . . ,1]^T, ⊗ is the Kronecker product,vis called the right ansatz vector for L∈L1(P) andωis called the left ansatz vector for L ∈ L2(P). Observe that C1(λ)∈ L1(P) with v = e1 and C2(λ) ∈ L2(P) with ω =e1. It is also shown that x ∈ Cⁿ is a right (resp. left) eigenvector of P corresponding to the eigenvalue λif and only if Λm−1⊗x(resp. Λm−1⊗x) is an eigenvector of L ∈L1(P)(resp.

L2(P)) corresponding to the eigenvalueλ.

Since the linearization is unique up to the choice of right/left ansatz vector, these spaces are too large to obtain a potential linearization. Keeping this in mind, define the double ansatz space DL(P) := L1(P)∩L2(P), see [64, 67]. It is also proved that for vectors v = [v1, v2, . . . , vm]^T andω = [ω1, ω2, . . . , ωm]^T,there exist anmn×mnmatrix pencil L(λ) = λX+Y ∈DL(P) that simultaneously satisfies

L(λ)·(Λm−1⊗In) =v⊗P(λ), (Λ^T_m−1⊗In)·L(λ) =ω^T ⊗P(λ)

if and only if v = ω, called the ansatz vector associated to L. Corresponding to an ansatz vector vectorv= [v1, v2, . . . , vm]^T ∈C^massociate the scalar polynomialp(x;v) =v1x^m−1+ v2x^m−2+. . .+vm−1x+vm referred to as the “v-polynomial” of the vector v, see [64, 67].

We adopt the convention that p(x;v) has a root at ∞ wheneverv1 = 0. Then L ∈ DL(P) corresponding to an ansatz vectorv ∈C^m is a linearization of P(λ) if and only if no root of thev-polynomial is an eigenvalue of P(λ),see [64, 67]. Note that this statement includes∞ as one of the possible roots ofp(x;v) or possible eigenvalue of P(λ).

Assume that L(λ) =λX+Y ∈L1(P) is a linearization of a polynomial P(z) =P_m

j=0z^jAj

with respect to the normalized right ansatz vector v ∈C^m.Then if x∈Cⁿ is a right eigen- vector of P corresponding to an eigenvalue λ then Λm−1 ⊗x is a right eigenvector of L corresponding to the eigenvalueλ.Recall from Lemma 1.2.19 that for L∈L1(P)/DL(P) the following hold for any x∈Cⁿ.

kL(λ)(Λ_m−1⊗x)k₂ = kvk₂kP(λ)xk₂ (4.7)

|(Λ_m−1⊗x)^TL(λ)(Λ_m−1⊗x)| = |Λ^T_m−1v| |x^TP(λ)x| (4.8)

|(Λ_m−1⊗x)^HL(λ)(Λ_m−1⊗x)| = |Λ^H_m−1v| |x^HP(λ)x| (4.9) It is also straightforward to verify that

√1

2 ≤ kΛ_mk₂

kΛm−1k2k(λ,1)k2 ≤1. (4.10)

Assume that P∈Pm(C^n×n) and (λ, x) is an approximate eigenpair of P.Now we compare η(λ, x,P) and η(λ,Λm−1⊗x,L;v) where L ∈ L1(P) is a linearization of P corresponding to an ansatz vector v and η(λ,Λm−1⊗x,L;v) denotes the unstructured backward error of the approximate eigenpair (λ,Λ_m−1⊗x) of L.It is needless to say that η(λ,Λ_m−1⊗x,L;v)

can easily be obtained from (4.1) by setting m= 1. Without loss of generality we carry the analysis by taking unit ansatz vector, that is,v∈C^mis an ansatz vector with kvk2= 1.

Theorem 4.4.1. Let P ∈ Pm(C^n×n) be a regular matrix polynomial and L ∈ L1(P) be a linearization of P corresponding to a right ansatz vector v ∈ C^m. Assume that (λ, x) is an approximate eigenpair of P. Then we have

√1

2 ≤η(λ,Λm−1⊗x,L;v) η(λ, x,P) ≤1.

Proof: By (4.1), we have

η(λ,Λm−1⊗x,L;v) = kL(λ)(Λ_m−1⊗x)k₂

k(Λm−1⊗x)k2k(λ,1)k2 = kvk₂kP(λ)xk₂ k(Λm−1⊗x)k2k(λ,1)k2

= kvk2kΛmk2kxk2

k(Λm−1⊗x)k2k(λ,1)k2η(λ, x,P)

= kΛmk2

kΛm−1k2k(λ,1)k2η(λ, x,P).

Now by (4.10) we obtain the desired result. ¥

Its evident that a matrix polynomial can have infinitely many linearizations. For a given structured polynomial P ∈S, it is necessary to detect a linearization L so that it preserves the spectral symmetry. Although, we have encountered three linear spaces of linearizations, namely L1(P),L2(P),DL(P), onlyL1(P) preserves most of the structures which we consider in this chapter and it is easy to obtain the right eigenvector of P from that of L ∈ L1(P).

In fact, for a T-symmetric polynomial P we have DL(P) = L1(P). As shown in [40, 68], structured linearization imposes a restriction on the ansatz vector. To obtain a potential structured linearization we restrict the ansatz vector into a particular subset ofC^mas shown by Mackey et al. [40, 68]. A list of structured linearizations and corresponding structure of ansatz vectors are given in the Table 4.2.

whereR=





1 . .. 1



, Σ = diag{(−1)^m−1,(−1)^m−2, . . . ,(−1)⁰}.

Table 4.2 shows that except forT-symmetric and T-skew-symmetric polynomials, for all other structured polynomials P we could have two types of structured linearizations having the same spectral symmetry as that of P.A crucial task is to choose the best possible type of structured linearization for a given structured polynomial. In this section we provide a recipe for structured linearization L of a given P∈S.

Recall thatη(λ, x,P)≤η^S_F(λ, x,P) andη(λ, x,P)≤η₂^S(λ, x,P).Hence for any structured linearization L we haveη(λ,Λm−1⊗x,L;v)≤η_F,2^S (λ,Λm−1⊗x,L;v),where the ansatz vector v is of the form as described in Table 4.2. To make the presentation simple, unless otherwise stated, we writeη^S(λ,Λm−1⊗x,L;v) for bothη_F^S(λ,Λm−1⊗x,L;v) andη^S₂(λ,Λm−1⊗x,L;v), in the rest of the chapter.

Lemma 4.4.2. Let P∈S. Let L∈L1(P)be a structured linearization of Pcorresponding to

S Structured Linearization ansatz vector

T-symm T-symm v∈C^m

T-skew-symm T-skew-symm v∈C^m

T-even T-even Σv=v

T-odd Σv=−v

T-odd T-even Σv=−v

T-odd Σv=v

H-Herm H-Herm v∈R^m

H-skew-Herm v∈iR^m

H-skew-Herm H-Herm v∈iR^m

H-skew-Herm v∈R^m

H-even H-even Σv=v

T-odd Σv=−v

H-odd H-even Σv=−v

H-odd Σv=v

Table 4.2: Table for the admissible ansatz vectors for structured polynomials.

an ansatz vector v.Then for both |||·||| ≡ |||·|||F and|||·||| ≡ |||·|||2 we have η^S(λ,Λm−1⊗x,L;v)

η(λ, x,P) ≥ 1

√2.

Proof: By Theorem 4.4.1 we haveη(λ,Λm−1⊗x,L;v)≥^√1₂η(λ, x,P).Therefore we obtain η^S(λ,Λm−1⊗x,L;v)

η(λ, x,P) ≥ η(λ,Λm−1⊗x,L;v) η(λ, x,P) ≥ 1

√2.

¥

Given an ansatz vectorv,for the rest of the chapter, we set δv :=kΛm−1k2

|Λ^∗_m−1v|,∗ ∈ {T, H}. (4.11) Obviouslyδv≥1.Note that Λ^∗_m−1v6= 0 and henceδv <∞when L∈DL(P) is a linearization of P corresponding to the ansatz vectorv.

For a T-symmetric matrix polynomial P, any ansatz vector v yields a T-symmetric lin- earization. Thus, we are free to choose v as followed by Table 4.2. Combined with Theo- rem 4.3.1, which shows that there is (almost) no difference between structured and unstruc- tured backward errors, we have the following result.

Theorem 4.4.3. LetS be the space ofT-symmetric matrix polynomials. Assume thatP∈S andL∈L1(P) =DL(P)is theT-symmetric linearization ofPwith respect to an ansatz vector v. Let(λ, x)∈C×Cⁿ be such thatkxk2= 1. Then we have

1. |||·||| ≡ |||·|||F :

p2−δ⁻²v

√2 ≤η_F^S(λ,Λm−1⊗x,L;v)

η(λ, x,P) ≤√

2

2. |||·||| ≡ |||·|||2: 1

√2≤ η^S2(λ,Λm−1⊗x,L;v) η(λ, x,P) ≤1

Proof: First consider|||·||| ≡ |||·|||F.By Theorem 4.3.1 we know that

η^S_F(λ,Λm−1⊗x,L;v) = q

2^kL(λ)(Λ_k(Λ ^{m−1⊗x)k22}

m−1⊗x)k²₂ −_k(Λ ¹

m−1⊗x)k⁴₂|(Λm−1⊗x)^TL(λ)(Λm−1⊗x)|² k(λ,1)k₂

= r

2kP(λ)xk²₂ −^|Λ_kΛ^Tm−1v|2

m−1k²₂|x^TP(λ)x|²

kΛm−1k2k(λ,1)k2 , by (1.6), (1.7).

Now applying p

2−δv⁻²kP(λ)xk2 ≤ q

2kP(λ)xk²₂−δv⁻²|x^TP(λ)x|² ≤ √

2kP(λ)xk2, (4.10) and (4.11) we obtain the desired result.

Next consider |||·||| ≡ |||·|||2. Since structured backward error and unstructured backward error are same for spectral norm, the desired result follows by Theorem 4.4.1 and (4.10).¥

Assuming |||·||| ≡ |||·|||F and |||·||| ≡ |||·|||2 the results discussed above present growth of un- structured backward error of an eigenpair of P∈Swhen P is linearized by a structured pencil in L1(P) = DL(P). However, comparing η^S(λ,Λm−1⊗x,L;v) with η^S(λ, x,P) we have the following results for|||·||| ≡ |||·|||F and|||·||| ≡ |||·|||2.

Theorem 4.4.4. LetS be the space ofT-symmetric matrix polynomials. Assume thatP∈S andL∈L1(P) =DL(P)is theT-symmetric linearization ofPwith respect to an ansatz vector v. Let(λ, x)∈C×Cⁿ be such thatkxk2= 1. Then we have

1. |||·||| ≡ |||·|||F : 1

√2≤ ηF^S(λ,Λm−1⊗x,L;v) η_F^S(λ, x,P) ≤√

2

2. |||·||| ≡ |||·|||2: 1

√2≤ η^S2(λ,Λm−1⊗x,L;v) η₂^S(λ, x,P) ≤1.

Proof: First consider|||·||| ≡ |||·|||F.Then by Theorem 4.3.1 we have η_F^S(λ,Λm−1⊗x,L;v)

η^S_F(λ, x,P) = q

2krk²₂−δv⁻²|x^Tr|²

p2krk²₂− |x^Tr|² · kΛmk2

kΛm−1k2k(λ,1)k2

, r=−P(λ)x.

Now using the inequality 0< δ_v⁻¹≤1 by (4.11), and by (4.10) we haveη^S_F(λ,Λm−1⊗x,L;v) η_F^S(λ, x,P) ≥

√1

2.Further notice that

η^S_F(λ,Λm−1⊗x,L;v)

η_F^S(λ, x,P) ≤η^S_F(λ,Λm−1⊗x,L;v) ηF(λ, x,P) . Now by Theorem 4.4.3 the desired result follows.

Next consider|||·||| ≡ |||·|||2. Then by Theorem 4.3.1 we have η_F^S(λ,Λm−1⊗x,L;v)

η^S_F(λ, x,P) = kΛmk2

kΛm−1k2k(λ,1)k2

. Therefore by (4.10) we obtain the desired result.¥

Now we consider T-skew-symmetric polynomials. It follows from the Table 4.2 that any ansatz vectorvyields aT-skew-symmetric linearization for aT-skew-symmetric matrix poly- nomial P. Combined with Theorem 4.3.1, which states that there is (almost) no difference between structured and unstructured backward errors, we have the following result.

Theorem 4.4.5. Let S be the space of T-skew-symmetric matrix polynomials. Assume that P ∈S andL ∈L1(P) is the T-skew-symmetric linearization of P with respect to the ansatz vector v.Let (λ, x)be an approximate eigenpair of P.Then we have

1. |||·||| ≡ |||·|||F : 1≤η_F^S(λ,Λm−1⊗x,L;v) η(λ, x,P) ≤√

2

2. |||·||| ≡ |||·|||2: 1

√2≤ η^S2(λ,Λm−1⊗x,L;v) η(λ, x,P) ≤1.

Proof: First consider|||·||| ≡ |||·|||_F.By Theorem 4.3.4 we have η_F^S(λ,Λm−1⊗x,L;v) = √

2 kL(λ)(Λm−1⊗x)k2

kΛm−1k2kxk2k(1, λ)k2

=

√2kΛmk2

kΛm−1k2k(λ,1)k2η(λ, x,P) for any ansatz vectorv.By (4.10) we obtain the desired result.

Since structured backward error and unstructured backward error are same for|||·||| ≡ |||·|||2, the desired result follows by Theorem 4.4.1 and (4.10).¥

Further, comparingη^S(λ,Λm−1⊗x,L;v) withη^S(λ, x,P) we have the following results for

|||·||| ≡ |||·|||F and|||·||| ≡ |||·|||2.

Theorem 4.4.6. Let S be the space of T-skew-symmetric matrix polynomials. Assume that P∈S andL∈L1(P) =DL(P)is theT-skew-symmetric linearization ofP with respect to an ansatz vector v.Let (λ, x)∈C×Cⁿ be such thatkxk2= 1. Then we have

1. |||·||| ≡ |||·|||F : 1≤η_F^S(λ,Λm−1⊗x,L;v) η_F^S(λ, x,P) ≤√

2

2. |||·||| ≡ |||·|||2: 1

√2≤ η^S₂(λ,Λm−1⊗x,L;v) η₂^S(λ, x,P) ≤1.

Proof: First consider|||·||| ≡ |||·|||F.Then by Theorem 4.3.4 it follows that η_F^S(λ,Λm−1⊗x,L;v)

η^S_F(λ, x,P) =

√2kΛmk2

kΛ_m−1k₂k(λ,1)k₂. Hence by (4.10) we obtain the desired result.

Next consider|||·||| ≡ |||·|||₂. Then by Theorem 4.3.4 we have η_F^S(λ,Λm−1⊗x,L;v)

η^S_F(λ, x,P) = kΛmk2

kΛm−1k2k(λ,1)k2

. Therefore by (4.10)the desired result follows.¥

Next considerT-even polynomials. Note thatT-even polynomials can have bothT-even andT-odd linearizations having the same eigen-symmetry as that ofT-even polynomial.

Theorem 4.4.7. Let S be the space of T-even matrix polynomials. Assume that P(z) = P_m

j=0z^jAj ∈S. Let Le (resp. Lo) from L1(P)be the T-even ( resp. T-odd) linearization of P with respect to the ansatz vector Σv = v (resp. Σv = −v). If (λ, x) is an approximate eigenpair ofP then we have the following.

1. |||·||| ≡ |||·|||F :If|λ| ≤1then q

2 + (|λ|²−1)δ⁻²v

√2 ≤η^S_F(λ,Λm−1⊗x,Le;v)

η(λ, x,P) ≤√

2and 1≤η^S_F(λ,Λm−1⊗x,Lo;v)

η(λ, x,P) ≤

q

2 + (|λ|⁻²−1)δv⁻². If|λ| ≥1then1≤ηF^S(λ,Λm−1⊗x,Le;v)

η(λ, x,P) ≤

q

2 + (|λ|²−1)δ⁻²v and q

2 + (|λ|⁻²−1)δv⁻²

√2 ≤ η^S_F(λ,Λm−1⊗x,Lo;v)

η(λ, x,P) ≤√

2

2. |||·||| ≡ |||·|||2: 1

√2≤ η^S2(λ,Λm−1⊗x,Le;v)

η(λ, x,P) ≤

q

1 +|λ|²δ⁻²v

√1

2 ≤η^S2(λ,Λm−1⊗x,Lo;v)

η(λ, x,P) ≤

q

1 +|λ|⁻²δv⁻² if λ6= 0,

√1

2 ≤η^S2(λ,Λm−1⊗x,Lo;v)

η(λ, x,P) ≤1, if λ= 0.

Proof: First consider theT-even linearization Leof P∈S.By Theorem 4.3.6 we know that

η^S_F(λ,Λm−1⊗x,Le;v) = q

2^kLe_k(Λ^{(λ)(Λm−1⊗x)k22}

m−1⊗x)k²₂ +_k(Λ^(|λ|2−1)

m−1⊗x)k⁴₂|(Λm−1⊗x)^TLe(λ)(Λm−1⊗x)|² k(λ,1)k2

= r

2kP(λ)xk²₂+^(|λ|2_kΛ^{−1)|ΛTm−1v|2}

m−1k²₂ |x^TP(λ)x|² kΛm−1k2k(λ,1)k2

.

For|λ| ≤1,we have q

2 + (|λ|²−1)δv⁻²kP(λ)xk2≤ q

2kP(λ)xk²₂+ (|λ|²−1)δv⁻²|x^TP(λ)x|²≤

√2kP(λ)xk2and for|λ|>1 we have

√2kP(λ)xk2≤ q

2kP(λ)xk²₂+ (|λ|²−1)δ⁻²v |x^TP(λ)x|²≤ q

2 + (|λ|²−1)δ⁻²v kP(λ)xk2. Hence by (4.10) we obtain the desired results for|||·||| ≡ |||·|||F.

Next, consider|||·||| ≡ |||·|||2. Then by Theorem 4.3.6 we have

η^S₂(λ,Λm−1⊗x,Le;v) =

qkLe(λ)(Λm−1⊗x)k²₂

k(Λm−1⊗x)k²₂ +_k(Λ ^|λ|2

m−1⊗x)k⁴₂|(Λm−1⊗x)^TLe(λ)(Λm−1⊗x)|² k(λ,1)k2

= r

kP(λ)xk²₂+^|λ|_kΛ^{2|ΛTm−1v|2}

m−1k²₂ |x^TP(λ)x|² kΛm−1k2k(λ,1)k2

.

Notice that kP(λ)xk2 ≤ q

kP(λ)xk²₂+|λ|²δ⁻²v |x^TP(λ)x|² ≤ q

2 +|λ|²δv⁻²kP(λ)xk2 for any λ∈C.Hence by (4.10) we obtain the desired result.

Next assume that Lo is theT-odd linearization of P∈S. By Theorem 4.3.8 we have the

following for|||·||| ≡ |||·|||F :

η^S_F(λ,Λm−1⊗x,Lo;v) =













r

2^{kLo(λ)(Λm−1⊗x)k22}

k(Λm−1⊗x)k2 2

+( ¹

|λ|2−1) ¹

k(Λm−1⊗x)k4 2

|(Λm−1⊗x)^TLo(λ)(Λm−1⊗x)|² k(Λm−1⊗x)k2k(1,λ)k2 ,

if λ6= 0.

√2_k(Λ^{kLo(λ)(Λm−1⊗x)k2}

m−1⊗x)k2k(1,λ)k2, if λ= 0.

=









s

2kP(λ)xk²₂+(_|λ|¹₂−1)^|Λ

Tm−1v|2 kΛm−1k2

2|x^TP(λ)x|²

kΛm−1k2kxk2k(1,λ)k2 , ifλ6= 0.

√2_kΛ ^kP(λ)xk2

m−1k2kxk2k(1,λ)k2, ifλ= 0.

Letλ6= 0.Then for|λ| ≤1,we have√

2kP(λ)xk2≤ q

2kP(λ)xk²₂+ (|λ|⁻²−1)δ⁻²v |x^TP(λ)x|²≤ q

2 + (|λ|⁻²−1)δv⁻²kP(λ)xk2and for|λ|<1 we have q

2 + (|λ|⁻²−1)δ⁻²v kP(λ)xk2≤ q

2kP(λ)xk²₂+ (|λ|⁻²−1)δv⁻²|x^TP(λ)x|²≤√

2kP(λ)xk2. Hence by (4.10), we obtain the desired result.

Now consider|||·||| ≡ |||·|||2.Then by Theorem 4.3.8 we have

η₂^S(λ,Λm−1⊗x,Lo;v) =













r

kLo(λ)(Λm−1⊗x)k2 2 k(Λm−1⊗x)k2

2

+ ¹

|λ|2k(Λm−1⊗x)k4 2

|(Λm−1⊗x)^TLo(λ)(Λm−1⊗x)|²

k(1,λ)k2 ,

if λ6= 0.

kLo(λ)(Λm−1⊗x)k2

k(Λm−1⊗x)k2k(1,λ)k2, if λ= 0

=









s

kP(λ)xk²₂+ ^|Λ

Tm−1v|2

|λ|2kΛm−1k2 2

|x^TP(λ)x|²

kΛm−1k2k(1,λ)k2 , ifλ6= 0.

kP(λ)xk2

kΛm−1k2k(1,λ)k2, if λ= 0 Forλ6= 0 notice thatkP(λ)xk2≤

q

kP(λ)xk²₂+|λ|⁻²δv⁻²|x^TP(λ)x|²≤ q

1 +|λ|⁻²δv⁻²kP(λ)xk2. Hence by (4.10) the desired result follows.¥

Corollary 4.4.8. For|||·||| ≡ |||·|||F, using that fact that δ_v⁻¹ >0 for any ansatz vector v,we have 1 ≤ η^S_F(λ,Λm−1⊗x,Le;v)

η(λ, x,P) ≤√

2 when |λ| ≤ 1 and 1 ≤ η_F^S(λ,Λm−1⊗x,Lo;v) η(λ, x,P) ≤√

2 when |λ| ≥1. For|||·||| ≡ |||·|||2, using the fact that |δ_v⁻¹| ≤1 for any ansatz vector v, we have

√1

2 ≤ η₂^S(λ,Λm−1⊗x,Le;v) η(λ, x,P) ≤√

2 when |λ| ≤ 1, and 1

√2 ≤ η₂^S(λ,Λm−1⊗x,Lo;v) η(λ, x,P) ≤√

2 when |λ| ≥1.

Now notice that the bound in Theorem 4.4.7 is true also forT-odd polynomials but with the roles ofT-even andT-odd exchanged.

Remark 4.4.9. Let P be a T-even or T-odd polynomial. Then if |λ| ≤1 pick the T-even linearization and for|λ|>1pick theT-odd linearization. The corrolarry 4.4.8 also ensure that the backward error is magnified at most by a factor of√

2 due to the structured linearization process.

Next we considerH-Hermitian polynomials. Note thatH-Hermitian polynomial can have bothH-Hermitian andH-skew-Hermitian linearizations preserving the eigen-symmetry of the original polynomial.

Theorem 4.4.10. LetSbe the space ofH-Hermitian matrix polynomials. Assume thatP∈S andL∈DL(P)is the H-Hermitian/H-skew-Hermitian linearization ofPwith respect to the ansatz vector v.If (λ, x) is an approximate right eigenpair ofP then we have the following:

1. |||·||| ≡ |||·|||F :

p2−δ⁻²v

√2 ≤η_F^S(λ,Λm−1⊗x,L;v)

η(λ, x,P) ≤√

2ifλ∈R,

√1

2 ≤η^SF(λ,Λm−1⊗x,L;v) η(λ, x,P) ≤√

2 q

τ²δ⁻²v k(1, λ)k²₂+ 2,ifλ∈C\R

2. |||·||| ≡ |||·|||2: 1

√2≤ η^S2(λ,Λm−1⊗x,L;v)

η(λ, x,P) ≤1ifλ∈R,

√1

2 ≤η^S₂(λ,Λm−1⊗x,L;v) η(λ, x,P) ≤√

2 q

τ²δ⁻²v k(1, λ)k²₂+ 1,ifλ∈C\R

where τ =

°°

°°

°°

"

1 re(λ) 0 im(λ)

#_†°

°°

°°

°2

for H-Hermitian linearization and τ =

°°

°°

°°

"

0 −im(λ) 1 re(λ)

#_†°

°°

°°

°2

for H-skew-Hermitian linearization.

Proof: First we consider the H-Hermitian linearization. Then the ansatz vector v ∈ R^m. Assume thatλ∈R.Then by Theorem 4.3.9 we have

η^S_F(λ,Λm−1⊗x,L;v) = q

2kP(λ)xk²₂−δv⁻²|x^HP(λ)x|²

kΛm−1k2k(1, λ)k2 , η^S₂(λ,Λm−1⊗x,L;v) = kP(λ)xk2

kΛm−1k2k(λ,1)k2. Now applyingp

2−δ⁻²v kP(λ)xk2≤ q

2kP(λ)xk²₂−δv⁻²|x^HP(λ)x|²≤√

2kP(λ)xk2and (4.10) we obtain the desired result.

Next considerλ∈C\R.Then for|||·||| ≡ |||·|||F,by Theorem 4.3.9 we have

η_F^S(λ,Λm−1⊗x,L;v) = 1 kΛm−1k2kxk2

vu ut kbrk²₂

kΛm−1k²₂+2kP(λ)xk²₂−2^|Λ_kΛ^Hm−1v|2

m−1k²₂|x^HP(λ)x|² k(1, λ)k²₂

≤ 1

kΛm−1k2

s kbrk²₂

kΛm−1k²₂+ 2krk²₂

k(1, λ)k²₂, r:=−P(λ)x

where br=

"

1 reλ 0 imλ

#_†"

re(Λ^H_m−1vx^HP(λ)x) im(Λ^H_m−1vx^HP(λ)x)

#

. For|||·||| ≡ |||·|||2,by Theorem 4.3.9 we have the following:

η^S₂(λ,Λm−1⊗x,L;v) = 1 kΛm−1k2kxk2

vu ut kbrk²₂

kΛm−1k²₂ +kP(λ)xk²₂−^|Λ_kΛ^Hm−1v|2

m−1k²₂|x^HP(λ)x|² k(1, λ)k²₂

≤ 1

kΛm−1k2

s kbrk²₂

kΛm−1k²₂ +kP(λ)xk²₂ k(1, λ)k²₂

Now notice that kbrk₂≤τ|Λ^H_m−1v|krk₂,whereτ=

°°

°°

°°

"

1 reλ 0 imλ

#_†°

°°

°°

°2

.

Therefore for|||·||| ≡ |||·|||Fwe haveη^S_F(λ,Λm−1⊗x,L;v)

η(λ, x,P) ≤ kΛmk2

kΛm−1k2

q

τ²δv⁻²+ 2k(1, λ)k⁻²₂ , and η₂^S(λ,Λm−1⊗x,L;v)

η(λ, x,P) ≤ kΛmk2

kΛm−1k2

q

τ²δv⁻²+k(1, λ)k⁻²₂ . Hence by Lemma 4.4.2 the re- sults follows.

Next assume that L is the H-skew-Hermitian linearization of P∈ S associated with the ansatz vector v ∈iR^m. Since the structured backward error expression of any approximate eigenpair is same for H-Hermitian andH-skew-Hermitian linear polynomial, we obtain the same bound.¥

Note that, forλ ∈R, it does not really make any difference in the magnified backward error due to H-Hermitian or H-skew-Hermitian linearization. Further for λ ∈ C\R two cases arise. Let us redefine τ :=τh for H-Hermitian linearization and τ :=τsh for H-skew- Hermitian linearization. Then from the bounds of Theorem 4.4.10 we conclude the following.

For a given (λ, x),ifτh≤τsh then chooseH-Hermitian linearization andH-skew-Hermitian linearization otherwise.

Next considerH-even polynomials. Note thatH-even polynomials can have bothH-even and H-odd types of linearizations having the same eigen-symmetry (λ,−λ).The main issue here is to decide which type of linearization is to be chosen to minimize the backward error.

Theorem 4.4.11. Let S be the space of H-even polynomials. Assume thatP ∈S andL ∈ L1(P)is the H-even/H-odd linearization ofPwith respect to the ansatz vector v.If(λ, x)is an approximate eigenpair ofP then we have the following:

1. |||·||| ≡ |||·|||F :

p2−δ⁻²v

√2 ≤η_F^S(λ,Λm−1⊗x,L;v)

η(λ, x,P) ≤√

2ifλ∈iR,

√1

2 ≤η^S_F(λ,Λm−1⊗x,L;v) η(λ, x,P) ≤√

2 q

τ²δ⁻²v k(1, λ)k²₂+ 2,ifλ∈C\iR

2. |||·||| ≡ |||·|||2: 1

√2≤ η^S2(λ,Λm−1⊗x,L;v)

η(λ, x,P) ≤1ifλ∈iR,

√1

2 ≤η^S2(λ,Λm−1⊗x,L;v) η(λ, x,P) ≤√

2 q

τ²δ⁻²v k(1, λ)k²₂+ 1,ifλ∈C\iR

where τ =

°°

°°

°°

"

1 −im(λ) 0 re(λ)

#_†°

°°

°°

°2

for H-even linearization and τ =

°°

°°

°°

"

0 re(λ) 1 im(λ)

#_†°

°°

°°

°2

for H-odd linearization.

Proof: The proof is similar as that of Theorem 4.4.10 and hence omitted.¥

The similar bound can be obtained if we replaceH-even polynomials by H-odd polyno- mials in Theorem 4.4.11.

Observe that, forλ∈iR,it does not really make any difference in the magnified backward error due to H-even or H-odd linearization. Further for λ∈C\iRtwo cases arise. Let us redefineτ :=τe forH-even linearization andτ:=τoforH-odd linearization. Then from the bounds that have been obtained in Theorem 4.4.11, we conclude the following. For a given (λ, x) and P ∈ S, pick an H-even linearization when τe ≤ τo and pickH-odd linearization otherwise.

structuredeigenvaluechoiceofβS :=ηS (λ,Λm−1⊗x,L;v)/η(λ,x,P)βS :=ηS (λ,Λm−1⊗x,L;v)/η(λ,x,P) polynomialofinterestlinearization|||·|||≡|||·|||F|||·|||≡|||·|||2 T-symmetricλ∈CT-symmetric√ 2−δ−2 v√ 2≤βS≤√ 21√ 2≤βS≤1 T-skew-symmetricλ∈CT-skew-symmetric1≤βS≤√ 21√ 2≤βS≤1 T-even/T-odd|λ|≤1T-even√ 2+(|λ|2−1)δ−2 v√ 2≤βS≤√ 21√ 2≤βS≤√ 2 |λ|≥1T-odd√ 2+(|λ|−2−1)δ−2 v√ 2≤βS≤√ 2 H-Hermitian/λ∈RH-Hermitian/√ 2−δ−2 v√ 2≤βS≤√ 21√ 2≤βS≤1 H-skew-HermitianH-skew-Hermitian λ∈C\RH-Hermitian1√ 2≤βS ≤√ 2q 2+τ2 hδ−2 vk(1,λ)k^{2 2}1√ 2≤βS ≤√ 2q 1+τ2 hδ−2 vk(1,λ)k^{2 2} H-skew-Hermitian1√ 2≤βS≤√ 2q 2+τ2 shδ−2 vk(1,λ)k^{2 2}1√ 2≤βS≤√ 2q 1+τ2 hδ−2 vk(1,λ)k^{2 2} H-even/H-oddλ∈iRH-even/H-odd√ 2−δ2 √ 2≤βS≤√ 21√ 2≤βS≤1 λ∈C\iRH-even1√ 2≤βS≤√ 2q 2+τ2 eδ−2 vk(1,λ)k^{2 2}1√ 2≤βS≤√ 2q 1+τ2 eδ−2 vk(1,λ)k^{2 2} H-odd1√ 2≤βS ≤√ 2q 2+τ2 oδ−2 vk(1,λ)k^{2 2}1√ 2≤βS ≤√ 2q 1+τ2 oδ−2 vk(1,λ)k^{2 2} Table4.3:Thechoiceofstructuredlinearizations,whereδv=kΛm−1k2/|Λ∗ m−1v|,∗∈{T,H}andvistheansatzvector. Herer=−P(λ)x,τh=° ° ° ° °· 1re(λ) 0im(λ)¸†° ° ° ° ° ² ,τsh=

° ° ° ° °·¸† 0−im(λ) 1re(λ)

° ° ° ° ° ² ,τe=

° ° ° ° °·¸† 1−im(λ) 0re(λ)

° ° ° ° ° ² ,τo=

° ° ° ° °·¸† 0re(λ) 1im(λ)

° ° ° ° ° ²

.

4.5 Structured pseudospectra of structured matrix poly-