Hermitian polynomials

3.4 T-alternating polynomials

4.1.1 Hermitian polynomials

LetP(z) =Pm

j=0z^jAj be a Hermitian polynomial i.e., Aj ∈Herm(n) forj = 0, . . . , mand λ ∈C\ {0}. We follow the notation from Chapter 2 and denote the Hermitian backward error with respect to |||·|||∞ norm by η^Herm_∞ (P, λ). Note that for a Hermitian polynomial P(z), if λ ∈R then there is no difference between structured and unstructured backward error, i.e. η_∞^Herm(P, λ) = η_∞(P, λ). This fact is shown in [42]. The situation is completely different for λ /∈ R. Thus in the following our attempt is to calculate η_∞^Herm(P, λ) when λ /∈R. In the view of Lemma 1.2.6, we have

η^Herm_∞ (P, λ) = infn

|||(∆₀, . . . ,∆_m)|||∞

(∆₀, . . . ,∆_m)∈(Herm(n))^m+1, ∃v₀, . . . , v_m ∈Cⁿ, vλ :=Pm

j=0λ^jvj 6= 0, vj = ∆jMvλ, j = 0, . . . , mo

. (4.1.5) The following result provides an expression for structured eigenvalue backward error η^Herm_∞ (P, λ) of a Hermitian polynomial P(z).

Theorem 4.1.2. Let P(z) = Pm

j=0z^jAj be a Hermitian polynomial i.e., Aj ∈ Herm(n) for j = 0, . . . , m. Let λ ∈C\R be such that det(P(λ))6= 0. Set M = (P(λ))⁻¹ and define vλ :=Pm

j=0λ^jvj for (v0, . . . , vm)∈(Cⁿ)^m+1. Then η_∞^Herm(P, λ)2

= inf

(v0,...,vm)∈K sup

γ₀²+···+γ_m²=1

gγ0,...,γm(v0, . . . , vm), (4.1.6) where

K=

(v0, . . . , vm)∈(Cⁿ)^m+1vλ 6= 0, Imv_j^∗(Mvλ) = 0, j = 0, . . . , m

(4.1.7)

and for γ0, . . . , γm ∈R, gγ0,...,γm :K →R is defined by

g_γ0,...,γm(v₀, . . . , v_m) = γ₀²kv₀k²+· · ·+γ_m²kv_mk²

kMvλk² . (4.1.8)

Proof. By Theorem 1.2.9, for x(6= 0), y ∈Cⁿthere exist ∆ ∈Herm(n) such that ∆x=yif and only if Imx^∗y= 0. Furthermore minimal 2-norm of such a ∆ is k∆k= ^k_k^y_x^k_k. Therefore in the view of (4.1.5)

η^Herm_∞ (P, λ)2

= inf

f(v0, . . . , vm) (v0, . . . , vm)∈ K , (4.1.9) where f(v0, . . . , vm) is defined for all (v0, . . . , vm)∈(Cⁿ)^m+1 with vλ 6= 0 by

f(v0, . . . , vm) = maxn kv0k²

kMvλk², . . . , kvmk² kMvλk²

o. (4.1.10)

Being the supremum of a continuous function on a compact set, sup

γ²₀+···+γ²_m=1

gγ0,...,γm(v0, . . . , vm) is a continuous function of (v0, . . . , vm). Also, for a fixed (v0, . . . , vm) ∈ K and for any γ0, . . . , γm ∈R with γ₀²+· · ·+γ_m² = 1, we have gγ0,...,γm(v0, . . . , vm)≤f(v0, . . . , vm). This implies that

sup

γ₀²+···+γ_m²=1

g_γ0,...,γm(v₀, . . . , v_m)≤f(v₀, . . . , v_m) (4.1.11) and

(v0,...,vinfm)∈K sup

γ²₀+···+γ²_m=1

g_γ0,...,γm(v₀, . . . , v_m)≤ η_∞^Herm(P, λ)2

<∞.

Therefore, by arguments similar to those in Lemma 3.1.4, the infimum is attained at some element of K.Now if maxn

kv0k²

kM v_λk², . . . ,_k^k_{M v}^vmk2

λk²

o

= _k^k_{M v}^vj0k2

λk² for somej0 ∈ {0, . . . , m}, then max

kv0k²

kMvλk², . . . , kvmk² kMvλk²

= Xm

j=0

ˆ

γ_j² kvjk²

kMvλk² =gˆγ0,...,ˆγm(v0, . . . , vm), (4.1.12) where ˆγj = 1 if j =j0 and ˆγj = 0, otherwise. Therefore from (4.1.11) and (4.1.12), we get

sup

γ₀²+···+γ_m²=1

gγ0,...,γm(v0, . . . , vm) =f(v0, . . . , vm) for each (v0, . . . , vm)∈ K. This implies

(v₀,...,vinf_m)∈K sup

γ₀²+···+γ²_m=1

gγ₀,...,γ_m(v0, . . . , vm) = inf

(v₀,...,v_m)∈Kf(v0, . . . , vm) = η_∞^Herm(P, λ)2

.

Note that the backward error obtained in Theorem 4.1.2 is not easy to compute. There- fore the main aim of this section is to derive computable tight bounds for η^Herm_∞ (P, λ) and also give sufficient conditions for the bounds to be equal to the exact backward error. We first state a lemma which is used to prove the main result.

Lemma 4.1.3. LetM ∈C^n×n be a nonsingular matrix andλ ∈Csuch that Imλ6= 0. Let Λm := [1, λ, . . . , λ^m]∈C^1×(m+1) and set

G:= (Λ^∗_mΛm)⊗(M^∗M), Hj :=i((ej+1Λm)⊗M −(Λ^∗_me^∗_j+1)⊗M^∗)

forj = 0, . . . , m,where ej denotes the j-th standard basis vector ofR^m+1.For nonzero real numbers γ0, . . . , γm let

Γ := diag γ0, . . . , γm

, G(Γ) := Γ⁻¹GΓ⁻¹ and Hj(Γ) := Γ⁻¹HjΓ⁻¹ for j = 0, . . . , m. Then the following statements hold.

(1) For each γ0, . . . , γm ∈R\ {0}, the function

Lγ0,...,γm(t0, . . . , tm) :=λmax G(Γ) +t0H0(Γ) +· · ·+tmHm(Γ)

(4.1.13) attains a global minimum at some t0, . . . , tm ∈R.

(2) Let Lγ0,...,γm(t^?₀, . . . , t^?_m) := min

t0,...,tm∈RLγ0,...,γm(t0, . . . , tm) for some t^?₀, . . . , t^?_m ∈ R. If Lγ0,...,γm(t^?₀, . . . , t^?_m) is a simple eigenvalue of G(Γ) +t^?₀H0(Γ) +· · ·+t^?_mHm(Γ) or m= 1, then

(v0,...,vinfm)∈K gγ0,...,γm(v0, . . . , vm) =

Lγ0,...,γm(t^?₀, . . . , t^?_m)₋1

, (4.1.14)

where gγ0,...,γm and K are defined by (4.1.8) and (4.1.7), respectively.

Proof. The proof of (1) follows immediately from Theorem 1.2.18 if we can show that any linear combination α0H0(Γ) +· · ·+αmHm(Γ) for (α0, . . . , αm)∈R^m+1 \ {0} is indefinite.

Since Hj and Hj(Γ), j = 0, . . . , mare congruent and the congruence transformation does not depend onj, by Sylvester’s Law of Inertia, it is sufficient to show thatα0H0+· · ·+αmHm

is indefinite for any (α0, . . . , αm) ∈ R^m+1 \ {0}. But this can be shown on the lines of Theorem 2.2.2.

Arguments similar to those used in Lemma 3.1.4 imply that for each (γ0, . . . , γm)∈Rⁿ\{0},

gγ0,...,γm attains its infimum over K. Also following the arguments on pages 2 and 3 in Chapter 2, we can write infimum of gγ0,...,γm over Kas

inf

gγ0,...,γm(v0, . . . , vm)|(v0, . . . , vm)∈ K

=

sup

u^∗(ΓGΓ)u u^∗u

u∈(Cⁿ)^m+1\ {0}, u^∗(ΓHjΓ)u= 0, j = 0, . . . , m −1

. Now the proof of (2) follows immediately by Theorem 1.2.18.

Theorem 4.1.4. Let P(z) = Pm

j=0z^jAj be a Hermitian polynomial i.e., Aj ∈ Herm(n) for j = 0, . . . , m.Let λ∈C\R be such that M = (P(λ))⁻¹ exists. For γ0, . . . , γm ∈R, let g_γ0,...,γm and K be as defined by (4.1.8) and (4.1.7) respectively. Then the following hold.

(a)

(η_∞(P, λ))² ≤ sup

γ₀²+···+γ_m²=1

(v0,...,vinfm)∈Kgγ0,...,γm(v0, . . . , vm)≤ η_∞^Herm(P, λ)2

≤ (m+ 1) sup

γ²₀+···+γ_m²=1

(v0,...,vinfm)∈Kgγ0,...,γm(v0, . . . , vm). (4.1.15) (b) Forγ₀, . . . , γ_m ∈R\ {0} and Γ = diag(γ₀, . . . , γ_m), let L_γ0,...,γm(t₀, . . . , t_m)be defined by (4.1.13) for Hermitian matrices G(Γ) and Hj(Γ), j = 0, . . . , m of Lemma 4.1.3.

Suppose there exist γ₀^?, . . . , γ^?_m where γ^?_i are all nonzero such that γ^?₀²+· · ·+γ_m^?2 = 1 and

v0,...,vinfm∈Kg_γ^?_0,...,γm^?(v₀, . . . , v_m) = sup

γ²₀+···+γ_m²=1

v0,...,vinfm∈Kg_γ0,...,γm(v₀, . . . , v_m).

Then

λ^?_max:= inf

t0,...,tm∈RLγ₀^?,...,γ_m^?(t0, . . . , tm) (4.1.16) is attained for some t^?₀, . . . , t^?_m ∈ R. If m = 1 or λ^?_max is a simple eigenvalue of G(Γ^?) +t0H0(Γ^?) +· · ·+tmHm(Γ^?) where Γ^? = diag (γ₀^?, . . . , γ_m^?), then

η_∞(P, λ)≤ 1

pλ^?_max ≤η_∞^Herm(P, λ) ≤

t0,...,tinfm∈RL_1,...,1(t₀, . . . , t_m) ₋¹₂

≤ p

(m+ 1) 1

pλ^?_max. (4.1.17)

(c) We have,

η_∞^Herm(P, λ) =

v0,...,vinfm∈Kgγ₀^?,...,γ_m^?(v0, . . . , vm) 1/2

= 1

pλ^?_max, (4.1.18)

if m = 1 or λ^?_max is a simple eigenvalue of G(Γ^?) +t0H0(Γ^?) +· · ·+tmHm(Γ^?), and either of the following statements hold.

(i) There exists a (ˆv0, . . . ,ˆvm) ∈ K with kˆvjk = kvˆmk for all j ∈ 0, . . . , m−1 such that

gγ₀^?,...,γ^?_m(ˆv0, . . . ,ˆvm) = inf

(v0,...,vm)∈Kgγ₀^?,...,γ_m^?(v0, . . . , vm).

(ii) sup

γ₀²+...+γ_m²=1

(v0,...,vinfm)∈Kg_γ0,...,γm(v₀, . . . , v_m) = inf

(v0,...,vm)∈K sup

γ²₀+···+γ_m²=1

g_γ0,...,γm(v₀, . . . , v_m).

Proof. By (4.1.9)

η^Herm_∞ (P, λ)2

= inf

f(v₀, . . . , v_m) (v₀, . . . , v_m)∈ K , (4.1.19) wheref(v0, . . . , vm) is defined by (4.1.10) andKis defined by (4.1.7). Now by Lemma 4.1.1, for eγj =q

|λ|^j

1+|λ|+···+|λ^m|, j ∈ {0, . . . , m} we have (η_∞(P, λ))² = inf

g_eγ0,...,eγm(v0, . . . , vm)|(v0, . . . , vm)∈(Cⁿ)^m+1, vλ 6= 0

≤ inf

g_eγ0,...,eγm(v0, . . . , vm)|(v0, . . . , vm)∈ K

≤ sup

γ₀²+···+γ_m²=1

(v0,...,vinfm)∈K gγ0,...,γm(v0, . . . , vm). (4.1.20) For all (γ0, . . . , γm)∈R^m+1 withγ₀²+· · ·+γ_m² = 1 and (v0, . . . , vm)∈(Cⁿ)^m+1 withvλ 6= 0, we have

gγ0,...,γm(v0, . . . , vm)≤f(v0, . . . , vm)≤g1,...,1(v0, . . . , vm).

Therefore

(v0,...,vinfm)∈K g_γ0,...,γm(v₀, . . . , v_m)≤ inf

(v0,...,vm)∈K f(v₀, . . . , v_m)≤ inf

(v0,...,vm)∈K g_1,...,1(v₀, . . . , v_m) for all (γ0, . . . γm)∈R^m+1 with γ₀²+· · ·+γ_m² = 1.Thus

sup

γ₀²+···+γ²_m=1

(v0,...,vinfm)∈Kgγ0,...,γm(v0, . . . , vm)≤ η_∞^Herm(P, λ)2

≤ inf

(v0,...,vm)∈Kg1,...,1(v0, . . . , vm).

(4.1.21) Also observe that

(m+1) sup

γ₀²+···+γm²=1

(v0,...,vinfm)∈Kgγ0,...,γm(v0, . . . , vm)

!

= sup

γ²₀+···+γm²=m+1

(v0,...,vinf1)∈Kgγ0,...,γm(v0, . . . , vm),

which implies

(v0,...,vinfm)∈Kg1,...,1(v0, . . . , vm)≤(m+ 1) sup

γ₀²+···+γ²_m=1

(v0,...,vinfm)∈Kgγ0,...,γm(v0, . . . , vm)

!

(4.1.22) Thus (4.1.15) follows immediately from (4.1.20), (4.1.21) and (4.1.22). Note that

(v0,...,vinfm∈K)g_γ0,...,γm(v₀, . . . , v_m) is a continuous function of γ_j for j = 0, . . . , m. Therefore its supremum over the compact set {(γ0, . . . , γm) ∈R^m+1|γ₀²+· · ·+γ²_m = 1} is attained by some (γ₀^?, . . . , γ_m^?). Thus with the assumption that γ_j^? are nonzero for all j = 0, . . . , m, inequality (4.1.17) follows immediately by using Lemma 4.1.3 in (4.1.21) and (4.1.22).

Again let (ˆv₀, . . . ,vˆ_m) be a minimizer of g_γ0^?_,...,γ^?_m(v₀, . . . , v_m) over K such that for each j ∈ {0, . . . , m},kvˆ_jk=kvˆ_mk.LetK⁰ :={(v₀, . . . , v_m)∈ K | kˆv_jk=kvˆ_mk, j = 0, . . . , m−1}. Then

sup

γ₀²+···+γ²_m=1

(v0,...,vinfm)∈Kg_γ0,...,γm(v₀, . . . , v_m) = g_γ0^?_,...,γm^?(ˆv₀, . . . ,ˆv_m) = kvˆ₀k² kMˆvλk²

≥ inf

(v0,...,vm)∈K⁰max

kv0k²

kMvλk², . . . , kvmk² kMvλk²

≥ inf

(v0,...,vm)∈Kmax

kv0k²

kMvλk², . . . , kvmk² kMvλk²

= inf

(v0,...,vm)∈Kf(v0, . . . , vm). (4.1.23) Therefore from (4.1.19), (4.1.21) and (4.1.23), we have

η^Herm_∞ (P, λ)2

= inf

(v0,...,vm)∈Kgγ₀^?,...,γ_m^?(v0, . . . , vm).

Hence (4.1.18) holds due to Lemma 4.1.3.

Remark 4.1.5. The function inf

(v0,...,vm)∈Kg_γ0,...,γm(v₀, . . . , v_m) is a continuous concave func- tion of (γ0, . . . , γm)∈R^m+1 and hence its supremum over the compact set

{(γ0, . . . , γm)∈R^m+1|γ₀²+· · ·+γ_m² = 1}

is attained for some (γ₀^?, . . . , γ_m^?). The assumption on (γ₀^?, . . . , γ_m^?) in the above theorem that γ_j^? are nonzero for all j ∈ {0, . . . , m}, is not a very strong assumption. This is because if γ_j^? = 0 for some j ∈ {0, . . . , m}, then for any given > 0 we can choose a

point (ˆγ0, . . . ,γˆm) ∈ R^m+1 close to (γ₀^?, . . . , γ_m^?) such that ˆγj 6= 0 for all j ∈ {0, . . . , m}, ˆ

γ₀²+· · ·+ ˆγ²_m= 1 and

inf

(v0,...,vm)∈Kg_γ^?_0,...,γm^?(v₀, . . . , v_m)− inf

(v0,...,vm)∈Kg_{ˆγ0,...,ˆγm}(v₀, . . . , v_m) < . This is possible due to the continuous dependence of the function inf

(v0,...,vm)∈Kg_γ0,...,γm(v0, . . . , v_m) on (γ0, . . . , γm). As a result tight lower and upper bounds are always possible to compute in terms of inf

t0,...,tm

Lˆγ0,...,ˆγm(t0, . . . , tm) .

Remark 4.1.6. Although, in the above theorem we cannot claim that µ^? := sup

γ₀²+···+γ²_m=1

v0,...,vinfm∈Kgγ0,...,γm(v0, . . . , vm)

is attained for some (γ^?₀, . . . , γ_m^?) ∈ R^m+1 where γ_j^? are all nonzero, and at least one of the sufficient conditions for (4.1.18) to hold is always satisfied, numerical experiments suggest that this hold generically. We have observed in all numerical experiments that µ^? is attained for some (γ₀^?, . . . , γ_m^?) ∈ R^m+1 where γ_j^? is very different from zero for each j. Also every minimizer (ˆv₀, . . . ,ˆv_m) of g_γ0^?_,...,γ^?_m(v₀, . . . , v_m) over K satisfies kvˆ_jk =kvˆ_mk, for j ∈0, . . . , m−1. Thus equality (4.1.18) is always satisfied in numerical experiments.

On the basis of numerical experiments we have the following conjecture.

Conjecture 4.1.7. Let P(z) =Pm

j=0z^jAj be a Hermitian polynomial i.e., Aj ∈Herm(n) for j = 0, . . . , m. Let λ ∈ C be such that Imλ 6= 0 and det(P(λ)) 6= 0. There exist γ₀^?, . . . , γ_m^? ∈R\ {0} and (ˆv0, . . . ,ˆvm)∈ K with kvˆjk= kˆvkk for all j, k ∈ {0, . . . , m} such that

g_γ^?_0,...,γm^?(ˆv₀, . . . ,ˆv_m) = inf

(v0,...,vm)∈Kg_γ0^?_,...,γm^?(v₀, . . . , v_m) = sup

γ²₀+···+γm²=1

(v0,...,vinfm)∈Kg_γ0,...,γm(v₀, . . . , v_m), where gγ0,...,γm and K are as defined in (4.1.8) and (4.1.7), respectively.

Remark 4.1.8. To obtain an optimal Hermitian perturbation (provided Conjecture 4.1.7 is true) with norm equals toη_∞^Herm(P, λ), we first compute an unit eigenvectorucorresponding toLγ₀^?,...,γ_m^?(t^?₀, . . . , t^?_m) =λmax G(Γ^?) +t^?₀H0(Γ^?) +· · ·+t^?_mHm(Γ^?)

satisfying u^∗G(Γ^?)u=Lγ₀^?,...,γ_m^?(t^?₀, . . . , t^?_m) and u^∗Hi(Γ^?)u= 0 for i= 0, . . . , m.

The vectors (v0, . . . , vm) ∈ K with kvjk = kvkk, j, k ∈ {0, . . . , m} are obtained by using the relation [v₀^T, . . . , v^T_m]^T = Γ^?−1u, where Γ^? = diag{γ₀^?, . . . , γ_m^?} ⊗In. The coefficients

∆j satisfying ∆j(M(v0 +λv1+· · ·+λ^mvm)) =vj for j = 0, . . . , m, and η_∞^Herm(P, λ)2

= max{k∆0k², . . . ,k∆mk²}= Xm

j=0

γ^?_j² kv_jk² kMvλk² =

Lγ₀^?,...,γ_m^?(t^?₀, . . . , t^?_m)₋1

may be constructed from Theorem 1.2.9.

Remark 4.1.9. In the view of the relations in Section 2.3, we can calculate the structured backward error η^S_∞(P, λ) for ∗-alternating and skew-Hermitian polynomials by converting the polynomial into an equivalent Hermitian polynomial.

Remark 4.1.10. Note that zero weights in the weight vector w can be allowed to re- strict the perturbations set. Thus counterparts of Theorem 4.1.4 and Conjecture 4.1.7, for η^Herm_w,b∞ (P, λ) with a restricted perturbation set can be obtained by following the strategy used in Section 2.4.