2.5 Numerical experiments
3.1.1 Reformulation of the problem
Here, we further reformulate the already reformulated problem in Lemma 3.1.2 of finding the eigenvalue backward errors for •-palindromic polynomials into an equivalent problem of maximizing the Rayleigh quotient of a Hermitian matrix with respect to specified con- straints. These constraints involve Hermitian matrices when•=∗, and symmetric matrices when•=T. Letλ∈C\ {0},letP(z) =Pm
j=0zjAj be a•-palindromic matrix polynomial such thatM = (P(λ))−1 exists and define vλ =Pm
j=0λjvj wherev0, . . . , vm ∈Cn.Also let k =bm−21c.
Due to Theorem 3.1.3, for any v0, . . . , vm ∈ Cn that satisfy vλ 6= 0, there exists a
∆= (∆0, . . . ,∆m)∈pal• such that
∆jMvλ =vj and ∆•jMvλ =vm−j, j = 0, . . . , k (3.1.3)
if and only if (Mvλ)•vj = vm•−j(Mvλ). For any ∆j ∈ Cn×n satisfying (3.1.3) which is minimal with respect to the 2-norm, we have k∆jk= maxn
kvjk
kM vλk, kkvM vm−jk
λk
o .
If m is even, the matrix ∆m2 of the tuple ∆ = (∆0, . . . ,∆m) is Hermitian when •=∗ and symmetric when •= T. In the case • =∗, the Hermitian matrix ∆m
2 may be chosen to satisfy ∆m
2Mvλ =vm
2, if and only if (Mvλ)∗vm
2 ∈R, (by Theorem 1.2.9). On the other hand, when • = T the symmetric matrix ∆m2 may be chosen to satisfy ∆m2Mvλ = vm2 without any restrictions on Mvλ and vm2 and in either case, any minimal 2-norm solution of this mapping problem satisfiesk∆m2k= kkvM vm/2k
λk,(see [33]). Therefore, if all the constraints are fulfilled, the minimal norm of∆ is given by
|||∆|||2w,2 =f(v0, . . . , vm), where
f(v0, . . . , vm) :=
Pk j=0
2w2jmaxn
kvjk2
kM vλk2,kvm−jk2
kM vλk2
o if m is odd,
Pk j=0
2w2jmaxn
kvjk2
kM vλk2,kvm−jk2
kM vλk2
o+w2m
2
kvm 2k2
kM vλk2 if m is even.
Thus, Lemma 3.1.2 yields ηw,2pal•(P, λ)2
= infn
f(v0, . . . , vm)(v0, . . . , vm)∈ Ko
, (3.1.4)
where K ⊆(Cn)m+1 is given by K:=n
(v0, . . . , vm)vλ 6= 0, (Mvλ)•vj =v•m−jMvλ, j = 0. . . , ko
(3.1.5) if •=T,or if m is odd and •=∗ and by
K:=n
(v0, . . . , vm)vλ 6= 0, (Mvλ)∗vm2 ∈R,(Mvλ)∗vj =vm∗−jMvλ, j = 0, . . . , ko
(3.1.6) otherwise (i.e., when • = ∗ and m is even). Observe that (Mvλ)•vj = vm•−j(Mvλ) for j = 0, . . . , k,if and only if
0 =
M(v0 +· · ·+λmvm)•
vj −vm•−j
M(v0+· · ·+λmvm)
=v•Cejv, where v := [vT0, . . . , vmT]T and
Cej := (Λ•me•j+1)⊗M•−(em−j+1Λm)⊗M, (3.1.7) with Λm := [1, λ, . . . , λm]∈C1×(m+1). Similarly (Mvλ)∗vm2 ∈R if and only if
0 =−2 Im
(Mvλ)∗vm2
=i v∗m
2(Mvλ)−(Mvλ)∗vm2
=v∗Cem2v,
where
Cem2 :=i
(Λ∗me∗m
2+1)⊗M∗−(em2+1Λm)⊗M
. (3.1.8)
Note that Cem2 is a Hermitian matrix but the matrices Cej, j = 0, . . . , k are not Hermitian.
Thus, from (3.1.5) if •=T, or if m is odd and •=∗ K=n
(v0, . . . , vm)vλ 6= 0, v•Cejv = 0, j = 0, . . . , ko
, (3.1.9)
and from (3.1.6) K=n
(v0, . . . , vm)vλ 6= 0, v∗Cem
2v = 0, v∗Cejv = 0, j = 0, . . . , ko
, (3.1.10) otherwise.
As already stated in the beginning of this section, our aim is to reformulate the com- putation of the structured eigenvalue backward error as an equivalent problem of maxi- mizing the Rayleigh quotient of a Hermitian matrix subject to specified constraints. The same strategy was applied in Chapter 2 to find the structured eigenvalue backward errors ηSw,2(P, λ) for Hermitian and related structures. But the reformulation was aided by the fact that ηw,2S (P, λ) satisfied
ηSw,2(P, λ) = sup
(PmkMvλk2
j=0w2jkvjk2
vλ 6= 0, v∗jMvλ ∈R )!−12
(3.1.11) for those structures, which made it possible to compute ηw,2S (P, λ) by minimizing the Rayleigh quotient of a particular Hermitian matrix G as given in Theorem 2.2.2 subject to certain conditions involving Hermitian matrices. However, this is not the case for the structured eigenvalue backward error ηw,2pal•(P, λ) for the •-palindromic structures, because the function f in the right hand side of (3.1.4) involves taking a maximum instead of a sum of squares. The following lemma is a key step towards establishing a relationship similar to (3.1.11) forηw,2pal•(P, λ),because it shows that computingηw,2pal•(P, λ) is equivalent to minimizing a function g related to f that can be interpreted as a Rayleigh quotient of a certain Hermitian matrix.
Lemma 3.1.4. Let P(z) =Pm
j=0zjAj be •-palindromic and λ∈C\ {0}. Assume further that M = P(λ)−1
exists and k =bm2−1c. Then ηw,2pal•(P, λ)2
= infn
g(v0, . . . , vm)(v0, . . . , vm)∈ Ko ,
where
g(v0, . . . , vm) :=
Pk j=0
2w2j(kvjk2+|λ|m−2jkvm−jk2)
(1+|λ|m−2j)kM vλk2 if m is odd, Pk
j=0
2w2j(kvjk2+|λ|m−2jkvm−jk2) (1+|λ|m−2j)kM vλk2 +w
2m 2kvm
2k2
kM vλk2 if m is even, and K is as defined in (3.1.9) and (3.1.10), respectively.
Proof. Set ν := inf
g(v0, . . . , vm)(v0, . . . , vm)∈ K . It is easily verified that g(v0, . . . , vm)≤f(v0, . . . , vm) for all (v0, . . . , vm)∈(Cn)m+1 with vλ 6= 0.
This together with (3.1.4) implies ν ≤ ηwpal•(P, λ)2
. The opposite inequality is an imme- diate consequence of the following facts:
(a) The infimum of g in the definition of ν is attained for some (ˆv0, . . . ,ˆvm)∈ K. (b) For every minimizer (ˆv0, . . . ,vˆm)∈ K of g, we have
g(ˆv0, . . . ,vˆm) =f(ˆv0, . . . ,vˆm).
Proof of (a): SinceKis closed under scalar multiplication and since for allt∈R\{0}and all (v0, . . . , vm)∈ Kwe haveg(v0, . . . , vm) =g(t v0, . . . , t vm), we obtain thatg(K) = g(K ∩ S), where S is defined as
S =
(v0, . . . , vm)∈(Cn)m+1
Xk j=0
kvjk2 = 1
.
Let (v0(`), . . . , vm(`)), ` ∈N, be a sequence in K ∩ S for which
`lim→∞g v(`)0 , . . . , vm(`)
=ν.
SinceSis compact, we may assume without loss of generality that the sequence (v0(`), . . . , v(`)m) has a limit (ˆv0, . . . ,ˆvm) ∈ S. Suppose that (ˆv0, . . . ,vˆm) does not belong to K. Then we have ˆvλ :=Pm
j=0λjvˆj = 0,as (ˆv0, . . . ,ˆvm) belongs to the closure of K. This implies
`lim→∞
M
v(`)0 +λv1(`)+· · ·+λmvm(`)−1 =∞ and hence
`lim→∞g(v0(`), . . . , vm(`)) =∞ 6=ν
which is a contradiction. Thus, (ˆv0, . . . ,vˆm)∈ K and g(ˆv0, . . . ,vˆm) =ν.
Proof of (b): Let (ˆv0, . . . ,vˆm) ∈ K be such that g(ˆv0, . . . ,vˆm) = ν. Observe that, to show that g(ˆv0, . . . ,vˆm) =f(ˆv0, . . . ,ˆvm), it is sufficient to show that kvˆjk=kvˆm−jk for all j = 0, . . . , k. Let
x0 =
M(ˆvλ)/kM(ˆvλ)k if •=∗, M(ˆvλ)/kM(ˆvλ)k if •=T
and for each j such that 0 ≤ j ≤ k, let yj, ym−j be the projections of ˆvj and ˆvm−j, respectively, onto the orthogonal complement of x0,for 0≤j ≤k. Then
ˆ
vj =yj+cjx0 and ˆvm−j =ym−j +cm−jx0
for somecj, cm−j ∈C.Since (ˆv0, . . . ,vˆm)∈ K we have ¯cj =cm−j when •=∗ andcj =cm−j
when •=T. Hence
kvˆjk2 =kyjk2+|cj|2 and kˆvm−jk2 =kym−jk2+|cj|2. (3.1.12) Let y= ¯λm−2jyj +|λ|m−2jym−j. Observe that
(ˆv0, . . . ,ˆvj +t λm−2jy, . . . ,ˆvm−j −t y, . . . ,vˆm)∈ K for all t∈ R. Thus as (ˆv0, . . . ,vˆm) is a minimizer of g over K, we have
0 = d
dtg vˆ0, . . . ,vˆj +t λm−2jy, . . . ,ˆvm−j −t y, . . . ,ˆvm
t=0
= d
dt
2wj2(kvˆj+t λm−2jyk2+|λ|m−2jkvˆm−j −t yk2) (1 +|λ|m−2j)kMˆvλk2
t=0
= 2w2jRe ˆvj∗(λm−2jy)− |λ|m−2jˆvm∗−jy (1 +|λ|m−2j)kMvˆλk2
since, d
dtkv+tyk2
t=0
= 2Re(v∗y)
= 2w2jRe yj∗(λm−2jy)− |λ|m−2jy∗m−jy (1 +|λ|m−2j)kMvˆλk2
= 2w2j|λ|m−2jRe |λ|m−2jkyjk2+λm−2jyj∗ym−j −¯λm−2jy∗m−jyj− |λ|m−2jkym−jk2 (1 +|λ|m−2j)kMˆvλk2
= 2w2j|λ|2(m−2j) kyjk2− kym−jk2 (1 +|λ|m−2j)kMˆvλk2 ,
which implies kyjk = kym−jk. This together with (3.1.12) yields kˆvjk = kvˆm−jk. Hence kvˆjk = kvˆm−jk for all j, and the latter implies g(ˆv0, . . . ,vˆm) = f(ˆv0, . . . ,ˆvm). This com- pletes the proof.
Recalling that k = bm2−1c, define γj1 = wj
q 2
1+|λ|m−2j, γj2 = wj
q 2|λ|m−2j 1+|λ|m−2j for j = 0, . . . , k,and
Γ :=
(diag(γ01, . . . , γk1, γk2, . . . , γ02)⊗In, if m is odd,
diag(γ01, . . . , γk1, wm2, γk2, . . . , γ02)⊗In if m is even. (3.1.13) Also recall that Λm = [1, λ, . . . , λm]∈C1×(m+1). Then we have
g(v0, . . . , vm) = v∗Γ2v
v∗Gve , where Ge := (Λ∗mΛm)⊗(M∗M), v = [v0T , . . . , vTm]T (3.1.14) and where v∗Gve =kMvλk2 6= 0, or, equivalently, vλ 6= 0. It follows that
ηw,2pal•(P, λ) = infn
f(v0, . . . , vm)(v0, . . . , vm)∈ Ko1/2
=
infn
g(v0, . . . , vm)(v0, . . . , vm)∈ Ko1/2
(by Lemma 3.1.4)
=
supn
g(v0, . . . , vm)−1 (v0, . . . , vm)∈ Ko−1/2
. Set u:= Γv and
G:= Γ−1GΓe −1, Cj := Γ−1CejΓ−1, Cm2 := Γ−1Cem2 Γ−1, (3.1.15) where G,e Cej and Cem2 are as defined in (3.1.14), (3.1.7) and (3.1.8) respectively.
By Lemma 3.1.4 and (3.1.14), for k=bm2−1c, we have ηw,2pal•(P, λ)−2
= supn v∗Gve v∗Γ2v
v ∈Cn(m+1)\ {0}, v•Cejv = 0, j = 0, . . . , ko
= supnu∗Gu u∗u
u∈Cn(m+1)\ {0}, u•Cju= 0, j = 0, . . . , ko , if •=T,or if m is odd and •=∗, and
ηw,2pal∗(P, λ)−2
= supnu∗Gu u∗u
u∈Cn(m+1)\{0}, u∗Cm2u= 0, u∗Cju= 0, j = 0, . . . , ko otherwise. Note that the condition vλ 6= 0 from the definition of K in (3.1.9) or (3.1.10), respectively or, equivalently, the conditions v∗Gve 6= 0 and u∗Gu 6= 0 can be dropped in the two expressions for ηw,2pal∗(P, λ)−2
, because Ge and G are semidefinite. This implies
u∗Gu
u∗u ≥ 0 and hence the supremum of this Rayleigh quotient over all nonzero vectors u satisfying some constraints will be the same with or without the additional condition u∗Gu6= 0.
In order to state the main result of this section, for each j = 0, . . . , k we define
Hj :=Cj+Cj∗, Hm−j := i(Cj −Cj∗), Hm2 :=Cm2, (3.1.16)
Sj := Cj +CjT, (3.1.17)
where Cj, forj = 0, . . . , k, and Cm2 , are as in (3.1.15).
Observe that for j = 0, . . . , k,
v∗Cejv = 0 ⇐⇒ u∗Hju= 0 andu∗Hm−ju= 0, vTCejv = 0 ⇐⇒ uTSju= 0,
v∗Cem2v = 0 ⇐⇒ u∗Hm2 u= 0.
Therefore we have proved the following theorem which gives the desired reformulation.
Theorem 3.1.5. Let P(z) = Pm
j=0zjAj be •-palindromic and λ∈ C\ {0}. Suppose that P(λ)is nonsingular and M = (P(λ))−1. Furthermore, letk :=bm−21c, G be as in (3.1.15), Hj, forj = 0, . . . , m,be defined by (3.1.16)and Sj, forj = 0, . . . , k,be defined by (3.1.17).
Then
ηw,2palT(P, λ) =
supnu∗Gu u∗u
u∈Cn(m+1)\{0}, uTSju= 0, j = 0, . . . , ko−12
(3.1.18) and
ηpalw,2∗(P, λ) =
supnu∗Gu u∗u
u∈ Cn(m+1)\{0}, u∗Hju= 0, j = 0, . . . , mo−12
. (3.1.19)