6.4 Structured condition numbers for linearizations
6.4.3 T -palindromic matrix polynomials
Again using (6.24) and noting that
1≤ s
1 +|λ|2(1− |xTy|2|ΛHm−1ΣΛm−1|2 kΛm−1k42 )≤√
2 when|λ| ≤1,and
q
2− |xTy|2≤ s
1 +|λ|2(1− |xTy|2|ΛHm−1ΣΛm−1|2
kΛm−1k42 )≤ k(1, λ)k2
when|λ|>1,the desired results follow. The proof is similar forT-odd linearization of P and follows from Lemma 6.2.10¥
We mention that the results in Theorem 6.4.7 hold when P isT-odd matrix polynomial.
linearization L∈L1(P)with ansatz vector v such that(R⊗I)L∈DL(P)satisfies κL(λ) =
p1 +|λ|2
|p(λ;Rv)| · kΛm−1k22
|yHP0(λ)x| = k(1, λ)k2kΛm−1k22
|p(λ;Rv)| kΛmk2
κP(λ),
provided that the perturbations 4L = 4X +λ4Y are measured in the norm k4Lk = pk4Xk2M +k4Yk2M for a unitarily invariant norm k · kM. Thus, we have
kΛm−1k2
|p(λ;Rv)| ≤ κL(λ) κP(λ) ≤√
2kΛm−1k2
|p(λ;Rv)|. Proof: The proof follows from Lemma 6.4.3, (6.35) and (6.24). ¥
Note thatδRv= kΛm−1k2
|p(λ;Rv)|serves as a growth factor for the unstructured condition number for a structured linearization L(λ)∈L1(P).Obviously δRv ≥1. Note thatp(λ;Rv)6= 0 and henceδRv<∞when L is structured linearization of P corresponding tov.For a finite simple eigenvalue of aT-palindromic matrix polynomial, we have the following.
Theorem 6.4.9. Let Sp andSa, respectively, denote the sets of T-palindromic and T-anti- palindromic polynomials. Let λ be a finite simple eigenvalue of a T-palindromic matrix polynomial P. Let Lp (resp. La) be T-palindromic (resp. T-anti-palindromic) linearization from L1(P) of P corresponding to the ansatz vectors v = Rv (resp. v = −Rv). Then for k · kM ≡ k · kF,we have the following.
1. Ifre(λ)>0 : δRv
√2 ≤ κSLp
p(λ) κP(λ) ≤√
2δRv and |1−λ| kΛm−1k2
√2kΛmk2
δRv≤κSLaa(λ) κP(λ) ≤δRv. 2. Ifre≤0 : |1 +λ| kΛm−1k2
√2kΛmk2
δRv≤κSLpp(λ)
κP(λ) ≤δRv and δRv
√2 ≤κSLa
a(λ) κP(λ) ≤√
2δRv. Proof: First, consider Lp.By Lemma 6.2.11 we have
κSLpp(λ) =
kΛm−1k22 r
1 +|λ|2+ 2re(λ)|yTx|2|ΛHm−1kΛRΛm−1|2
m−1k42
√2|p(λ;Rv)| kΛmk2
κP(λ).
Ifre(λ)>0 then
k(1, λ)k2≤ s
1 +|λ|2+ 2re(λ)|yTx|2|ΛHm−1RΛm−1|2 kΛm−1k42 ≤√
2k(1, λ)k2.
Hence the result for Lp follows from Lemma 6.4.8 and (6.24). Similarly we obtain the result forre(λ)≤0.
Next consider La.Then by Lemma 6.2.13
κSLaa(λ) =
kΛm−1k22 r
1 +|λ|2−2re(λ)|yTx|2|ΛHm−1kΛRΛm−1|2
m−1k42
√2|p(λ;Rv)| kΛmk2
κP(λ).
Ifre(λ)≤0 then
k(1, λ)k2≤ s
1 +|λ|2−2re(λ)|yTx|2kΛHm−1RΛm−1|2 kΛm−1k42 ≤√
2k(1, λ)k2.
Hence the result for La follows from Lemma 6.4.8 and (6.24). Similarly we obtain the result forre(λ)>0.¥
In view of Table 6.5, the ansatz vectorvfor aT-palindromic linearization of aT-palindromic polynomial should satisfyRv=v with the flip permutationR defined in (6.34). By Proposi- tion 6.4.4,
maxv=Rv kvk2≤1
|p(λ;Rv)|=kΠ+(Λm−1)k2, where the maximum is attained by v+ defined via
v±=
£λm−1±1
2 , . . . , λm/2+12±λm/2, λm/2+12±λm/2, . . . , λm−12±1¤T kΠ±(Λm−1)k2
(6.36) ifmis even and as
v±=
£λm−1±1
2 , . . . , λ(m−1)/2±λ2 (m−1)/2, . . . , λm−12±1¤T
kΠ±(Λm−1)k2 (6.37)
ifmis odd. Similarly,
v=−Rvmax
kvk2≤1
|p(λ;Rv)|=kΠ−(Λm−1)k2, with the maximum attained by v−.
Theorem 6.4.10. Let Sp andSa, respectively, denote the sets ofT-palindromic andT-anti- palindromic polynomials. Let λ be a finite or infinite, simple eigenvalue of a T-palindromic matrix polynomial P. Let Lp (resp. La) be T-palindromic (resp. T-anti-palindromic) lin- earization from L1(P) of P corresponding to the ansatz vectors v = Rv (resp. v = −Rv).
Then the following statements hold for k · kM ≡ k · kF. 1. Ifmis odd: κSLp
p(λ) κSPp(λ) ≤p
2(m+ 1)δRv.
2. Ifmis even and re(λ)≥0: δRv
√2 ≤κSLpp(λ) κSPp(λ) ≤p
2(m+ 1)δRv.
3. Ifmis even and re(λ)≤0: δRv
√2 ≤κSLaa(λ) κSPp(λ) ≤p
2(m+ 1)δRv.
Proof: SinceκSL(λ)/κP(λ)≤κSL(λ)/κSP(λ) holds for any structureS,the desired lower bounds follow from Theorem 6.4.9.
1. If m is odd and re(λ) ≥ 0, Lemma 6.2.11 implies – together with Lemma 6.6.1.(1)
and (6.28) – the inequality
κSLpp(λ) κSPp(λ) =
r
1− |yTx|2|1+λ|2(1+|λ|2−|1−λ|2) 2
|ΛHm−1RΛm−1|2 kΛm−1k42
q
1− |yTx|2kΠ+(Λm)kkΛ22−kΠ−(Λm)k22
mk22
· κLp(λ) κP(λ)
≤ 1
q
1−kΠ+(Λm)kkΛ22−kΠ−(Λm)k22
mk22
·κLp(λ) κP(λ)
= kΛmk2
√2kΠ+(Λm)k2
· κLp(λ) κP(λ) ≤√
m+ 1κLp(λ) κP(λ). Hence the desired result follows from Lemma 6.3.2.
2. The proof of the second part follows along the lines of the first part. For even m, Lemma 6.6.1(3) implies
κSLp
p(λ)
κSPp(λ) ≤ kΛmk2
√2kΠ+(Λm)k2
·κLp(λ) κP(λ) ≤√
m+ 1κLp(λ) κP(λ). Hence the result follows from Lemma 6.3.2.
3. The proof of the third part also follows along the lines of the first part. Lemmas 6.2.11, 6.2.13 and 6.6.1(3) reveal – for evenmand aT-anti-palindromic linearization – the inequality
κSLaa(λ) κSPp(λ) =
r
1− |yTx|2|1−λ|2(1+|λ|2−|1+λ|2) 2
|ΛHm−1RΛm−1|2 kΛm−1k42
q
1− |yTx|kΠ+(Λm)kkΛ22−kΠ−(Λm)k22
mk22
· κLa(λ) κP(λ) ≤√
m+ 1κLa(λ) κP(λ). The desired result follows from Lemma 6.3.2. ¥
Corollary 6.4.11. Let Sp andSa denote the sets of T-palindromic andT-anti-palindromic polynomials, respectively. Let λ be a finite or infinite, simple eigenvalue of a T-palindromic matrix polynomial P. Consider the T-palindromic (resp. T-anti-palindromic) linearizations Lp,La ∈ L1(P) belonging to the ansatz vectors v+ and v− defined in (6.36)–(6.37), respec- tively. Then the following statements hold for k · kM ≡ k · kF.
1. Ifmis odd: κSLpp(λ)≤2(m+ 1)κSPp(λ).
2. Ifmis even and re(λ)≥0: κSLpp(λ)≤2(m+ 1)κSPp(λ).
3. Ifmis even and re(λ)≤0: κSLaa(λ)≤2(m+ 1)κSPp(λ).
Proof: The desired results follow from Theorem 6.4.10 and Lemma 6.6.1. ¥
Theorem 6.4.10 and Corollary 6.4.11 admit a simple interpretation. If eithermis odd or mis even andλhas nonnegative real part, it is OK to use aT-palindromic linearization; there will be no significant (a small constant multiple ofδRv) increase of the structured condition number. In the exceptional case, when m is even and λ has negative real part, a T-anti- palindromic linearization should be preferred. This is especially true for λ= −1, in which case there is noT-palindromic linearization.
The upper bounds in Corollary 6.4.10 are probably too pessimistic; at least they do not fully reflect the optimality of the choice of v+ and v−. In comparison, the heuristic choices listed in Table 6.6 yield almost the same bounds! These bounds are proven in the following lemma. To provide recipes for evenm larger than 2, one would need to discriminate further between|λ|close to 1 and|λ|far away from 1, similar as for oddm.
m λof v Bound on struct. cond. Example interest of linearization
odd |λ| ≥ αm
|λ| ≤α−1m 2 66 66 4
1 0... 0 1
3 77 77 5 κSLp
p(λ)≤2√
2(m+ 1)κSPp(λ) 2
4 A0 0 A0
A1−AT0 A0−AT1 0 AT1 A1−AT0 A0
3
5 +
λ 2
4 AT0 AT1 −A0 A1
0 AT0 −A1 AT1 −A0
AT0 0 AT0 3 5
odd |λ| ≤ αm
|λ| ≥α−1m
em−1 2
κSLp
p(λ)≤2(m+ 1)κSPp(λ) 2
4 0 A0 0 0 A1 A0
−AT0 0 0 3 5+λ
2
4 0 0 −A0
AT0 AT1 0 0 AT0 0
3 5
m= 2 re(λ)≥0
» 1 1
–
κSpLp(λ)≤2√ 3κSpP(λ)
» A0 A0
A1−AT0 A0
– +λ
» AT0 AT1 −A0
AT0 AT0 –
m= 2 re(λ)≤0
» 1
−1 –
κSLaa(λ)≤2√ 3κSpP(λ)
» −A0 A0
−A1−AT0 −A0
– +λ
» AT0 AT1+A0
−AT0 AT0 –
Table 6.6: Recipes for choosing the ansatz vectorvfor aT-palindromic orT-anti-palindromic linearization Lp or La of a T-palindromic matrix polynomial of degree m. Note thatαm = 21/(m−1).
Lemma 6.4.12. The upper bounds onκSLp
p(λ)andκSLa
a(λ)listed in Table 6.6 are valid.
Proof: It suffices to derive an upper bound on kΛp(λ;v)m−1k2. Multiplying such a bound by p2(m+ 1) then gives the coefficient in the upper bound on the structured condition number of the linearization, see the proof of Theorem 6.4.10.
1. For oddmand |λ| ≥αm or|λ| ≤1/αm, the bound κSLpp(λ)≤√
2(m+ 1)κSPp(λ) follows from kΛm−1k22
|p(λ;v)|2 ≤1 +α2m+· · ·+α2m−2m
|1−αm−1m |2 = 1 +α2m+· · ·+α2m−2m ≤4m.
2. For oddmand 1/αm≤ |λ| ≤αm, the boundκSLpp(λ)≤2(m+ 1)κSPp(λ) follows from kΛm−1k22
|p(λ;v)|2 ≤1 +α2m+· · ·+α2m−2m αm−1m
=1
2(1 +α2m+· · ·+α2m−2m )≤2m.
3. Form= 2 andre(λ)≥0, the bound κSLp
p(λ)≤2(m+ 1)κSPp(λ) follows for|λ| ≤1 from kΛm−1k22
|p(λ;v)|2 =1 +|λ|2
|1 +λ|2 ≤2 and for|λ| ≥1 from
kΛm−1k22
|p(λ;v)|2 =|λ|2
|λ|2
1
|λ|2 + 1
|1λ+ 1|2 ≤2.
4. The proof form= 2 andre(λ)≤0 is analogous to Part 3.¥
For T-anti-palindromic polynomials, the results of Theorems 6.4.10 and 6.4.9, Corol- lary 6.4.11 and Table 6.6 hold, but with the roles of T-palindromic and T-anti-palindromic exchanged. For example, if either m is odd or m is even and re(λ) ≥ 0, there is always a goodT-anti-palindromic linearization. Otherwise, ifmis even andre(λ)≤0, there is a good T-palindromic linearization.