Comparison of Condition Numbers for QEP - Condition number of Fiedler pencils

6.2 Condition number of Fiedler pencils

6.2.1 Comparison of Condition Numbers for QEP

Consider the quadratic eigenvalue problem (QEP) Q(λ)x= λ²A₂+λA₁ +A₀

x= 0, (6.11)

where A₂, A₁, and A₀ ∈Cⁿ^×ⁿ. The companion linearizations of Q(λ) is given by L(λ)v =



λ



 A2 0 0 I_n



+



 A1 A0

−I_n 0











 λx x



= 0 (6.12)

and the left eigenvector of L(λ) is defined byu=



 In

(λA2+A1)^∗



y, wherey is the left eigenvector of P(λ). Let

A0 =CB

be the full-rank decomposition, where C ∈ R^n×r, B ∈ R^r×n and r is the rank of A0. Then the QEP in (6.11) can be written as [37]

G(λ)x=

λA2+A1+C(λIn−0)⁻¹B

x= 0. (6.13)

Then we have

C(λ)˜v =



λ



 A2

−In



+



 A1 C

B 0











 x s



= 0, (6.14)

where ev =



 x λ⁻¹Bx



 is a right eigenvector of C(λ) and ue =



 y (λ⁻¹C)^∗y



 is a left eigenvector of C(λ).

The following result compares condition number of the quadratic eigenvalue problem with that of the companion form given in (6.14).

Theorem 6.2.8. Let Q(λ) be as in (6.11) and C(λ) be as in (6.14). Then

|λ|

p1 +|λ|²ku˜k2kv˜k2 ≤ cond(λ,C)

cond(λ, Q) ≤ ku˜k2kv˜k2

and

|λ| p1 +|λ|²

1 + 1

|λ|²σ_min² (B) 1 + 1

|λ|²σ²_min(C) 1/2

≤ cond(λ,C) cond(λ, Q)≤

1 + 1

|λ|²σ²_max(B) 1 + 1

|λ|²σ_max² (C) 1/2

Proof. Letxand ybe the left and right eigenvector ofQ(λ) corresponding to the eigenvalue λ withkxk= 1 and kyk= 1. Then

cond(λ,C)

cond(λ, Q) = k(1, λ)k2ku˜k2kv˜k2

|y^∗∂λG(λ)x|

|y^∗∂λQ(λ)x| k(1, λ, λ²)k²kxk²kyk²

= k(1, λ)k²ku˜k²kv˜k² k(1, λ, λ²)k2

|y^∗∂λQ(λ)x|

|y^∗∂_λG(λ)x|

= k(1, λ)k2ku˜k2kv˜k2

k(1, λ, λ²)k2 |λ|

= k(λ, λ²)k2ku˜k2kv˜k2

k(1, λ, λ²)k²

Now k(λ, λ²)k²2

k(1, λ, λ²)k²2

= |λ|²+|λ|⁴

1 +|λ|²+|λ|⁴ ≤1 and

k(λ, λ²)k²2

k(1, λ, λ²)k²2

= |λ|²+|λ|⁴

1 +|λ|²+|λ|⁴ = 1− 1

1 +|λ|²+|λ|⁴ ≥1− 1

1 +|λ|² = |λ|² 1 +|λ|².

So |λ|

p1 +|λ|² ≤ k(λ, λ²)k2

k(1, λ, λ²)k² ≤1.

Hence

|λ|

p1 +|λ|²ku˜k2kv˜k2 ≤ cond(λ,C)

cond(λ, Q) ≤ ku˜k2k˜vk2.

Since ˜u and ˜v are left and right eigenvector of C(λ) given in (6.14), we have s

1 + 1

|λ|²σ_min² (C)≤ ku˜k² ≤ s

1 + 1

|λ|²σ²_max(C), s

1 + 1

|λ|²σ_min² (B)≤ kv˜k2 ≤ s

1 + 1

|λ|²σ_max² (B).

Consequently,

p |λ|

1 +|λ|²

1 + 1

|λ|²σ_min² (B) 1 + 1

|λ|²σ_min² (C) 1/2

≤ cond(λ,C) cond(λ, Q)≤

1 + 1

|λ|²σ²_max(B) 1 + 1

|λ|²σ²_max(C) 1/2

Theorem 6.2.9. [15] Let λ be a simple, finite, and nonzero eigenvalue of the quadratic matrix polynomialQ(λ) given in (6.11)andL be the corresponding companion linearization given in (6.12). Then

kuk²

≤ cond(λ, L)

≤ 2

√ kuk².

Theorem 6.2.10. LetG(λ)be given in (6.13) andC(λ)be the corresponding companion linearization given in (6.14). Let λ be a simple, finite, and nonzero eigenvalue.Then

[(|λ|²+σ_min² (B))(|λ|²+σ_min² (C))]^1/2

[(|λ|²+σ²_max(B))(|λ|²+σ_max² (C))]^1/2 ≤ cond(λ,C)

cond(λ, G) ≤ [(|λ|²+σ_max² (B))(|λ|²+σ_max² (C))]^1/2 [|λ|⁴+σ_min² (B)σ_min² (C)]^1/2 . Proof. By Theorem 6.2.7, we have

s.ku˜k2kv˜k2≤ cond(λ,C)

cond(λ, G) ≤s1.ku˜k2kv˜k2, where

s= k(1, λ)k²

[k(1, λ)k²2+σ_max² (λ⁻¹B) +σ_max² (λ⁻¹C) +k(1, λ)k²2(σ_max² (λ⁻¹B)σ²_max(λ⁻¹C))]^1/2,

s1 = k(1, λ)k2

[k(1, λ)k²2+σ_min² (λ⁻¹B) +σ_min² (λ⁻¹C) +k(1, λ)k²2(σ²_min(λ⁻¹B)σ²_min(λ⁻¹C))]^1/2. (6.15) Now

s= 1

1 + ^σ²^max^(λ⁻¹_k^B)+σ_(1,λ)_k²^max2 ^(λ⁻¹^C)

2 +σ_max² (λ⁻¹B)σ_max² (λ⁻¹C)i1/2

≥ 1

[1 +σ_max² (λ⁻¹B) +σ_max² (λ⁻¹C) +σ_max² (λ⁻¹B)σ_max² (λ⁻¹C)]^1/2

= 1

[(1 +σ²_max(λ⁻¹B))(1 +σ_max² (λ⁻¹C))]^1/2 and

ku˜k2kv˜k2 ≥

1 +σ_min² (λ⁻¹B)

1 +σ_min² (λ⁻¹C)1/2

. Hence

cond(λ,C)

cond(λ, G) ≥ [(1 +σ_min² (λ⁻¹B))(1 +σ_min² (λ⁻¹C))]^1/2 [(1 +σ²_max(λ⁻¹B))(1 +σ_max² (λ⁻¹C))]^1/2

⇒ cond(λ,C)

cond(λ, G) ≥ [(|λ|²+σ_min² (B))(|λ|²+σ²_min(C))]^1/2 [(|λ|²+σ_max² (B))(|λ|²+σ²_max(C))]^1/2. Now, from equation (6.15) we have

h 1

1 + ^σ²^min^(λ⁻¹_k^B)+σ_(1,λ)_k²^min2 ^(λ⁻¹^C)

2 +σ_min² (λ⁻¹B).σ_min² (λ⁻¹C)i1/2.

Note that f(t) := _[(1+α)t^t₂_+β]1/2 is strictly increasing function on [0,∞) and f(t)→ ^√_1+α¹ as t→ ∞. Hence

(1 +α+β)^1/2 ≤f(t)≤ 1 (1 +α)^1/2.

This shows that

[1 +σ²_min(λ⁻¹B)σ²_min(λ⁻¹C) +σ²_min(λ⁻¹B) +σ²_min(λ⁻¹C)]^1/2 ≤s1≤ 1 q

1 +σ²_min(λ⁻¹B)σ_min² (λ⁻¹C)

Now

ku˜k2kv˜k2 ≤p

(1 +σ_max² (λ⁻¹B)) (1 +σ²_max(λ⁻¹C)).

Consequently,

cond(λ,C)

cond(λ, G) ≤ [(1 +σ_max² (λ⁻¹B)) (1 +σ_max² (λ⁻¹C))]^1/2 [1 +σ²_min(λ⁻¹B)σ_min² (λ⁻¹C)]^1/2

= [(|λ|²+σ_max² (B)) (|λ|²+σ_max² (C))]^1/2 [|λ|⁴+σ_min² (B)σ_min² (C)]^1/2 . This completes the proof.

Theorem 6.2.11. LetG(λ)be given in (6.13) andC(λ)be the corresponding companion linearization given in (6.14). Let λ be a simple, finite, and nonzero eigenvalue. Then

1≤ cond(λ,C) cond(λ, G) ≤

1 +

|λ|²kBxk²2+ _|_λ¹_|2kC^∗yk²2

1 + _|_λ¹_|2kBxk²2 1

|λ|²kC^∗yk²2

#1/2

Proof. Letxand ybe the left and right eigenvector ofG(λ) corresponding to the eigenvalue λ withkxk= 1 and kyk= 1. Then

cond(λ,C)

cond(λ, G) = k(1, λ)k2ku˜k2kv˜k2

[k(1, λ)k²2+k(λ⁻¹Bx)^∗k²2+k(λ⁻¹C)^∗yk²2+k(1, λ)k²2(k(λ⁻¹Bx)^∗k²2k(λ⁻¹C)^∗yk²2)]^1/2. Nowkeuk2 = [kxk²2+kλ⁻¹Bxk²2]^1/2 =h

1 + _|_λ¹_|2kBxk²2

i1/2

andkevk2 =h

1 + _|_λ¹_|2kC^∗yk²2

i1/2

cond(λ,C)

cond(λ, G) = k(1, λ)k2

1 + _|_λ¹_|2kBxk²2

1/2

1 + _|_λ¹_|2kC^∗yk²2

1/2

hk(1, λ)k²2+_|λ|¹2kBxk²2+_|λ|¹2kC^∗yk²2+ (1 +|λ|²)

|λ|²kBxk²2._|λ|¹2kC^∗yk²2

i1/2

1 + _|λ|¹2kBxk²2

1/2

1 + _|λ|¹2kC^∗yk²2

1/2

1 +

|λ|2kBxk²₂+ ¹

|λ|2kC^∗yk²₂ k(1,λ)k²₂ +

|λ|²kBxk²2._|λ|¹2kC^∗yk²2

^1/2 (6.16)

≥

1 + _|λ|¹2kBxk²2

1/2

1 + _|λ|¹2kC^∗yk²2

1/2

1 + ¹ kBxk²+ ¹ kC^∗yk²+

1 kBxk². ¹ kC^∗yk²i1/2

1 + _|_λ¹_|2kBxk²2

1/2

1 + _|_λ¹_|2kC^∗yk²2

1/2

1 + _|λ|¹2kBxk²2

1/2

1 + _|λ|¹2kC^∗yk²2

1/2 = 1.

So cond(λ,C)

cond(λ, G) ≥1.

Again from equation (6.16) we have cond(λ,C)

cond(λ, G) ≤

1 + _|_λ¹_|2kBxk²2

1/2

1 + _|_λ¹_|2kC^∗yk²2

1/2

h1 + _|λ|¹2kBxk²2._|λ|¹2kC^∗yk²2

i1/2

= h

1 + _|_λ¹_|2kBxk²2+_|_λ¹_|2kC^∗yk²2+_|_λ¹_|2kBxk²2 1

|λ|²kC^∗yk²2

i1/2

h1 + _|_λ¹_|2kBxk²2 1

|λ|²kC^∗yk²2

i1/2

1 +

|λ|²kBxk²2+_|_λ¹_|2kC^∗yk²2

1 + _|_λ¹_|2kBxk²2 1

|λ|²kC^∗yk²2

#1/2

The following result compares the condition number of Q(λ) given in (6.11) with that of G(λ) given in (6.13).

Theorem 6.2.12. Let Q(λ) be given in (6.11) and G(λ) given in (6.13). Let λ be a simple, finite, and nonzero eigenvalue. Then

pσ_min² (C) +σ_min² (B)

k(1, λ, λ²)k2 ≤ cond(λ, G) cond(λ, Q) ≤

1 +σ_max² (λ⁻¹B)

1 +σ_max² (λ⁻¹C)1/2

. Proof. We have ^cond(λ,G)_cond(λ,Q)

= [k(1, λ)k²2+k(λ⁻¹Bx)^∗k²+k(λ⁻¹C)^∗yk²+ (1 +|λ|²)(k(λ⁻¹Bx)^∗k²k(λ⁻¹C)^∗yk²)]^1/2 k(1, λ, λ²)k2kxk2kyk2

.|y^∗∂λQ(λ)x|

|y^∗∂λG(λ)x| (6.17)

= [k(1, λ)k²2+kλ⁻¹Bxk²2+k(λ⁻¹C)^∗yk²2+ (1 +|λ|²)(kλ⁻¹Bxk²2k(λ⁻¹C)^∗yk²2)]^1/2

k(1, λ, λ²)k2 |λ|.

Hence

s≤ cond(λ, G) cond(λ, Q) ≤s1,

where

s= |λ|[k(1, λ)k²2+σ_min² (λ⁻¹B) +σ²_min(λ⁻¹C) + (1 +|λ|²)(σ_min² (λ⁻¹B)σ_min² (λ⁻¹C))]^1/2

k(1, λ, λ²)k² ,

s1 = |λ|[k(1, λ)k²2+σ_max² (λ⁻¹B) +σ_max² (λ⁻¹C) + (1 +|λ|²)(σ_max² (λ⁻¹B)σ_max² (λ⁻¹C))]^1/2 k(1, λ, λ²)k2

= |λ|k(1, λ)k2

1 + ^σ^max² ^(λ⁻¹_k(1,λ)k^B)+σ²^max2 ^(λ⁻¹^C)

2 + (σ_max² (λ⁻¹B)σ_max² (λ⁻¹C))i1/2

k(1, λ, λ²)k2

≤ |λ|k(1, λ)k2

(1, λ, λ²)k²

1 +σ_max² (λ⁻¹B) +σ²_max(λ⁻¹C) + (σ_max² (λ⁻¹B)σ²_max(λ⁻¹C))1/2

= |λ|k(1, λ)k2

(1, λ, λ²)k2

(1 +σ_max² (λ⁻¹B))(1 +σ_max² (λ⁻¹C))1/2

. Now

|λ|k(1, λ)k2

(1, λ, λ²)k² = |λ|p

1 +|λ|² p1 +|λ|²+|λ|⁴ =

|λ|²+|λ|⁴ 1 +|λ|²+|λ|⁴

1/2

≤1

So cond(λ, G)

cond(λ, Q) ≤

(1 +σ_max² (λ⁻¹B))(1 +σ_max² (λ⁻¹C))1/2

Now

s= k(λ, λ²)k2

k(1, λ, λ²)k2

1 + σ_min² (λ⁻¹B) +σ²_min(λ⁻¹C) k(1, λ)k²2

+σ²_min(λ⁻¹B).σ²_min(λ⁻¹C) 1/2

. Thusp

σ²_min(C) +σ_min² (B)

k(1, λ, λ²)k2 ≤ cond(λ, G) cond(λ, Q) ≤

1 +σ²_max(λ⁻¹B)

1 +σ_max² (λ⁻¹C)1/2

Example 6.5 (Loaded elastic string, [37]). Consider the rational eigenvalue problem

A1−λA2 + λ λ−σE

x= 0, (6.18)

where A₁ and A₂ are tridiagonal and symmetric positive definite, andE =e_ne^T_n is the last column of the identity matrix and σ >0 is a parameter. Then we have

G(λ)x=

A₁+e_ne^T_n −λA₂ +e_n λ

σ −1 −1

e^T_n

x = 0, (6.19)

and C1(λ)v = (λX − Y)v =



λ



 A2



−



 −(A1+ene^T_n) −en











 x



= 0. (6.20)

The quadratic eigenvalue problem of (6.18) is given by

Q(λ)x= [λ²A2−λ(A1+σA2+ene^T_n) +σA1]x= 0, σ > 0, (6.21) Here

σmin

λ δ −1

−1

e^T_n

=σmax

λ δ −1

−1

e^T_n

= 1

^λ_δ −1 =σmin en

λ δ −1

−1!

=σmax en

λ δ −1

₋1!

By Theorem 6.2.7, we have s=s1. So cond(λ, C1)

cond(λ, G) =s.kuk2

kyk² kvk2

kxk², where

s= k(1, λ)k^q

[k(1, λ)k^q^q+|^λ_δ −1|⁻^q+|^λ_δ −1|⁻^q+ (1 +|λ|^q) |^λ_δ −1|⁻^2q ]^1/q

= (1 +|λ|^q)^1/q

[1 +|λ|^q+_|λ−δ|^δ^q q + _|λ−δ|^δ^q q + (1 +|λ|^q)

δ^2q

|λ−δ|^2q

]^1/q

= (1 +|λ|^q)^1/q

[(1 +|λ|^q) + 2|_λ₋^δ_δ|^q+ (1 +|λ|^q) |_λδ^δ|^2q|

]^1/q ≤1

and 1 ≤ ^kuk_k_y_k²₂_k^kvk_x_k²₂ ≤ σmax







 I_n (en(^λ_δ −1)⁻¹)^∗







σmax







 I_n (^λ_δ −1)⁻¹e^T_n



v



 ≤ 1 +

|λ/δ−1|⁻². This shows that

cond(λ, C1)

cond(λ, G) ≤1 +|λ/δ−1|⁻².

Next, consideringG(λ) andQ(λ) given in (6.19) and (6.21), respectively, by Theorem 6.2.12, we have

cond(λ, G) cond(λ, Q) ≤

1 + σ²

|λ−σ|²

7

Conclusions

In this thesis, we have developed a framework for solutions of rational eigenvalue problems. We have achieved this goal by reformulating a rational eigenvalue problem G(λ)u = 0 to that of computing transmission zeros and zero directions of a linear time invariant (LTI) system Σ given by

Ex(t) =˙ Ax(t) +Bu(t) y(t) =Cx(t) +P(_dt^d)u(t)

for which G(λ) is the transfer function. Since eigenvalues of G(λ) form a subset of the transmission zeros of the LTI system Σ and the transmission zeros form a subset of the invariant zeros of the system Σ, we have focused on computing invariant zeros and zero directions of the LTI system which are in fact eigenvalues and eigenvectors of the Rosenbrock system matrix S(λ) given by

S(λ) =



 P(λ) C B (A−λE)



.

Hence we have developed a framework for computing eigenvalues and eigenvectors of the Rosenbrock system matrix S(λ).For this purpose, we have introduced three families of linearizations - which we referred to as Fiedler pencils, Generalized Fiedler (GF) pencils and Generalized Fiedler pencils with repetition (GFPR) - of the Rosenbrock system matrix S(λ). Thus invariant zeros of the LTI system Σ could be computed by solving generalized eigenvalue problems for the Fiedler pencils.

We have described construction of Fiedler pencils of the Rosenbrock system matrix S(λ) and have shown that these Fiedler pencils are linearizations for S(λ).Further, we have shown that the Fiedler pencils of S(λ) are linearizations of G(λ) when the LTI

realization of the transfer function G(λ).Furthermore, we have shown that GF pencils and GFPR could be utilized to obtain structure (such as symmetric and Hermitian) preserving linearizations of the Rosenbrock system matrix.

We have also described eigenvector recovery of G(λ) from that of Fiedler pencils of the system matrix S(λ). In fact, we have shown that the eigenvectors of G(λ) could be recovered from those of the Fiedler pencils of S(λ) without incurring any additional computational cost. Further, we have shown that the state-space framework so developed could be utilized to linearizea higher order LTI state-space system so as to analyze and solve the higher order LTI system. Furthermore, we have shown that the linearized systems so obtained are strict system equivalent to the higher order systems and hence preserve system characteristics of the original systems.

We have also developed a framework for sensitivity analysis of eigenvalues of the system matrixS(λ) and the transfer functionG(λ).We have defined condition numbers for simple eigenvalues ofS(λ) andG(λ),and have obtained explicit computable expres- sions for the condition numbers. We have also analyzed the effect of linearization on the conditioning of the eigenvalues of the system matrix S(λ).

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Dalam dokumen Fiedler line arizations for LTI state-space systems and for rational eigenvalue problems (Halaman 178-190)