Journal of Computational and Applied Mathematics

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Journal of Computational and Applied Mathematics

journal homepage:www.elsevier.com/locate/cam

A structure-preserving doubling algorithm for quadratic eigenvalue problems arising from time-delay systems

Tie-xiang Li

^a

, Eric King-wah Chu

^b,∗

, Wen-Wei Lin

^c

aDepartment of Mathematics, Southeast University, Nanjing, 211189, People’s Republic of China

bSchool of Mathematical Sciences, Building 28, Monash University, VIC 3800, Australia

cDepartment of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan

a r t i c l e i n f o

Article history:

Received 10 September 2008 Keywords:

Doubling

Quadratic eigenvalue problem Palindromic eigenvalue problem Structure-preserving

Time-delay system Unimodular eigenvalue

a b s t r a c t

We propose a structure-preserving doubling algorithm for a quadratic eigenvalue problem arising from the stability analysis of time-delay systems. We are particularly interested in the eigenvalues on the unit circle, which are difficult to estimate. The convergence and backward error of the algorithm are analyzed and three numerical examples are presented.

Our experience shows that our algorithm is efficient in comparison to the few existing approaches for small to medium size problems.

©2009 Elsevier B.V. All rights reserved.

1. Introduction

In this paper, we consider the numerical solution of the quadratic eigenvalue problem (QEP)

Q

(λ)

^x

≡ (λ

^2B

+ λ

^C

+

A

)

^x

=

0

,

^(1.1a)

whereA

,

^B

,

^C

∈

_C^n×nsatisfy

PBP

= ε

^A

,

^PAP

= ε

^B

,

^{PC P}

= ε

^{C (1.1b)}

with

ε = ±

1,P

∈

_C^n×nbeing an idempotent matrix (i.e.,P²

=

I_n), andBdenoting the complex conjugate ofB. In our application for the time-delay system(1.3), the matrixP is a real involuntary matrix (more details are given later). The scalar

λ ∈

_Cand the vectorx

∈

_Cⁿ

\ {

0

}

are the eigenvalue and the associated eigenvector of the quadratic pencilQ

(λ)

^{, and} the pair

(λ,

^x

)

is called aneigenpairofQ

(λ)

^.

We shall propose a structure-preserving doubling algorithm (SDA; [1–5]) for the solution of (1.1). We define the conjugateand thereverseof a quadratic pencil, respectively, by

Q

(λ) ≡ λ

^2B

+ λ

^C

+

A

,

rev

(

^Q

(λ)) ≡ λ

^2A

+ λ

^C

+

B

.

It is easy to see that QEP in(1.1)satisfies

Q

(λ) = ε

^Prev

(

^Q

(λ))

^P

.

^(1.2)

From(1.2), we see that

(λ,

^x

)

is an eigenpair ofQ

(λ)

if and only if

(

¹

/λ,

^Px

)

is also an eigenpair ofQ

(λ)

^[6].

∗Corresponding author. Tel.: +61 412 596430; fax: +61 3 99054403.

E-mail addresses:[email protected](T.-x. Li),[email protected](E.K.-w. Chu),[email protected](W.-W. Lin).

0377-0427/$ – see front matter©2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.cam.2009.09.010

As in [7], a quadratic pencilQ

(λ)

^in(1.1a)is said to be palindromic ifQ

(λ) =

rev

(

^Q

(λ))

. Similar to the terminology of

(?, ε)

-palindromic QEPs (

? =

H or

>

) in [7], the QEP(1.1)is referred to as

(ε = +

1

)

P-Conjugate-P Palindromic QEP

(

^PCP_PQEP

),

^or

(ε = −

1

)

anti-P-Conjugate-P Palindromic QEP

( −

PCP_PQEP

).

PCP_PQEPs as in(1.1)were first proposed in [8] from the stability analysis ofretardedtime-delay systems (TDS), and generalized in [6] the more generalneutralTDS (see [9–16] and the references therein). Consider a neutral linear time-delay system withmconstant delaysh₀

=

0

<

^h1

< · · · <

^hm:

m

X

k=0

D_kx

˙ (

^t

−

h_k

) =

m

X

k=0

A_kx

(

^t

−

h_k

) (

^t

>

⁰

) ;

x

(

^t

) = ϕ(

^t

) (

^t

∈ [−

h_m

,

⁰

] )

^(1.3) withx

: [−

h_m

, ∞ ) →

_R^N,

ϕ ∈

_C¹

( [−

h_m

,

⁰

] )

^{, andA}k

,

^Dk

∈

_R^N×N(k

=

1

, . . . ,

^{m). WhenD}0

=

I_NandD_k

=

0

(

^k

>

⁰

)

^{, we} have the retarded time-delay systems.

The stability of the TDS(1.3)can be determined by its characteristic equation s

m

X

k=0

D_ke^−hks

−

m

X

k=0

A_ke^−hks

!

v =

0

(v 6=

0

)

^(1.4)

with eigenpairs

(

^s

, v)

from the nonlinear eigenvalue problem.

A TDS is said to becriticalif and only if some eigenvaluessis purely imaginary. The set of all points

(

^h1

, . . . ,

^hm

)

^{in the} delay-parameter space for which the TDS(1.3)is critical are called critical curves (m

=

2) or surfaces (m

>

^{2). Under} certain continuity assumptions, the boundary of the stability domain of a TDS is a subset of the critical curves/surfaces.

Consequently, purely imaginary eigenvalues of(1.4)are of great interest. See [8] and the references therein for approaches to compute critical surfaces. In [6], the following parameterization of critical surfaces gives rise to an associated PCP_PQEP.

Detailed discussion on palindromic linearizations, a Schur-like canonical form and other useful results can also be found in [6].

For a given eigenpair

(

^s

, v)

^of(1.4)with

k v k

₂

=

1, a point

(

^h1

, . . . ,

^hm

)

in the delay-parameter space is critical if and only if there exist

ϕ

k

∈ [− π, π ]

(k

=

1

, . . . ,

^m

−

1) and

ω ∈

_Rsuch that

h_k

= ϕ

k

+

2p_k

π

ω ,

^pk

∈

_Z

(

^k

=

1

, . . . ,

^m

−

1

) ;

h_m

= −

Argz

+

2p_m

π

ω ,

^pm

∈

_Z

;

(1.5)

giving rise to the QEP

(

^z2E

+

zF

+

G

)

^u

=

0

,

^(1.6)

where the unimodular eigenvaluez

=

e^−iωhm, and the corresponding eigenvectoru

=

vec

vv

^∗

= v ⊗ ¯ v

^,

ω =

−

i

w

^∗

w

w

^{∗ A}mz

+

m−1

X

k=0

A_ke^−iϕk

!

v, w =

D_mz

+

m−1

X

k=0

D_ke^−iϕk

!

v,

^(1.7)

and E

=

m−1

X

k=0

D_ke^iϕk

!

⊗

A_m

+

m−1

X

k=0

A_ke^iϕk

!

⊗

D_m

∈

_C^N2×N2

,

F

=

m−1

X

k=0

D_ke^iϕk

!

⊗

m−1

X

k=0

A_ke^−iϕk

!

+

m−1

X

k=0

A_ke^iϕk

!

⊗

m−1

X

k=0

D_ke^−iϕk

!

+

D_m

⊗

A_m

+

A_m

⊗

D_m

∈

_C^N2×N2

,

G

=

D_m

⊗

m−1

X

k=0

A_ke^−iϕk

!

+

A_m

⊗

m−1

X

k=0

D_ke^−iϕk

!

∈

_C^N2×N2

.

Here

⊗

denotes the usual Kronecker product. AsM₁

⊗

M₂andM₂

⊗

M₁both contain products of elements ofM₁andM₂at different positions, we can find an involuntary matrixP

∈

_R^N2×N2 (P⁻¹

=

P^>

=

P) such thatM₁

⊗

M₂

=

P

(

^M2

⊗

M₁

)

^P [17, Corollary 4.3.10]. Here,P

=

PN

i,j=1E_ij

⊗

E_ij^>

= [

E_ij^>

]

^N_i,j=1,E_ij

=

e_i

⊗

e^>_j

∈

_R^N×Nande_jis thejth column ofI_N. Consequently from the structures inE,FandG, we can easily show that(1.6)is a PCP_PQEP becauseE

=

PGPandF

=

PF P.

Remark 1.1. In [18], it has been proved that unimodular eigenvalues occur quite often for PCP_PQEPs (and other palindromic eigenvalue problems with eigenvalue pairs

(λ, ε/λ)

). These eigenvalues can stay unimodular under perturbation, thus are numerically stable to compute. Furthermore, the probability of having too many of them is low and multiple unimodular eigenvalues are rare. This makes our problem of computing the unimodular eigenvalueszof(1.6)well-posed, unlike for complex-Tpalindromic eigenvalue problems.

Before proceeding further, we would like to point out some recent developments in the numerical solution of palin- dromic eigenvalue problems. The QEP(1.1a)is said to be

?

-palindromic ifB^?

=

AandC^?

=

C, with

? = >

orH. The train vibration problem and the associated

>

-palindromic QEP were discussed in [19,20] and structure-preserving palindromic linearizations for(1.1a)were suggested in [7]. An SDA algorithm [3] and a backward stable generalized (Arnoldi) Patel al- gorithm [21] were proposed for the

>

-palindromic QEP, and can be extended toH-palindromic QEPs. For other approaches for

?

-palindromic QEPs, see [22–24]. In [2], the SDA algorithm was generalized for theg-palindromic QEPs, which do not include the PCP_PQEP. For further information on the numerical solution of palindromic EVPs, see [4]. For general survey of matrix polynomials, the associated eigenvalue problems and their applications, see [25,26].

Throughout this paper,C^n×mis the set of alln

×

mcomplex matrices,Cⁿ

=

_C^n×1, andC

=

_C¹;I_n(orI if there is no confusion) is then

×

nidentity matrix; andX^H

=

X^>denote the Hermitian (conjugate transpose) ofX. We shall also adopt MATLAB-like convention to access the entries of vectors and matrices—X

(

ⁱ

,

^j

)

^{is the}

(

ⁱ

,

^j

)

^{th entry,X}

(

^k

: `,

ⁱ

:

j

)

^the sub-matrix from rowsk

: `

and columnsi

:

j,X

( : ,

ⁱ

:

j

)

from columnsi

:

jandX

(

^k

: `, : )

^{from rowsk}

: `

^.

In Section2, we develop an SDA algorithm for solving the

ε

PCP_PQEP. Convergence of the SDA is proved in Section3.

For the PCP_PQEP arisen from the stability analysis of TDSs, we develop a deflation technique for finding all unimodular eigenvalues in Section4. A structured backward error analysis for PCP_PQEPs is presented in Section5. Numerical examples of PCP_QEPs arising from TDS are given in Section6. Concluding remarks are given in Section7.

2. SDA algorithm forε^PCP_PQEP

For a given PCP_PQEP(1.1), we define M

=

A 0

−

C

−

I

,

^L

=

D I B 0

.

^(2.1)

WithD

=

0 in(2.1), we have A 0

−

C

−

I x₁ x₂

= λ

0 I

B 0 x₁ x₂

,

leading to

Ax₁

= λ

^x2

,

^(2.2a)

−

Cx₁

−

x₂

= λ

^Bx1

.

^(2.2b)

Multiplying(2.2b)by

λ

and substituting(2.2a)into it, we obtain

(λ

^2B

+ λ

^C

+

A

)

^x1

=

0

.

We have shown that the pencilM

− λ

^Lis a linearization of PCP_PQEP(1.1)withD

=

0. Based on the SDA algorithm proposed in [5], we develop a new SDA for solving the PCP_PQEP.

For the pencilM

− λ

^{Ldefined in}(2.1), we compute M∗

=

−

AK⁻¹ 0 BK⁻¹ I

,

^L∗

=

I AK⁻¹ 0

−

BK⁻¹

,

whereK

≡

C

−

Dis assumed to be invertible. It is easy to check thatM∗L

=

_L∗M. Direct calculation gives rise to Mb

=

_M∗M

=

bA 0

−b

C I

,

Lb

=

_L∗L

=

bD I bB 0

,

^(2.3)

where

bA

≡ −

AK⁻¹A

,

bB

≡ −

BK⁻¹B

,

^(2.4a)

bC

≡

C

−

BK⁻¹A

,

bK

≡

K

− (

^BK−1A

+

AK⁻¹B

),

^(2.4b)

and

bD

=

bC

−

bK

.

^(2.4c)

Theorem 2.1. The pencilMb

− λ

Lbhas the doubling property, i.e., if M

X₁ X₂

=

_L X₁

X₂

S

,

where X₁

,

^X2

∈

_C^n×mand S

∈

_C^m×m, then Mb

X₁ X₂

=

Lb X₁

X₂

S²

.

Proof. From(2.3)and the relationM∗L

=

_L∗M, we have Mb

X₁ X₂

=

_M∗M X₁

X₂

=

_M∗L X₁

X₂

S

=

_L∗M X₁

X₂

S

=

_L∗L X₁

X₂

S²

=

Lb X₁

X₂

S²

.

The iteration in(2.4)is structure-preserving for

ε

PCP_PQEP, as shown in the following theorem:

Theorem 2.2. For a pencilM

− λ

^{Lgiven in}(2.1), suppose that

PAP

= ε

^B

,

^PBP

= ε

^A

,

^{PK P}

= ε

^K

,

^(2.5)

where K

=

C

−

D. Then it holds that

PbAP

= ε

bB

,

^PbBP

= ε

bA

,

^PbK P

= ε

bK

,

wherebA

,

bB

,

bK are defined in(2.4).

Proof. From(2.4b)and(2.5), we have PbK P

=

PK P

−

PBP

(

^{PK P}

)

^−1PAP

+

PAP

(

^{PK P}

)

^−1PBP

= ε

^K

− ε

³

(

^AK−1B

+

BK⁻¹A

) = ε

bK

.

Similarly, from(2.4a)and(2.5), we have

PbAP

= −

PAP

(

^{PK P}

)

^−1PAP

= − ε

^3BK−1B

= ε

bB and then

PbBP

= ε

bA

.

FromTheorems 2.1and2.2, we restate the SDA for finding a basis for the stable invariant subspace of

(

^M

,

^L

)

^. Algorithm SDA

Input:A

,

^B

,

^C

,

^P

∈

_C^n×nwithPAP

= ε

^B

,

^PBP

= ε

^A

,

^{PC P}

= ε

^C

,

^P2

=

P,

τ

(a small tolerance);

Output: a basisX_s

= [

X_s1^>

,

^Xs2^>

]

^>

∈

_C^2n×mwithX_s1^HX_s1

=

I_mfor the stable invariant subspace of

(

^M

,

^L

)

^in(2.1);

Setk

=

0

,

^Ak

=

A

,

^Bk

=

B

,

^Ck

=

C

,

^Kk

=

C

,

^Dk

=

0;

Dountil convergence:

Compute

A_k+1

= −

A_kK_k⁻¹A_k B_k+1

=

PA_k+1P W_k

=

B_kK_k⁻¹A_k

K_k+1

=

K_k

− (

^Wk

+

PW_kP

)

C_k+1

=

C_k

−

W_k

(

^Dk+1

=

C_k+1

−

K_k+1

)

k

=

k

+

1

If dist

(

Null

(

^Ak

),

Null

(

^Ak+1

)) < τ

^,Stop End

Computean orthonormal basisX_s1such that

k

A_k+1X_s1

k ≤ τ k

A_k+1

k

; setX_s2

= −

C_k+1X_s1.

3. Convergence of SDA

Consider the matrix pair

(

^M

,

^L

)

^{as in}(2.1). In order to ensure that the SDA converges to a basis of the stable invariant subspace of

(

^M

,

^L

)

, we suppose that all eigenvalues of

(

^M

,

^L

)

on the unit circle are semisimple (generically, multiple unimodular eigenvalues are rare; see [18]).

From the Kronecker Canonical form [6,27], there exist nonsingular matricesQ andZsuch that QMZ

=

J₁ 0 0 I_n

≡

J_M

,

^(3.1a)

QLZ

=

I 0

0 J₂

≡

J_L

,

^(3.1b)

where

J₁

=

_Ω1

⊕

J_s

,

^J2

=

_Ω2

⊕ ε

^Js

,

Ω1

=

diag

{

e^iω1

, . . . ,

^eiω`

} ,

Ω2

=

diag

{

e^iω`+1

, . . . ,

^eiω2`

} ,

andJ_sis the stable Jordan block of sizem(i.e.,

ρ(

^Js

) <

^{1) withm}

=

n

− `

^{. Here}

ρ

denotes the spectral radius and

⊕

the direct sum of matrices. SinceJ_MandJ_Lcommute with each other, it follows from(3.1)that

MZJ_L

=

Q⁻¹J_LJ_M

=

_LZJ_M

.

^(3.2)

Let

{ (

Mk

,

Lk

) }

^∞_k=0be the sequence with Mk

=

A_k 0

−

C_k

−

I_n

,

^Lk

=

D_k I_n B_k 0

,

^(3.3)

whereA_k

,

^Bk

,

^CkandD_kare generated by the SDA. WithM0

=

_MandL0

=

_L, it follows from(3.2)andTheorem 2.1that

MkZJ_L^2k

=

_LkZJ_M^2k

.

^(3.4)

Theorem 3.1. Let

(

^M

,

^L

)

be given in(2.1)with all its unimodular eigenvalues being semisimple. Write Z in(3.1)in the form Z

=

Z₁ Z₃ Z₂ Z₄

,

^Zi

∈

_C^n×n

,

ⁱ

=

1

, . . . ,

⁴

.

^(3.5)

Suppose that the sequence

{

A_k

,

^Bk

,

^Ck

,

^Dk

}

^∞_k=1generated by the SDA is well defined and

{

A_k

}

^∞_k=1is uniformed bounded on k. If Z₁ and Z₃are invertible, we have

(i) A_kZ₁₂

=

O

(ρ(

^Js^2k

)) →

0as k

→ ∞

;

(ii) C_kZ₁₂

= −

Z₂₂

+

O

(ρ(

^Js^2k

)) → −

Z₂₂as k

→ ∞

; (iii) D_kZ₁₂

= −

_Z42

+

_O

(ρ(

^J2s^k

)) → −

_{Z42as k}

→ ∞

_,

where Z₁₂

=

Z₁

( : , ` +

1

, . . . ,

ⁿ

),

^Z22

=

Z₂

( : , ` +

1

, . . . ,

ⁿ

),

^Z32

=

Z₃

( : , ` +

1

, . . . ,

ⁿ

),

^Z42

=

Z₄

( : , ` +

1

, . . . ,

ⁿ

)

^and

ρ( · )

is the spectrum radius.

Proof. Substituting

(

^Mk

,

^Lk

)

^from(3.3),JMandJ_Lin(3.1)as well asZin(3.5)into(3.4), we have

A_kZ₁

= (

^DkZ₁

+

Z₂

)(

Ω1^2k

⊕

J_s^2k

),

^(3.6a)

A_kZ₃

(

Ω2^2k

⊕

J²

k

s

) =

D_kZ₃

+

Z₄

,

^(3.6b)

− (

^CkZ₁

+

Z₂

) =

B_kZ₁

(

Ω1^2k

⊕

J_s^2k

),

^(3.6c)

− (

^CkZ₃

+

Z₄

)(

Ω₂^2k

⊕

J²

k

s

) =

B_kZ₃

.

^(3.6d)

From(3.6b), it follows that

D_k

= −

Z₄Z₃⁻¹

+

A_kZ₃

(

Ω2^2k

⊕

J²

k

s

)

^Z3⁻¹

.

^(3.7)

Substituting(3.7)into(3.6a), we get A_k

I

−

Z₃

Ω2^2k

⊕

J²

k s

Z₃⁻¹Z₁

Ω1^2k

⊕

J_s^2k

Z₁⁻¹

= −

Z₄Z₃⁻¹Z₁

+

Z₂

Ω1^2k

⊕

J_s^2k

Z₁⁻¹

.

^(3.8)

SinceΩ1^2kandΩ2^2kare uniformly bounded (independent ofk), and

ρ(

^Js

) <

^1,(3.8)can be rewritten as A_k

h I

−

z₃Ω2^2k

ω

3^> z₁Ω1^2k

ω

^>1

+

O

ρ(

^Js^2k

)

i

= −

Z₄Z₃⁻¹z₁

+

z₂

Ω1^2k

ω

^>1

+

O

ρ(

^Js^2k

) ,

where

z₁

=

Z₁

( : ,

¹

: `),

^z2

=

Z₂

( : ,

¹

: `),

^z3

=

Z₃

( : ,

¹

: `), ω

1^>

=

Z₁⁻¹

(

¹

: `, : ), ω

^>3

=

Z₃⁻¹

(

¹

: `, : ).

By the assumption thatA_kis uniformly bounded onk, we have A_k

=

a_k

ω

^>1

+

O

(ρ(

^Js^2k

))

for some suitablea_k

∈

_C^n×`with

k

a_k

k

being uniformly bounded onk. Thus we have A_kZ₁₂

=

O

(ρ(

^Js^2k

)) →

0

,

^ask

→ ∞ ,

whereZ₁₂

=

Z₁

( : , ` +

1

, . . . ,

ⁿ

)

being orthogonal to

ω

1. This proves (i).

SinceB_k

=

PA_kPbyTheorem 2.2, from(3.6c)we obtain C_k

= −

Z₂Z₁⁻¹

+

c_k

ω

^>1

+

O

(ρ(

^Js^2k

))

for some suitablec_k

∈

_C^n×`with

k

c_k

k

being uniformly bounded onk.

Thus, we have

C_kZ₁₂

= −

Z₂Z₁⁻¹Z₁₂

+

c_k

ω

^>1Z₁₂

+

O

(ρ(

^Js^2k

))

= −

Z₂ 0

I_n−`

+

O

(ρ(

^Js^2k

)) → −

Z₂₂

,

^ask

→ ∞ ,

whereZ₂₂

=

Z₂

( : , ` +

1

, . . . ,

ⁿ

)

. This proves (ii).

Similarly from(3.7), we see that

D_kZ₃₂

= −

Z₄Z₃⁻¹Z₃₂

+

d_k

ω

^>3Z₃₂

+

O

(ρ(

^J2s^k

))

→ −

Z₄₂

→

0

,

^ask

→ ∞ ,

whereZ₃₂

≡

Z₃

( : , ` +

1

, . . . ,

ⁿ

),

^Z42

≡

Z₄

( : , ` +

1

, . . . ,

ⁿ

)

^anddk

∈

_C^n×`is some suitable matrix which is uniformly bounded onk.

4. Application to TDS

We are ready to solve PCP_PQEPs from TDSs. For a given PCP_PQEP orQ

(λ)

^{as in}(1.1), we are interested in finding all eigenvalues on the unit circle and the associated eigenvectors. We first run the SDA until

dist

(

Null

(

^Ak−1

),

Null

(

^Ak

)) <

Tolerance

.

Then we compute the SVD decomposition ofA_k, such that

U_k^HA_kV_k

=

Σk 0 0 Σk⁰

,

Σk

_Σk⁰

≈

O

(τ),

^(4.1)

where

τ

is a small tolerance.

Consequently, with

`

denoting the number of unimodular eigenvalues, we have A_k 0

−

C_k

−

I V_k2

−

C_kV_k2

≈

O

(τ),

^(4.2)

whereV_k2

=

V_k

( : , ` +

1

: . . . ,

ⁿ

)

^{. From}Theorems 2.1and3.1, it follows that span

X₁ X₂

≡

V_k2

−

C_kV_k2

'

span Z₁₂

Z₂₂

.

^(4.3)

That is, X₁^>

,

^X2^>

>

approximates a basis of the stable invariant subspace of

(

^M

,

^L

)

^{. From(4.2)and(3.1)}we compute an approximate stable eigenmatrix associated with

[

X₁^>

,

^X2^>

]

^>for

(

^M

,

^L

)

^by

M X₁

X₂

=

_L X₁

X₂

S

.

This implies that

S

= (

^X2^>X₂

)

^−1X2^>AX₁

.

^(4.4)

LetS

ξ

j

= λ

j

ξ

j

,

^j

= ` +

1

, . . . ,

^{n. Then}

{ (λ

j

,

^X1

ξ

j

) }

ⁿ_j=`+1are the computed stable eigenpairs ofQ

(λ)

. Furthermore,

{ (

¹

/λ

j

,

^PX1

ξ

j

) }

ⁿ_j=`+1are the computed unstable eigenpairs ofQ

(λ)

. Again from(3.1),(4.1)–(4.2)andTheorem 2.2, we have

D_k I B_k 0

PV_k2

−

D_kPV_k2

≈

O

(τ).

Thus, from(4.3),

(

^PX1

)

^>

, − (

^DkPX₁

)

^>>

forms a basis for the unstable invariant subspace.

Table 1

Flop counts for the SDA+Newton algorithm.

Task ≈Flops

(a) SDA ⁸⁰₃n³×8= 213n³

(b) SVD in(4.1)and(4.4) ¹⁶⁰₃ n³+32n³= 85n³

(c)eig(S) 100n³

(d) Compute(^R0,^T0)^{in(4.5) 75n3}

(e) Newton’s method in(4.6) 5.3`n³

Compute all eigenvalues (a)–(e) (473+5.3`)n³

`unimodular eigenvalues only (a), (b), (d) & (e) (<sup>373</sup>+5.<sup>3</sup>`)ⁿ³

LetΦ1be an orthonormal basis of h _X

1 PX₁

−C_kX₁ −D_kPX₁

i

. Then, applying deflation, there are unitary matricesΦ

= [

_Φ1

,

Φ0

]

andΨ

= [

_Ψ1

,

Ψ0

]

such that

"

Ψ₁^H Ψ0^H

#

M

[

_Φ1

,

Φ0

] =

R₁ R₂ 0 R₀

,

^(4.5a)

"

Ψ₁^H Ψ0^H

#

L

[

_Φ1

,

Φ0

] =

T₁ T₂

0 T₀

.

^(4.5b)

Approximations to the unimodular eigenvalues can be obtained by solving the matrix pair

(

^R0

,

^T0

) ≡ (

Ψ0^HMΦ0

,

Ψ0^HLΦ0

)

(e.g. usingeigin MATLAB). To refine an approximate unimodular eigenvalue

λ

0ofQ

(λ)

^from

(

^R0

,

^T0

)

^{, we can} apply Newton’s method proposed by [28]:

λ

k+1

= λ

k

−

1

/

^x(n^k)

,

^k

=

0

,

¹

, . . . ,

^(4.6)

where x⁽_n^k)

=

"

L⁻_k¹Θk

dQ d

λ

_λ

=λk

! Q_k

#

nn

andΘkQ

(λ

k

) =

L_kQ_kis a QL-factorization with row pivotingΘk. If the Newton process in(4.6)converges, i.e.,

| λ

k+1

− λ

k

| <

^{Tol or}

|[

L_k

]

_nn

| <

^Tol

,

then we stop(4.6)and setb^x

=

Q_ke_n (the last column ofQ_k) to be the associated eigenvector ofQ

(λ)

corresponding to b

λ ≡ λ

k+1.

We now study the total flop counts of the SDA

+

Newton algorithm as described in (4.1)–(4.6) for computing all eigenvalues of a PCP_PQEP. Since all computations are in complex arithmetic, the addition and the multiplication of two complex numbers require 2 flops and 6 flops (4 multiplications and 2 additions), respectively. The flop counts of the SDA

+

Newton algorithm are listed inTable 1. From experience, at most eight iterations are required for the SDA to converge and only one iteration is required for the Newton refinement in(4.6).

We shall compare the SDA

+

Newton algorithm with the QZ algorithm [29]. In [6], a ‘‘good’’ structured linearization for (1.1)has been proposed:

C

−

B A

A A

+ λ

B B B C

−

A

≡

X

+ λ

_PX

˜

_P

˜ ,

where_P

˜ =

h_{0 P}

P 0

i

. Thus, the eigenpairs for(1.1)can then be computed by the QZ algorithm on

(

^X

, − ˜

PX_P

˜ )

^or

(

^M

,

^L

)

^{. In} general, the QZ algorithm applied to

(

^M

,

^L

)

^or

(

^X

, − ˜

PX_P

˜ )

requires about 960n³flops for the computation of all eigenvalues and about 1600n³flops if, in additional, eigenvectors are needed.

Consequently, computing only unimodular eigenpairs of PCP_PQEPs, the number of flops required by the SDA

+

Newton algorithm and the QZ

+

Newton algorithm are summarized inTable 2(with the last column containing ratios between the flop counts of the SDA and the QZ algorithms). For a fairer comparison, Newton refinement is applied at the end of both algorithms to achieve a similar accuracy. The SDA

+

Newton algorithm needs about one-half of the flop count of the QZ

+

Newton algorithm

(

⁴⁴¹ⁿ³

:

1013n³

)

^when

` =

10. In general, the SDA

+

Newton algorithm is always more efficient, even more so for smaller values of

`

. Note from [18] that the expected value of

`

^equalsEn

(`) =

√ 10n

4 for random palindromic eigenvalue problems. We haveE₁₀₀₀

(`) ≈

25,E_{10 000}

(`) ≈

79 andE_{100 000}

(`) ≈

250, so even for very large TDSs,

`

is manageably small and the SDA

+

Newton algorithm will always be substantially more efficient that the QZ

+

_Newton algorithm.

Table 2

Relative flop counts vs.`^.

` SDA+Newton QZ+Newton SDA:QZ

10 428n³ 1013n³ 0.42

20 481n³ 1067n³ 0.45

30 535n³ 1120n³ 0.48

40 588n³ 1173n³ 0.50

50 641n³ 1227n³ 0.52

100 908n³ 1493n³ 0.61

500 3041n³ 3627n³ 0.84

1000 5708n³ 6293n³ 0.91

5. A Structured backward error analysis of PCP_PQEP

Let

( λ, ˆ

x

ˆ )

be an approximate eigenpair ofQ

(λ)

ⁱⁿ(1.1). A natural definition of the normwise backward error of

( λ, ˆ

x

ˆ )

^for (1.1)is

η( λ, ˆ

_x

ˆ ) =

min

δ

Q

( λ) ˆ +

₁Q

( λ) ˆ

x

ˆ =

0

, k

₁B

k

₂

≤ δ k

B

k

₂

, k

₁C

k

₂

≤ δ k

C

k

₂

, k

₁A

k

₂

≤ δ k

A

k

₂

,

^(5.1a)

where

k · k

₂is the spectrum norm and

1^Q

(λ) = λ

²1^B

+ λ

1^C

+

₁A (5.1b)

with1^A

,

1^B

,

1^C

∈

_C^n×nbeing the perturbation matrices.

An explicit expression for

η( λ, ˆ

_x

ˆ )

with respect to the residualr

=

Q

( λ) ˆ

_x

ˆ

is given by [30]:

Theorem 5.1 (Tisseur). The normwise backward error

η( λ, ˆ

_x

ˆ )

is given by

η( λ, ˆ

x

ˆ ) = k

r

k

₂

α ˆ kˆ

x

k

₂

,

^(5.2)

where r

=

Q

( λ) ˆ

x and

ˆ α ˆ = | ˆ λ |

²

k

B

k

₂

+ | ˆ λ |k

C

k

₂

+ k

A

k

₂.

It is of interest to consider a backward error in which the perturbations

{

₁B

,

1^C

,

1^A

}

preserve the PCP-structure in

{

B

,

^C

,

^A

}

. Therefore, we define the structured backward error of

( λ, ˆ ˆ

_x

)

^by

η

S

( λ, ˆ ˆ

x

) =

min

δ

Q

( λ) ˆ +

₁Q

( λ) ˆ

ˆ

x

=

0

,

^P1^BP

=

₁A

,

^P1^{C P}

=

₁C

, k

₁B

k

₂

≤ δ k

B

k

₂

, k

₁C

k

₂

≤ δ k

C

k

₂

, k

₁A

k

₂

≤ δ k

A

k

₂

.

Note, for convenience, that we only consider the case of

ε =

1 in(1.1).

It is clear that

η

S

( λ, ˆ ˆ

x

) ≥ η( λ, ˆ ˆ

x

)

. The optimal perturbations in(5.1)do not, in general, have the PCP-Structure. We first prove the following property.

Theorem 5.2. Let

(λ,

^x

)

be an eigenpair of Q

(λ)

^in(1.1)with

ε =

1. If

λ

is a simple unimodular eigenvalue, then Px

=

x.

Proof. From(1.1), we have

(λ

^2B

+ λ

^C

+

A

)

^x

=

0

(

^by(1.1a)

)

⇔

λ

²

(

^PBP

) + λ(

^PCP

) +

PAP Px

=

0

⇔ (λ

^2A

+ λ

^C

+

B

)

^Px

=

0

(

^by(1.1b)

)

⇔ (λ

^2B

+ λ

^C

+

A

)

^Px

=

0

(

^by

λ = λ

⁻¹

).

Since

λ

is simple, it follows thatPx

=

x.

Based on the assertion ofTheorem 5.2, the next theorem shows that requiring the perturbations to possess the PCP- structure has no effect on the backward error, provided that

λ ˆ

is on the unit circle and

ˆ

xsatisfiesP

ˆ

x

= ˆ

x.

Theorem 5.3. Let

( λ, ˆ

x

ˆ )

be an approximate eigenpair of Q

(λ)

^{as in(1.1)with}

λ ˆ =

e^iθand P

ˆ

x

= ˆ

x. Then

η

S

( λ, ˆ ˆ

x

) = η( λ, ˆ

x

ˆ ).