Recent Advances in the Numerical Solution of Quadratic ...ftisseur/talks/talk_ala09.pdf · Appears in many practical applications. Recent work on quadratization: convert degree ℓ

Recent Advances in the

Numerical Solution of

Quadratic Eigenvalue Problems

Françoise Tisseur

School of Mathematics

The University of Manchester

[email protected]

http://www.ma.man.ac.uk/~ftisseur/

SIAM Conference on Applied Linear Algebra

October 2009


http://www.ma.man.ac.uk

http://www.man.ac.uk

mailto:[email protected]


Nonlinear Eigenproblems

Let F : Ω → Cn×n be analytic on open set Ω ⊆ C.

The nonlinear eigenvalue problem: Find scalars λ and

nonzero x , y ∈ Cn satisfying F (λ)x = 0 and y∗F (λ) = 0.

λ is an e’val, x , y are corresponding right and left e’vecs.

E’vals are solutions of det(F (λ)) = 0.

In practice, elements of F most often polynomial, rational or

exponential functions of λ.

Can be very difficult to solve (poor conditioning, algebraic

structure to be preserved).

Françoise Tisseur Quadratic eigenproblem 2 / 29

NLEVP Toolbox

Collection of Nonlinear Eigenvalue Problems :

T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder, F. T.

Quadratic, polynomial, rational and other nonlinear

eigenproblems.

Provided in the form of a MATLAB Toolbox.

Problems from real-life applications + specifically

constructed problems.

http://www.mims.manchester.ac.uk/research/

numerical-analysis/nlevp.html

New release to come. Further contributions welcome.


http://www.mims.manchester.ac.uk/research/numerical-analysis/nlevp.html

Sample of Problems

Bicycle (pep,qep,real,parameter-dependent)

2 × 2 quadratic poly. arising in study of bicycle self-stability.

Railtrack (pep,qep,t-palindromic,sparse)

T-palindromic quadratic of size 1005. Stems from a model of

vibration of rail tracks under the excitation of high speed trains.

Butterfly (pep,real, T-even,scalable)

quartic matrix polynomial with T-even structure.

Loaded string (rep,real,symmetric,scalable)

rational eigenvalue problem describing eigenvibration of a

loaded string.

Gun (nep,sparse) nonlinear eigenvalue problem modeling a

radio-frequency gun cavity.


Quadratic Eigenvalue Problem (QEP)

Concentrate on

Q(λ) = λ2M + λD + K ,

with M , D, K ∈ Cn×n.

Appears in many practical applications.

Recent work on quadratization: convert degree ℓ matrix

polynomials to degree 2 (Al-Ammari & T, 2009; Huang,

Lin & Su, 2008)

Mainly for structured matrix polynomials.

Use new efficient algorithms for quadratics.

Theoretical reasons.


Outline

Review of recent progress.

Structure Preserving Transformations.


Linearization

Standard way of treating QEPs both theoretically and

numerically.

Convert Q(λ) = λ2M + λD + K into a linear pencil such as

C1(λ) = λ

[M 0

0 I

]+

[D K

−I 0

].

(first companion form)

C1(λ) is a linearization of Q, i.e, there exist E(λ) and

F (λ) with constant, nonzero determinants s.t.

[Q(λ) 0

0 I

]= E(λ)C1(λ)F (λ).


Work on Linearizations

New (structure preserving) linearizations derived along with

algorithms preserving spectral properties in finite precision

arithmetic.

[Antoniou, Higham, Lin, Mackey, Mackey, Mehl, Mehrmann,

T., Vologiannidis, . . . ]

Mackey, Mackey, Mehl & Mehrmann (2006) introduce

L1(Q) =

L(λ) : L(λ)

[λI

I

]=

[v1Q(λ)v2Q(λ)

], v ∈ C

2

.

Almost all pencils in L1 are linearizations.

E’vecs of Q easily recovered from e’vecs of L ∈ L1

([M4, 2006], [Higham, Li, T., 2007]).

L1(Q) is a rich source of interesting linearizations.


Block Symmetric Linearizations

Higham, Mackey, Mackey and T. (2006) define

B(Q) :=

λX + Y ∈ L1(Q) : XB = X , Y B = Y

=

v1L1(λ) + v2L2(λ), v ∈ C2 ,

where

L1(λ) = λ

[M 0

0 −K

]+

[D K

K 0

], L2(λ) = λ

[0 M

M D

]+

[−M 0

0 K

].


Block Symmetric Linearizations

Higham, Mackey, Mackey and T. (2006) define

B(Q) :=

λX + Y ∈ L1(Q) : XB = X , Y B = Y

=

v1L1(λ) + v2L2(λ), v ∈ C2 ,

where

L1(λ) = λ

[M 0

0 −K

]+

[D K

K 0

], L2(λ) = λ

[0 M

M D

]+

[−M 0

0 K

].

L ∈ B(Q) with vector v is a linearization of Q iff e’val of Q

is not a root of p(x ; v) = v1x + v2.

[Higham, Mackey, T., 2009] Identified definite pencils in

B(Q) for hyperbolic (overdamped) Q.


Linearization Process

Better understanding of linearization process and effects of

scaling on

conditioning of eigenvalues,

backward error of computed eigenpairs.

[Adhikari, Alam, Betcke, Higham, Kressner, Li, Mackey, Mehl,

Mehrmann, T., . . . ]


Linearization Process

Better understanding of linearization process and effects of

scaling on

conditioning of eigenvalues,

backward error of computed eigenpairs.

[Adhikari, Alam, Betcke, Higham, Kressner, Li, Mackey, Mehl,

Mehrmann, T., . . . ]

Illustration: free vibrations of aluminium beam.

M > 0, K > 0, D ≥ 0 ⇒ all ei’vals have Re(λ) ≤ 0.

D is rank 1. Can show n pure imaginary e’vals.


Eigenvalues of Q(λ) = λ2M + λD + K

C1(λ) = λ

[M 0

0 I

]+

[D K

−I 0

].

L1(λ) = λ

[M 0

0 −K

]+

[D K

K 0

], L2(λ) = λ

[0 M

M D

]+

[−M 0

0 K

].

coeffs = nlevp(’damped−beam’,100);

K = coeffs1; D = coeffs2; M = coeffs3;

I = eye(2*nele); O = zeros(2*nele);

eval = eig([D K; -I 0],-[M O; O I]; % C1

%eval = eig([D K; -I 0],-[M O; O I]; % L1

%eval = eig([D K; -I 0],-[M O; O I]; % L2

plot(eval,’.r’);


Computed Spectrum of C1, L1 and L2

−16 −14 −12 −10 −8 −6 −4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4x 10

6

−16 −14 −12 −10 −8 −6 −4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4x 10

6

−16 −14 −12 −10 −8 −6 −4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4x 10

6


Comments

Show practical value of condition numbers and backward

errors for understanding the quality of computed results.

Importance of scaling QEPs before computing e’vals via

linearization.

Results not confined to the beam problem but apply to

any QEP.

[Grammont, Higham, T., 2009] General framework for

analyzing the linearization process (see Grammont’s talk,

MS12, Monday).


Spectrum of C1, L2 before/after Scaling

−16 −14 −12 −10 −8 −6 −4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4x 10

6

−16 −14 −12 −10 −8 −6 −4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4x 10

6

−16 −14 −12 −10 −8 −6 −4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4x 10

6

−16 −14 −12 −10 −8 −6 −4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4x 10

6


A Black Box (Dense Case)

[Hammarling, Higham, Munro, T. ]

step 1: Apply Fan, Lin and Van Dooren scaling.

step 2: Construct “a" companion linearization.

step 3: Deflate zeros and ∞ e’vals.

step 4: Solve generalized eigenproblem with QZ.

Optional:

step 5: Recover right/left e’vecs of Q from those of

companion form.

step 6: Compute e’vals condition number.


Basic Facts

Triples (M , D, K ) cannot , in general, be simultaneously

diagonalized.

(See Lancaster, MS35, Wednesday)

No analog of generalized Schur form for matrix triples.

Cannot simultaneously tridiagonalize (M , D, K ).


Diagonalization of Quadratics

For n × n symmetric Q(λ) = λ2M + λD + K with all e’vals

distinct, there exists U ∈ R2n×2n s.t.

UT

(λ

[0 M

M D

]+

[−M 0

0 K

])U = λ

[0 ΛM

ΛM ΛD

]+

[−ΛM 0

0 ΛK

],

where ΛM , ΛD and ΛK are diagonal matrices.

(Garvey, Friswell, Prells, 2002)

New diagonal quadratic matrix polynomial

QΛ(λ) = λ2ΛM + λΛC + ΛK .

Q(λ) and QΛ(λ) have the same e’vals.

U is a structure preserving transformation.


Structure Preserving Transformation (SPT)

Recall: L(λ) ∈ B(Q) ⇐⇒ L(λ) = v1L1(λ) + v2L2(λ),

L1(λ) = λ

[M 0

0 −K

]+

[D K

K 0

], L2(λ) = λ

[0 M

M D

]+

[−M 0

0 K

].

Assume M nonsingular.

Definition

SL, SR nonsingular define an SPT for Q(λ) if

STL L2(λ)SR = λ

[0 M

M D

]+

[−M 0

0 K

]=: L2(λ).

L(λ) is a linearization of Q(λ) = λ2M + λD + K .


Some Properties

Let (SL, SR) be an SPT mapping Q(λ) to Q(λ).

SPTs preserve block structure of pencils in B(Q) .

STL B(Q)SR = B(Q). Moreover, L(λ) ∈ B(Q) with vector v

⇔ L(λ) = STL L(λ)SR ∈ B(Q) with vector v .

Well-defined relations between e’vecs of Q and Q .

Let (λ, x , y) be an eigentriple of Q(λ). Then

[λx

x

]= SR

[λx

x

],

[λy

y

]= SL

[λy

y

],

where x , y such that Q(λ)x = 0, y∗Q(λ) = 0.


Constraints

S =

[S11 S12

S21 S22

], T =

[T11 T12

T21 T22

], Sij , Tij ∈ R

n×n.

(S, T ) defines a SPT iff

ST11MT21 + ST

21MT11 + ST21CT21 = 0, (1)

−ST11MT12 + ST

21KT22 = 0, (2)

−ST12MT12 + ST

22KT22 = 0, (3)

ST11MT22 + ST

21MT12 + ST21DT22 = ST

11MT11 − ST21KT21, (4)

ST22MT11 + ST

12MT21 + ST22DT21 = ST

11MT11 − ST21KT21. (5)

5n2 constraints .


Elementary SPTs (symmetric case)

Let T = I2n +

[abT adT

af T ahT

]∈ R

2n×2n, a, b, d , f , h ∈ Rn.

Rank-2 modification of I2n.

Let V = [ b d f h ] ∈ Rn×4.

For almost all a ∈ Rn, any solution V to VA = B defines

an SPT T .

A ∈ R4×3, B ∈ R

n×3 depend on a, M, D and K .

If (M , D, K )T

7−→ (M, D, K ) then M, D, K are low rank

modifications of M , D, K .


An Application: Deflation of Eigenvalues

Let (λj , xj), j = 1, 2 be two given eigenpairs of Q(λ).

Want to transform n × n Q(λ) into

Q(λ) =

[Qd(λ) 0

0 q(λ)

]n − 1

1

such that

Λ(Q) = Λ(Q), (same spectrum)

q(λj) = 0, j = 1, 2.

Deflation procedure decoupling Q(λ) into two quadratics.


Decoupling by Similarity

Suppose there exist nonsingular SL, SR s.t.

SLQ(λ)SR =

[Qd(λ) 0

0 q(λ)

]= Q(λ).

The roots λ1, λ2 of q(λ) are e’vals of Q and Q

Q(λj)xj = 0, Q(λj)en = 0, j = 1, 2

with e’vecs related by S−1R [ x1 x2 ] = [ en en ] .

Decoupling possible only if e’vecs x1 and x2 are parallel.


Decoupling by Similarity

Suppose there exist nonsingular SL, SR s.t.

SLQ(λ)SR =

[Qd(λ) 0

0 q(λ)

]= Q(λ).

The roots λ1, λ2 of q(λ) are e’vals of Q and Q

Q(λj)xj = 0, Q(λj)en = 0, j = 1, 2

with e’vecs related by S−1R [ x1 x2 ] = [ en en ] .

Decoupling possible only if e’vecs x1 and x2 are parallel.

Is there an elementary SPT mapping Q to Q s.t. e’vecs of

Q with e’vals λ1, λ2 are parallel?


Constraints

(λ1, x1), (λ2, x2) to be deflated with λ1 6= λ2 and x1 6= αx2.

Aim: construct T = I2n +[

abT

af TadT

ahT

]and nonzero z ∈ R

n s.t.

Q(λ)T

7−→ Q(λ) with Q(λj)z = 0, j = 1, 2.

Yields a, z and zT [ b d f h ] = zT V .

T is an SPT ⇐⇒ VA = B .


Existence of SPT T

Theorem

Eigenpairs (λ1, x1), (λ2, x2) with λ1 6= λ2 either real or complex

conjugate can be mapped to (λ1, z), (λ2, z) by elementary

SPTs if

xTj Q′(λj)xj 6= 0, j = 1, 2,

real eigenpairs have opposite type:

sign(xT1 Q′(λ1)x1) = −sign(xT

2 Q′(λ2)x2).

Can generate a family of SPTs mapping (λj , xj) to (λj , z),j = 1, 2.


Deflation of (λ1, z), (λ2, z)

Lemma

If (λ2j M + λjC + K )z = 0, j = 1, 2 with λ1 6= λ2 then

(M , C, K )z = (mp, cp, kp), p ∈ Rn, pT z = 1,

c = −m(λ1 + λ2), k = mλ1λ2.


Deflation of (λ1, z), (λ2, z)

Lemma

If (λ2j M + λjC + K )z = 0, j = 1, 2 with λ1 6= λ2 then

(M , C, K )z = (mp, cp, kp), p ∈ Rn, pT z = 1,

c = −m(λ1 + λ2), k = mλ1λ2.

Let nonsingular G be such that

Gen = z, GT p = en.

Then GT MGen = GT Mz = mGT p = men and Lemma ⇒

GT (M , C, K )G =

([M 0

0 m

],

[C 0

0 c

],

[K 0

0 k

]).


Example 1

Q(λ) = λ2

[2 −1

−1 3

]+ λ

[0 1

1 0

]+

[3 2

2 3

]

Given λ1,2 = −0.34 ± 1.84i and associated e’vecs, our

deflation procedure yields

λ2[

5.6 2.0e-162.0e-16 −1.4e-1

]+ λ

[−1.6 −9.4e-16

−9.4e-16 −9.3e-2

]+

[1.6 −9.8e-17

−9.8e-17 −4.8e-1

],

with κ2(T ) = 7.9 and κ2(G) ≈ 1.

Decoupling accomplished to within the working precision.


Example 2: Damped Beam Problem

M, C, K generated by nlevp(’damped−beam’,nele).

Q(λ) = λ2M + λC + K and undamped Qu(λ) = λ2M + K have

n e’vals in common that we deflate by

our decoupling procedure: QS

7−→[

Q1(λ)0

0Q2(λ)

],

using special property of Q(λ) to orthogonally block

diag’lize it and then diag’lize one block with Cholesky-QR

(transformation W ).

n κ2(S) κ2(W )16 4.47e1 3.79e1

32 9.57e1 7.84e1

64 1.95e2 1.57e2

κ2(E) = ‖E‖2‖E−1‖2

κ2(S) not much larger that κ2(W ).


Concluding Remarks

Deflation procedure extends to nonsymmetric quadratics.

First attempt at defining an SPT with a well-defined

action.

Deflation procedure finds application in

second-order model reduction,

model updating with no spill-over.

For papers and Eprints,




Documents

Recent Advances in the Numerical Solution of Quadratic ...ftisseur/talks/talk_ala09.pdf · Appears in many practical applications. Recent work on quadratization: convert degree ℓ