institut für atheon m mathematics for key technologies ...b58/talks/mehrmann.pdfdfg research center...

DFG Research Center MATHEON Institut für

Mathematics for key technologies Mathematik

Staircase forms for structured matrix pencils (and matrix polynomials)

Volker MehrmannTU Berlin, Inst. f. Mathematik

http://www.math.tu-berlin.de/˜mehrmann/

Work supported by MATHEON, the DFG research centerMathematics for key technologies in Berlin.

Joint work with Ralph Byers and Hongguo Xualso results by Christian Schröder and Lena Wunderlich

Volker Mehrmann Structured Staircase Forms 1

Overview

• Notation.

• (Structured) staircase forms: Why bother?

• Applications.

• Structured canonical form for even/palindromic pencils.

• Structured staircase form for even/palindromic pencils.

• Computational issues.

• Conclusion and open problems.


Notation

Definition A matrix pencil of the form

λN +M, M,N ∈ Cn,n

is called

• even if λN +M = (−λN +M)H, i.e. M = MH and N = −NH;

• palindromic if λN +M = (N + λM)H, i.e., M = NH.

Definition Let J =

[

0 I−I 0

]

. A matrix M is called

• Hamiltonian if (JM) = (JM)H;

• symplectic if MHJM = M.

Analogous definitions for the real and complex T case.


Observations:

Even matrix pencils generalize Hamiltonian matrices

Palindromic matrix pencils (almost) generalize symplectic matrices.

The generalized Cayley transformation C (λN +M) = µ(M−N)+(M +N)of an even pencil is palindromic and that of a palindromic pencil is even.

Generalization to even and palindromic matrix polynomials,Mackey2/Mehl/M. 2006 (IWASEP V).


(Structured) staircase forms. Why bother?

• Numerical methods to compute reliable information aboutJordan/Kronecker structure, sizes of blocks, eigenvalues, deflating andreducing spaces.

• Error and condition estimates.

• Deflation of singular blocks, and blocks associated with eigenvalues∞,0 in the even case, 1,−1 in the palindromic case.


General Philosophy in Numerical Analysis

A numerical algorithm should

• be numerically (backward) stable, i.e. the computed solution isthe exact solution of a nearby problem;

• be as efficient as possible, i.e. for dense eigenvalue problemsthe complexity should be O(n3) or better;

• reflect the structure of the physical problem and the mathematical model,i.e. in our case preserve the even or palindromic structure;

• be as accurate as possible even in the extreme (ill-conditioned) cases.


Applications.

• Singularities (cracks) in anisotropic materials as functions of material orgeometry parameters. Apel/M./Watkins 2002.

• Optimal control of continuous and discrete time systems, M. 1991.

• Optimal control of variable coefficient systems, Kunkel/M. 2006.

• Robust H∞ control, Benner/Byers/M./Xu 2005.

• Optimal waveguide design, Schmidt/Friese/Zschiedrich/Deuflhard 2003.

• Resonance phenomena in tracks excited by high speed trains,Hilliges/Mehl/M. 2005.

• Passivity checking, and passivization of reduced order models inelectrical field computation.Recent project with company CST in Darmstadt.


Optimal L2 control for variable coefficient DAEs.

In the γ-iteration for the solution of the continuous time optimal H∞ controlproblem one has to solve a boundary value problem with a pencil of matrixvalued functions

λ

0 E(t) 0−ET(t) 0 0

0 0 0

+

0 A(t) B(t)(A+ E)T(t) Q(t) S(t)

BT(t) ST(t) R(t)

Q = QT ,R = RT .

Goal: Determine nullspaces, and singular subspaces, index as functions of t.Structure preserving methods, Lena Wunderlich 2006.


Robust H∞ control.

In the γ-iteration for the solution of the continuous time optimal H∞ controlproblem one has to solve the parametric even eigenvalue problem

λ

0 E 0−ET 0 0

0 0 0

+

0 A BAT Q SBT ST R(γ)

for a sequence of parameters γ. (Blocks are real.)

Goal: Determine γ ∈ [0,∞], where there is a multiple eigenvalue on theimaginary axis, the multiplicity at ∞ changes or the pencilbecomes singular.


Passivity enforcing. Project with company CST (Darmstadt)

Rational matrix valued function Y (s) (admittance matrix), arising from reducedorder modelling of network equations including electrical field computation.

A network with admittance matrix Yp(s), depending on severalparameters p is called passive if Yp(s) = Y H

p (s) and if zH(Yp(s)H +Yp(s))z ≥ 0for all s ∈ C and all z ∈ C

n.

To check passivity, one performs a minimal realization

Yp(s) = C(p)(sI−A(p))−1B(p)+D(p)

and one computes the purely imaginary eigenvalues, infinite eigenvaluesand singular blocks of the even pencil

λ

0 I 0−I 0 00 0 0

+

0 A(p) B(p)AT(p) 0 CT(p)BT(p) C(p) D(p)+D(p)T

for sets of parameters.


Properties of even (palindromic) pencils

Proposition Consider a real even pencil λN +M.Then (λN +M)x = 0 if and only if xT(−λN +M) = 0,i.e., the eigenvalues occur in pairs λ, −λ (Hamiltonian spectrum.)

Consider a real palindromic pencil λAT +A.Then (λAT +A)x = 0 if and only if xT(AT +λA) = 0,i.e., the eigenvalues occur in pairs λ, 1/λ (symplectic spectrum.)

• In many applications the current methods fail near the extreme cases.

• These are the eigenvalues on or near the imaginary axis(unit circle) and in particular the eigenvalues 0 and ∞, (1 and −1).

• The (structured) perturbation theory near the extreme cases is mostly open,Results of Ran/Rodman 1988, Godunov/Sadkane 1997 imply thatnon-structured methods will usually fail.


Structure preserving equivalence transformations

To preserve the even(palindromic) structure, we use congruence transformations

λN + M = λUT NU +UT MU,

λAT + A = λUT ATU +UT AU,

with nonsingular (unitary) U .

What is the structured canonical form under this transformation?

What is the structured condensed form under unitary transformations?

Can we compute it numerically?


A structured Kronecker form for even pencils

Theorem: Thompson 1991 If N, M ∈ Rn,n with N = −NT ,M = MT , then thereexists a nonsingular matrix X ∈ C

n,n such that

XT (λN +M)X = diag(BS ,BI ,BZ,BF ),

is in structured Kronecker form, where

BS = diag(Oη,Sξ1, . . . ,Sξk

),

BI = diag(

I2ε1+1, . . . ,I2εl+1,I2δ1, . . . ,I2δm

)

,

BZ = diag(

Z2σ1+1, . . . ,Z2σr+1,Z2ρ1, . . . ,Z2ρs

)

,

BF = diag(Rφ1, . . . ,Rφt,Cψ1, . . . ,Cψu)

This structured Kronecker canonical form is unique up to permutation of theblocks, i.e., the kind, size and number of the blocks as well as the signcharacteristics are characteristic of the pencil λN +M.

The blocks have the following properties.


1. Oη = λ0η +0η;

2. Each Sξ j is a (2ξ j +1)× (2ξ j +1) block that combines a right singular blockand a left singular block, both of minimal index ξ j. It has the form

λ

1 0. . . . . .

1 0−1

. . . 0−1 . . .

0

+

0 1. . . . . .

0 10

. . . 10 . . .

1

;

3. Each I2ε j+1 is a (2ε j +1)× (2ε j +1) block that contains a single blockcorresponding to the eigenvalue ∞ with index 2ε j +1. It has the form

λ

1 0. . . . . .

1 0−1 0

. . . 0−1 . . .

0

+

0 1. . . . . .

0 10 s

. . . 10 . . .

1

,

where s ∈ 1,−1 is the sign-index or sign-characteristic of the block;


4. Each I2δ j is a 4δ j ×4δ j block that combines two 2δ j ×2δ j infiniteeigenvalue blocks of index δ j. It has the form

λ

1 0. . . . . .

1 . . .

0−1 0

. . . . . .

−1 . . .

0

+

1. . .

11

. . .

1

;

5. Each Z2σ j+1 is a (4σ j +2)× (4σ j +2) block that combines two(2σ j +1)× (2σ j +1) Jordan blocks corresponding to the eigenvalue 0.It has the form

λ

1. . .

1−1

. . .

−1

+

1 0. . . . . .

1 . . .

01 0

. . . . . .

1 . . .

0

;


6. Each Z2ρ j is a 2ρ j ×2ρ j block that contains a single Jordan blockcorresponding to the eigenvalue 0. It has the form

λ

1. . .

1−1

. . .

−1

+

1 0. . . . . .

1 . . .

s 01 0

. . . . . .

1 . . .

0

,

where s ∈ 1,−1 is the sign characteristic of this block;7. Each Rφ j is a 2φ j ×2φ j block that combines two φ j ×φ j Jordan blockscorresponding to nonzero real eigenvalues a j and −a j. It has the form

λ

1. . .

1−1

. . .

−1

+

1 a j. . . . . .

1 . . .

a j

1 a j. . . . . .

1 . . .

a j

.


8. a. Either Cψ j is a 2ψ j ×2ψ j block combining two ψ j ×ψ j Jordan blockswith purely imaginary eigenvalues ib j,−ib j (b j > 0). It has the form

λ

1. . .

1−1

. . .

−1

+ s

1 b j. . . . . .

1 . . .

b j

1 b j. . . . . .

1 . . .

b j

,

where s ∈ 1,−1 is the sign characteristic.8. b. or Cψ j is a 4ψ j ×4ψ j block combining ψ j ×ψ j Jordan blocks for each

of the complex eigenvalues a j + ib j,a j − ib j,−a j + ib j,−a j − ib j(with a j 6= 0 and b j 6= 0). In this case it has form

λ

Ω. . .

Ω−Ω

. . .

−Ω

+

Ω Λ j. . . . . .

Ω . . .

Λ j

Ω Λ j. . . . . .

Ω . . .

Λ j

with Ω =

[

0 11 0

]

and Λ j =

[

−b j a j

a j b j

]

.


A structured Kronecker form for palindromic pencils

Theorem: Schröder, Horn/Sergejchuk 2006If A ∈ R

n,n, then there exists a nonsingular matrix X ∈ Rn,n such that

λXT AT X +XT AX = diag(λA1 +AT1 , . . . ,λAT

` +A`)

is in structured Kronecker form.

This structured Kronecker canonical form is unique up to permutation of theblocks, i.e., the kind, size and number of the blocks as well as the signcharacteristics are characteristic of the pencil λAT +A.

The blocks have the following forms:


Sp =

00p+1 ... 1

0 ...

11 0

... ... 0p

1 0

∈ R2p+1,2p+1, p ∈ N0;

L1,p(λ) =

λ0p ... 1

... ...

λ 11

...

... 0p

1

∈ R2p,2p,

where p ∈ N,λ ∈ R, |λ| < 1;


L2,p(α,β) =

Λ02p ... I2

... ...

Λ I2

I2

...

... 02p

I2

∈ R4p,4p,

where p ∈ N,Λ =

[

α −ββ α

]

,α,β ∈ R\0,β < 0, |α+ iβ| < 1;

σU1,p = σ

10b p

2c... 1

1 ...

1 11

... 0b p2c

1

∈ Rp,p, where p ∈ N is odd, σ ∈ 1,−1;


U2,p = Lp(1) =

10p ... 1

... ...

1 11

...

... 0p

1

∈ R2p,2p,

where p ∈ N is even;

U3,p = Lp(−1) =

−10p ... 1

... ...

−1 11

...

... 0p

1

∈ R2p,2p, where p ∈ N is odd;


σU4,p = σ

−10 p

2... 1

... ...

−1 11 1

...

...

1

∈ Rp,p,

where p ∈ N is even, σ ∈ 1,−1;


σU5,p(α,β) = σ

Λ02b p

2c... I2

Λ ...

Λ12 I2

I2

... 02b p2c

I2

∈ R2p,2p,

where p ∈ N is odd, |α+ iβ| = 1,β < 0,Λ =

[

α −ββ α

]

,

Λ12 is defined as rotation matrix with rotation angle φ

2 ∈ (0,π),where φ = arctan(β

α) is the rotation angle of the rotation matrix Λ, σ ∈ 1,−1;


σU6,p(α,β) = σ

Λ02 p

2... I2

... ...

Λ I2

I2 I2

...

...

I2

∈ R2p,2p,

where p ∈ N is even, |α+ iβ| = 1,β < 0,Λ =

[

α −ββ α

]

,σ ∈ 1,−1.

Similar results als in complex ∗ and complex T case.

Corollary: Exact characterization when a symplectic matrix S has afactorization S = A−T A in terms of dimensions of blocks associatedwith eigenvalues 1 and −1. Schröder 2006.


Consequences of even (palindromic) Kronecker form

• The even (palindromic) Kronecker forms are nice theoretical results.

• The transformation matrix X may be arbitrarily ill-conditioned.

• The even (palindromic) Kronecker cannot be computed well with finiteprecision algorithms.

• The information given in the even (palindromic) Kronecker formis essential for the understanding of the computational problems.

• We need alternatives, from which we can derive the information, that allowsthe deflation of singular blocks and blocks associated with 0,∞ (1 and −1).

We need structured staircase forms!


Why not just use standard equivalence λQNU +QMU ?

Example Consider a 3×3 even pencil with matrices

N = Q

0 1 0−1 0 00 0 0

QT , M = Q

0 0 10 1 01 0 0

QT ,

where Q is a random real orthogonal matrix. The pencil is congruent to

λ

0 1 00 0 10 0 0

−

1 0 00 1 00 0 1

It has a triple eigenvalue ∞ with geom. multiplicity 1 and algebr. multiplicity 3.

For different randomly generated orthogonal matrices Q the QZ algorithmin MATLAB produced all variations of eigenvalues that are possible in a general3×3 pencil.


A structured staircase form for even pencils, Byers/M./Xu 2005

For λN +M with N = −NT ,M = MT ∈ Rn,n, there exists a real orthogonalmatrix U ∈ R

n,n, such that

UT NU =

N11 . . . . . . N1,m N1,m+1 N1,m+2 . . . N1,2m 0... . . . ... ... ... ... ...... . . . ... ... Nm−1,m+2 ...

−NT1,m · · · · · · Nm,m Nm,m+1 0

−NT1,m+1 . . . . . . −NT

m,m+1 Nm+1,m+1

−NT1,m+2 · · · −NT

m−1,m+2 0... ... ...

−NT1,2m ...

0

n1......

nm

lqm...

q2

q1

UT MU =

M11 · · · · · · M1,m M1,m+1 M1,m+2 . . . . . . M1,2m+1... . . . ... ... ... ...... . . . ... ... ... ...

MT1,m . . . . . . Mm,m Mm,m+1 Mm,m+2

MT1,m+1 . . . . . . MT

m,m+1 Mm+1,m+1

MT1,m+2 . . . . . . MT

m,m+2... ...... ...

MT1,2m+1

n1......

nm

lqm......

q1

,

where q1 ≥ n1 ≥ q2 ≥ n2 ≥ . . . ≥ qm ≥ nm,

N j,2m+1− j ∈ Rn j,q j+1, 1 ≤ j ≤ m−1,

Nm+1,m+1 =

[

∆ 00 0

]

, ∆ = −∆T ∈ R2p,2p,

M j,2m+2− j =[

Γ j 0]

∈ Rn j,q j, Γ j ∈ R

n j,n j, 1 ≤ j ≤ m,

Mm+1,m+1 =

[

Σ11 Σ12

ΣT12 Σ22

]

, Σ11 = ΣT11 ∈ R

2p,2p, Σ22 = ΣT22 ∈ R

l−2p,l−2p,

and the blocks Σ22 and ∆ and Γ j, j = 1, . . . ,m are nonsingular.

The middle block

The middle block

λNm+1,m+1 +Mm+1,m+1 = λ[

∆ 00 0

]

+

[

Σ11 Σ12

ΣT12 Σ22

]

,

contains all the blocks associated with finite eigenvalues and 1×1 blocksassociated with the eigenvalue ∞.

With this, all the dynamics and constraints can be identified.


—————————————————————————

The finite spectrum in the middle block

The finite spectrum of is obtained from the even pencil

λ∆+Σ = λ∆+(Σ11−Σ12Σ−122 ΣT

12)

with ∆ invertible.

The matrix ∆ has a skew-Cholesky factorization ∆ = LJLT , with

J =

[

0 I−I 0

]

,

Bunch 78, Benner/et al 2000.

Thus, the spectral information can be obtained from the Hamiltonian matrix

H = JL−1ΣL−T .


What can we do with the staircase form

• All the information about the invariants (Kronecker indices) can be read off.Formulas are given in Byers/M./Xu 2005.

• Singularities and high order blocks to the eigenvalue ∞ can be deflated off.

• The best treatment of infinite eigenvalue in the middle blockλNm+1,m+1 +Mm+1,m+1 is unclear.Is the use of skew-Cholesky better than projecting out the nullspacewith unitary (symplectic) transformations?

• The Hamiltonian part associated with the finite eigenvalues in the middle blockcan be treated with the structured methods for Hamiltonian problemsBenner/M./Xu 1998, Byers/Benner/M./Xu 2002, Chu/Liu/M. 2004.See Benner/Kressner’s HAPACK.


Computational procedure

• The procedure consists of a recursive sequence of singular valuedecompositions.

• The staircase form essentially determines a least generic even pencilwithin the rounding error cloud surrounding λN +M.


Computational difficulties

• Rank decisions face the usual difficulties and have to be adapted to therecursive procedure.

• Similar difficulties as in standard staircase form, GUPTRIDemmel/Kågström 1993.

• What to do in case of doubt? In applications, assume worst case, seeMattheij/Wijckmans 1998.

• Perturbation analysis is essentially open for singular and higherorder blocks associatedwith ∞.

• Perturbation theory for Hamiltonian matrices, Ran/Rodman 1988,Godunov/Sadkane 1997.

• Extension to regular even pencils Bora/M. 2005.


Example revisited

Our MATLAB implementation of the structured staircase Algorithm determinedthat in the cloud of rounding-error-small perturbations of each even λN +M,there is an even pencil with structured staircase form

λ

0 1 0−1 0 0

0 0 0

−

0 0 10 1 01 0 0

,

with one block I3 with sign-characteristic 1.

The algorithm successfully located a least generic even pencil within the cloud.


Conclusions

• Palindromic/even matrix pencils appear in many applications.

• Palindromic/even matrix polynomials can be linearized topalindromic/even matrix pencils, Mackey2/Mehl/M. 2006.

• Singular and high index blocks should be deflated in a structured way.

• There exist structure preserving staircase forms containing all the information.

• This can be used to deflate all the infinite and singular blocks.

• After deflation, the structured methods for Hamiltonian eigenvalue problemscan be applied.


Open problems

• Structured perturbation theory mostly open.

• Rank decisions in process.

• Practical implemention of structured staircase algorithms. Schröder Phd

• Extension of deflation procedures to structured matrix polynomialswithout linearization. Schröder Phd

• Structured tuples of matrix functions. Wunderlich PhDNonstructured case M./Shi 2006.


Recent References

• R. Byers, V. Mehrmann and H. Xu. A structured staircase algorithm forskew-symmetric/symmetric pencils.MATHEON PREPRINT url: http://www.matheon.de/ 2005

• R. Horn and V. Sergeichuk. Canonical forms for complex congruence and∗ congruence. To appear in Lin. Alg. Appl. 2006.

• D.S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann. PalindromicPolynomial Eigenvalue Problems: Good Vibrations fromGood Linearizations. To appear in SIMAX 2006.

• V. Mehrmann und C. Shi. Transformation of high order lineardifferential-algebraic systems to first order.Numerical Algorithms 2006 to appear.

• C. Schröder. A canonical form for palindromic pencils and palindromicfactorizations, MATHEON PREPRINT , url: http://www.matheon.de/ 2006

• L. Wunderlich. Structure preserving canonical forms for pairs of Hermitianmatrices and matrix functions.MATHEON PREPRINT url: http://www.matheon.de/ 2006.

institut für atheon m mathematics for key technologies ...b58/talks/mehrmann.pdfdfg research center...

Documents