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Structured condition numbers for eigenvalue problems D. Kressner ETH Zurich, Seminar for Applied Mathematics The Second Najman Conference, Dubrovnik, 12.05.2009 Matrices Matrix Pencils Matrix Polynomials Example Complex skew-symmetric

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Structured condition numbers for eigenvalue problems

  • D. Kressner

ETH Zurich, Seminar for Applied Mathematics

The Second Najman Conference, Dubrovnik, 12.05.2009

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Matrices Matrix Pencils Matrix Polynomials

Example

Complex skew-symmetric matrix: A =   1 − ϕ −1 + ϕ i −i   , 0 ≤ ϕ ≤ 1. Eigenvalues of A: 0,

  • 2ϕ − ϕ2, −
  • 2ϕ − ϕ2.

For ϕ = 0, zero is a triple eigenvalue with geometric mult. 1.

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  • p. 2
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Matrices Matrix Pencils Matrix Polynomials

Example: (Un)structured pseudospectra, ϕ = 0.01

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

General perturbations

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

Skew-symmetric perturbations (Structured pseudospectra based on [Karow’07, Karow/K.’09].)

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  • p. 3
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Matrices Matrix Pencils Matrix Polynomials

Example: (Un)structured pseudospectra, ϕ = 0

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

General perturbations

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

Skew-symmetric perturbations (Structured pseudospectra based on [Karow’07, Karow/K.’09].)

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Matrices Matrix Pencils Matrix Polynomials

Overview

Structured condition numbers for . . . . . . simple eigenvalues of matrices . . . multiple eigenvalues of matrices . . . multiple eigenvalues of matrix pencils . . . simple eigenvalues of matrix polynomials and linearizations Motivation: structured stability radii; evaluate benefits/limitations of structure-preserving numerical methods; a posteriori estimates for structured subspace projection methods.

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  • p. 5
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Matrices Matrix Pencils Matrix Polynomials

Simple Eigenvalues

  • f Matrices
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  • p. 6
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Matrices Matrix Pencils Matrix Polynomials

Simple eigenvalues

Setting: λ simple eigenvalue of A ∈ Cn×n; x right eigenvector, y left eigenvector; normalized such that x, y = 1; ˆ λ eigenvalue of perturbed A + ǫE with ǫ ≪ 1. Well-known perturbation expansion: ˆ λ = λ + (yHEx) ǫ + O(ǫ2)

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Matrices Matrix Pencils Matrix Polynomials

Simple eigenvalues

Condition number of λ: κ(λ) := lim

ǫ→0+

1 ǫ sup

  • |ˆ

λ − λ| : E ∈ Cn×n, E ≤ 1

  • Pert. expansion

⇒ = sup

  • |yHEx| : E ∈ Cn×n, E ≤ 1
  • =

x2 y2 provided that · is unitarily invariant.

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  • p. 8
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Matrices Matrix Pencils Matrix Polynomials

Simple eigenvalues

Condition number of λ: κ(λ) := lim

ǫ→0+

1 ǫ sup

  • |ˆ

λ − λ| : E ∈ Cn×n, E ≤ 1

  • Pert. expansion

⇒ = sup

  • |yHEx| : E ∈ Cn×n, E ≤ 1
  • =

x2 y2 provided that · is unitarily invariant. Structured condition number w.r.t. smooth manifold S ⊂ Cn×n κ(λ, S) := lim

ǫ→0+

1 ǫ sup

  • |ˆ

λ − λ| : A + E ∈ S, E ≤ 1

  • Pert. expansion

⇒ = sup

  • |yHEx| : E ∈ TAS, E ≤ 1
  • Explicit formulas for various S listed, e.g., in

[Karow/K./Tisseur’06].

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  • p. 8
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Matrices Matrix Pencils Matrix Polynomials

S = complex skew-symmetric matrices

κ(λ) = x2y2 κ(λ, S) =

  • x2y2 − |yT x|2

For intro example:

10

−3

10

−2

10

−1

10 10 10

1

10

2

10

3

φ unstructured cond.

  • struct. cond.

Rump’s bound on struct.cond.

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Matrices Matrix Pencils Matrix Polynomials

Multiple Eigenvalues

  • f Matrices
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Matrices Matrix Pencils Matrix Polynomials

Example: Λ     1 1 ǫ     =

  • e−2πij/3ǫ1/3 : j = 0, 1, 2
  • Eigenvalues not Lipschitz continuous, only Hölder continuous

[Langer/Najman’89 and’92]: Puiseux expansions for perturbed eigenvalues of analytic matrix functions. [Vishik/Lyusternik’60, Lidskii’65]: Cover special case of A − λI; summarized and extended in [Moro/Burke/Overton’97].

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Matrices Matrix Pencils Matrix Polynomials

Puiseux expansions in a nutshell

λ multiple eigenvalue of A ∈ Cn×n; n1 size of largest Jordan block. Ordered Jordan decomposition P−1AP = Jλ ∗

  • ,

s.t. Jλ gathers all r1 Jordan blocks of size n1 belonging to λ. P =

  • x1

n1−1 cols

∗ · · · ∗

  • · · ·
  • xr1

n1−1 cols

∗ · · · ∗

  • ∗
  • P−H

= n1−1 cols ∗ · · · ∗ y1

  • · · ·
  • n1−1 cols

∗ · · · ∗ yr1

  • ∗
  • .

Then X = [x1 · · · xr1], Y = [y1 · · · yr1] collect all eigenvectors belonging to largest Jordan blocks.

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Matrices Matrix Pencils Matrix Polynomials

Puiseux expansions in a nutshell, ctd.

A perturbation A + ǫE causes the n1r1 copies of λ sitting in the largest Jordan blocks bifurcate into λ + (ξk)1/n1ǫ1/n1 + o(ǫ1/n1), where ξk are the eigenvalues of Y HEX. See, e.g., [Langer/Najman’92, Moro/Burke/Overton’97]. Remarks: Other copies of λ bifurcate into λ + o(ǫ1/n1). Some or even all eigenvalues of Y HEX can be zero!

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Matrices Matrix Pencils Matrix Polynomials

Meaning of Y HEX

If A was already in Jordan form . . . A + ǫE =      

λ 1 λ 1 λ λ 1 λ 1 λ ∗

      + ǫ Y HEX =

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Matrices Matrix Pencils Matrix Polynomials

Unstructured Hölder condition number κ(λ) = (n1, α) where n1 = size of largest Jordan block α = sup

E∈Cn×n E2≤1

ρ(Y HEX) (ρ denotes spectral radius)

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Matrices Matrix Pencils Matrix Polynomials

Unstructured Hölder condition number κ(λ) = (n1, α) where n1 = size of largest Jordan block α = sup

E∈Cn×n E2≤1

ρ(Y HEX) (ρ denotes spectral radius) ρ(Y HEX) = ρ(EXY H) ≤ E2 XY H2 = XY H2

  • α ≤ XY H2,

E0 = YX H/(XY H2), ρ(Y HE0X) = XY H2

  • α ≥ XY H2,

α = XY H2.

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Matrices Matrix Pencils Matrix Polynomials

Unstructured Hölder condition number κ(λ) = (n1, α) where n1 = size of largest Jordan block, α = XY H2 Structured Hölder condition number αS = sup

E∈TAS E2≤1

ρ(Y HEX) If αS = 0, all perturbation expansions take the form λ + o(ǫ1/n1). If αS > 0, define structured Hölder condition number as κ(λ, S) = (n1, αS). Difficulty: Evaluation of αS.

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Matrices Matrix Pencils Matrix Polynomials

A useful class of perturbations

u, v := left/right singular vectors belonging to largest s.v. of XY H. Let E0 ∈ Cn×n such that E0u = βv, β ∈ C, |β| = 1. Then ρ(E0XY H) ≥ XY H2. Sketch of proof: ρ(E0XY H) = ρ(E0UΣV H) = ρ(V HE0UΣ) = ρ

  • βXY H2

∗ ∗

  • ≥ XY H2.
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Matrices Matrix Pencils Matrix Polynomials

A useful class of perturbations, ctd.

Structured Jordan form for A ∈ S.

(see [Thompson’91], [Mehl’06], ...)

⇓ ⇓ ⇓ Induced structure in XY H. ⇓ ⇓ ⇓ Mapping theorems: E0u = βv with |β| = 1, E02 ≈ 1, E0 ∈ TAS.

(see [Rump’03], [Mackey/Mackey/Tisseur’06], ...)

⇓ ⇓ ⇓ Unstructured ≈ structured Hölder condition number Works well for complex Toeplitz, Hankel, persymmetric, Hermitian, symmetric, Hamiltonian, skew-Hamiltonian: κ(λ) = κ(λ, S).

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  • p. 18
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Matrices Matrix Pencils Matrix Polynomials

S = complex skew-symmetric matrices

Case I: Nonzero eigenvalue, r1 = 1. κ(λ, S) =

  • n1,
  • X2Y2 − |Y TX|2
  • Case IIa: Zero eigenvalue, r1 = 1, n1 odd.

Y = X Y HEX = 0 for all E ∈ S αS = 0. Case IIb: Zero eigenvalue, r1 > 1, n1 odd. κ(λ, S) = (n1, √σ1σ2), where σ1, σ2 are the two largest singular values of XX T. Case III: Zero eigenvalue, n1 even. κ(λ) = κ(λ, S).

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Matrices Matrix Pencils Matrix Polynomials

Multiple Eigenvalues

  • f Matrix Pencils
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Matrices Matrix Pencils Matrix Polynomials

Puiseux expansions for matrix pencils

[Langer/Najman’89–93]: Eigenvalue perturbation expansion for analytic matrix functions. [de Terán, Dopico, Moro’07]: Results by Langer/Najman for A − λB in terms of Kronecker-Weierstraß form. QAP = Jλ ∗

  • ,

QBP = I ∗

  • ,

where Jλ gathers all r1 Jordan blocks of largest size n1 belonging to λ. Partition P =

  • x1

n1−1 cols

∗ · · · ∗

  • · · ·
  • xr1

n1−1 cols

∗ · · · ∗

  • ∗
  • QH

= n1−1 cols ∗ · · · ∗ y1

  • · · ·
  • n1−1 cols

∗ · · · ∗ yr1

  • ∗
  • ,

and set X = [x1 · · · xr1], Y = [y1 · · · yr1].

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Matrices Matrix Pencils Matrix Polynomials

Puiseux expansions for matrix pencils, ctd.

Perturbation (A, B) → (A+ǫE, B+ǫF) yields perturbed eigenvalues λ + (ξk)1/n1ǫ1/n1 + o(ǫ1/n1), where ξk are eigenvalues of Y H(E − λF)X. Unstructured Hölder condition number κ(λ) = (n1, α), where α = sup

E,F∈Cn×n E2≤1,F2≤1

ρ(Y H(E − λF)X) = (1 + |λ|)XY H2. Structured Hölder condition number αS analogously defined.

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Matrices Matrix Pencils Matrix Polynomials

Palindromic matrix pencils, S = {(A, AT) : A ∈ Cn×n}

Structured canonical forms in [Horn/Sergeichuk’06], [Rodman’06], [Schröder’06]. Case Ia: λ = 1 finite and r1 = 1; αS = |1 − λ| X2Y2 + |1 + λ|

  • X2

2Y2 2 − |Y TX|2.

Case Ib: λ = 1 finite and r1 > 1; |1 − λ| 1 + |λ|α ≤ αS ≤ α. Case II: λ = ∞; αS = α.

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Matrices Matrix Pencils Matrix Polynomials

Palindromic matrix pencils, ctd.

Case IIIa: λ = 1, r1 = 1, and n1 odd; αS = 0. Case IIIb: λ = 1, r1 > 1, and n1 odd; αS = 2√σ1σ2, where σ1, σ2 are the two largest singular values of XY H. Case IIIc: λ = 1, and n1 even; αS = α.

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Matrices Matrix Pencils Matrix Polynomials

Matrix Polynomials and Linearizations

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Matrices Matrix Pencils Matrix Polynomials

Simple eigenvalues of matrix polynomials

Consider matrix polynomial (here only of degree 2) P(λ) = A0 + λA1 + λ2A2, with simple eigenvalue λ and right/left eigenvectors x, y. Perturbed matrix polynomial P(λ) → (P+△P)(λ) = A0+ǫE0 + λ(A1+ǫE1) + λ2(A2+ǫE2) has perturbed eigenvalue ˆ λ with perturbation expansion ˆ λ = λ − 1 yHP′(λ)x yH△P(λ)x ǫ + O(ǫ2).

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Matrices Matrix Pencils Matrix Polynomials

Unstructured condition number With △P :=

  • E02, E12, E22
  • 2, define

κP(λ) = 1 |yHP′(λ)x| sup

  • |yH△P(λ)x| : △P ≤ 1
  • =

1 |yHP′(λ)x|

  • 1 + |λ| + |λ|2x2y2.

Structured condition number w.r.t. S ⊂ Cn×n × Cn×n × Cn×n: κP(λ, S) = 1 |yHP′(λ)x| sup

  • |yH△P(λ)x| : △P ∈ TPS, △P ≤ 1
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Matrices Matrix Pencils Matrix Polynomials

Example: Palindromic matrix polynomials

Matrix polynomials of the form P(λ) = A0 + λA1 + λ2AT

0 ,

A1 = AT

1

P(λ) = A0 + λA1 + λ2AT

1 + λ3AT

. . . are called palindromic. Explicit expressions for structured condition number in [Adhikari/Alam/K.’09]. Example: P(λ) =

  • 1

1 − φ −1 + φ 1 i −i 1

  • + λI + λ2I − λ3
  • 1

1 − φ −1 + φ 1 i −i 1

  • with 0 < φ < 1. As φ → 0:

κP(λ) ≫ κP(λ, palindromic) for λ ≈ −1.

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Matrices Matrix Pencils Matrix Polynomials

Linearizations

Given P(λ) = A0 + λA1 + λ2A1 + λ3A0, many possibilities for linearization: Lc(λ) = companion linearization L1(λ) =

  • A0

A0 A1 − AT A0 − AT

1

AT

1

A1 − AT A0

  • + λ
  • AT

AT

1 − A0

A1 AT

0 − A1

AT

1 − A0

AT AT

  • L2(λ)

=

  • A0

A1 A0 −AT

  • + λ
  • −A0

AT AT

1

AT

  • Which one should be chosen?
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Matrices Matrix Pencils Matrix Polynomials

Choice of Linearization

[Adhikari/Alam/K.’09] provides recipes to choose structured linearization L s.t. κL(λ, structure) ≈ κP(λ, structure) accuracy benefits from structure are preserved.

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Matrices Matrix Pencils Matrix Polynomials

Choice of Linearization

[Adhikari/Alam/K.’09] provides recipes to choose structured linearization L s.t. κL(λ, structure) ≈ κP(λ, structure) accuracy benefits from structure are preserved. For palindromic example: If 1/ √ 2 ≤ |λ| ≤ √ 2 choose L2. Otherwise choose L1. Then κL(λ, palindromic) ≤ 8 √ 2 κP(λ, palindromic).

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Matrices Matrix Pencils Matrix Polynomials

Conclusions

In many cases, unstructured ≈ structured condition number. Notable exceptions are skew-symmetric matrices (λ ≈ 0) and palindromic matrix pencils/polynomials. Structured linearizations can be chosen to preserve structured condition numbers. This talk incomplete summary of joint work with several others:

  • B. Adhikari, R. Alam, and D. Kressner. Structured eigenvalue condition numbers

and linearizations for matrix polynomials. Technical report 2009-01, Seminar for applied mathematics, ETH Zurich. 2009.

  • M. Karow, D. Kressner, and F. Tisseur. Structured eigenvalue condition numbers.

SIAM J. Matrix Anal. Appl., 28(4):1052-1068, 2006.

  • D. Kressner, M. J. Peláez, and J. Moro. Structured Hölder condition numbers for

multiple eigenvalues. SIAM J. Matrix Anal. Appl., 31(1):175–201, 2009.

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  • p. 31