Structured condition numbers for eigenvalue problems D. Kressner - PowerPoint PPT Presentation

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 matrix:   0 1 − ϕ 0 A =  , − 1 + ϕ 0 0 ≤ ϕ ≤ 1 . i  0 − i 0 Eigenvalues of A : � � 0 , 2 ϕ − ϕ 2 , − 2 ϕ − ϕ 2 . For ϕ = 0, zero is a triple eigenvalue with geometric mult. 1. D. Kressner p. 2

Matrices Matrix Pencils Matrix Polynomials Example: (Un)structured pseudospectra, ϕ = 0 . 01 0.5 0.5 0.4 0.045 0.4 0.045 0.04 0.04 0.3 0.3 0.2 0.035 0.2 0.035 0.1 0.03 0.1 0.03 0 0.025 0 0.025 −0.1 0.02 −0.1 0.02 −0.2 −0.2 0.015 0.015 −0.3 0.01 −0.3 0.01 −0.4 −0.4 0.005 0.005 −0.5 −0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 General perturbations Skew-symmetric perturbations (Structured pseudospectra based on [Karow’07, Karow/K.’09].) D. Kressner p. 3

Matrices Matrix Pencils Matrix Polynomials Example: (Un)structured pseudospectra, ϕ = 0 0.5 0.5 0.4 0.045 0.4 0.045 0.04 0.04 0.3 0.3 0.2 0.035 0.2 0.035 0.1 0.03 0.1 0.03 0 0.025 0 0.025 −0.1 0.02 −0.1 0.02 −0.2 −0.2 0.015 0.015 −0.3 0.01 −0.3 0.01 −0.4 −0.4 0.005 0.005 −0.5 −0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 General perturbations Skew -symmetric perturbations (Structured pseudospectra based on [Karow’07, Karow/K.’09].) D. Kressner p. 4

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. D. Kressner p. 5

Matrices Matrix Pencils Matrix Polynomials Simple Eigenvalues of Matrices D. Kressner p. 6

Matrices Matrix Pencils Matrix Polynomials Simple eigenvalues Setting: λ simple eigenvalue of A ∈ C n × n ; x right eigenvector, y left eigenvector; normalized such that � x , y � = 1; ˆ λ eigenvalue of perturbed A + ǫ E with ǫ ≪ 1. Well -known perturbation expansion: λ = λ + ( y H Ex ) ǫ + O ( ǫ 2 ) ˆ D. Kressner p. 7

Matrices Matrix Pencils Matrix Polynomials Simple eigenvalues Condition number of λ : 1 λ − λ | : E ∈ C n × n , � E � ≤ 1 � � | ˆ κ ( λ ) := lim ǫ sup ǫ → 0 + Pert. expansion | y H Ex | : E ∈ C n × n , � E � ≤ 1 � � ⇒ = sup � x � 2 � y � 2 = provided that � · � is unitarily invariant. D. Kressner p. 8

Matrices Matrix Pencils Matrix Polynomials Simple eigenvalues Condition number of λ : 1 λ − λ | : E ∈ C n × n , � E � ≤ 1 � � | ˆ κ ( λ ) := lim ǫ sup ǫ → 0 + Pert. expansion | y H Ex | : E ∈ C n × n , � E � ≤ 1 � � ⇒ = sup � x � 2 � y � 2 = provided that � · � is unitarily invariant. Structured condition number w.r.t. smooth manifold S ⊂ C n × n 1 � λ − λ | : A + E ∈ S , � E � ≤ 1 � | ˆ κ ( λ, S ) := lim ǫ sup ǫ → 0 + Pert. expansion � � | y H Ex | : E ∈ T A S , � E � ≤ 1 ⇒ = sup Explicit formulas for various S listed, e.g., in [Karow/K./Tisseur’06]. D. Kressner p. 8

Matrices Matrix Pencils Matrix Polynomials S = complex skew -symmetric matrices � x � 2 � y � 2 κ ( λ ) = � x � 2 � y � 2 − | y T x | 2 � κ ( λ, S ) = For intro example: 3 10 unstructured cond. struct. cond. Rump’s bound on struct.cond. 2 10 1 10 0 10 −3 −2 −1 0 10 10 10 10 φ D. Kressner p. 9

Matrices Matrix Pencils Matrix Polynomials Multiple Eigenvalues of Matrices D. Kressner p. 10

Matrices Matrix Pencils Matrix Polynomials Example:     0 1 0 e − 2 π i j / 3 ǫ 1 / 3 : j = 0 , 1 , 2 � �  = Λ 0 0 1    ǫ 0 0 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 − λ I ; summarized and extended in [Moro/Burke/Overton’97]. D. Kressner p. 11

Matrices Matrix Pencils Matrix Polynomials Puiseux expansions in a nutshell λ multiple eigenvalue of A ∈ C n × n ; n 1 size of largest Jordan block. Ordered Jordan decomposition � J λ � 0 P − 1 AP = , 0 ∗ s.t. J λ gathers all r 1 Jordan blocks of size n 1 belonging to λ . n 1 − 1 cols n 1 − 1 cols � � � � � ∗ � P x 1 � �� � � x r 1 � �� � � � � = ∗ · · · ∗ � · · · ∗ · · · ∗ � n 1 − 1 cols n 1 − 1 cols � � � ∗ � � P − H � �� � ∗ · · · ∗ y 1 ∗ · · · ∗ y r 1 � �� � � � � = � · · · . � Then X = [ x 1 · · · x r 1 ] , Y = [ y 1 · · · y r 1 ] collect all eigenvectors belonging to largest Jordan blocks. D. Kressner p. 12

Matrices Matrix Pencils Matrix Polynomials Puiseux expansions in a nutshell, ctd. A perturbation A + ǫ E causes the n 1 r 1 copies of λ sitting in the largest Jordan blocks bifurcate into λ + ( ξ k ) 1 / n 1 ǫ 1 / n 1 + o ( ǫ 1 / n 1 ) , where ξ k are the eigenvalues of Y H EX . See, e.g., [Langer/Najman’92, Moro/Burke/Overton’97]. Remarks: Other copies of λ bifurcate into λ + o ( ǫ 1 / n 1 ) . Some or even all eigenvalues of Y H EX can be zero! D. Kressner p. 13

Matrices Matrix Pencils Matrix Polynomials Meaning of Y H EX If A was already in Jordan form . . . A + ǫ E =   λ 1 λ 1   λ   λ 1 + ǫ     λ 1 λ ∗   Y H EX = D. Kressner p. 14

Matrices Matrix Pencils Matrix Polynomials Unstructured Hölder condition number κ ( λ ) = ( n 1 , α ) where n 1 = size of largest Jordan block ρ ( Y H EX ) α = sup ( ρ denotes spectral radius) E ∈ C n × n � E � 2 ≤ 1 D. Kressner p. 15

Matrices Matrix Pencils Matrix Polynomials Unstructured Hölder condition number κ ( λ ) = ( n 1 , α ) where n 1 = size of largest Jordan block ρ ( Y H EX ) α = sup ( ρ denotes spectral radius) E ∈ C n × n � E � 2 ≤ 1 ρ ( Y H EX ) = ρ ( EXY H ) ≤ � E � 2 � XY H � 2 = � XY H � 2 α ≤ � XY H � 2 , � E 0 = YX H / ( � XY H � 2 ) , ρ ( Y H E 0 X ) = � XY H � 2 α ≥ � XY H � 2 , � � α = � XY H � 2 . D. Kressner p. 15

Matrices Matrix Pencils Matrix Polynomials Unstructured Hölder condition number κ ( λ ) = ( n 1 , α ) where n 1 = size of largest Jordan block, α = � XY H � 2 Structured Hölder condition number ρ ( Y H EX ) α S = sup E ∈ TAS � E � 2 ≤ 1 If α S = 0, all perturbation expansions take the form λ + o ( ǫ 1 / n 1 ) . If α S > 0, define structured Hölder condition number as κ ( λ, S ) = ( n 1 , α S ) . Difficulty: Evaluation of α S . D. Kressner p. 16

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 E 0 ∈ C n × n such that E 0 u = β v , β ∈ C , | β | = 1 . Then ρ ( E 0 XY H ) ≥ � XY H � 2 . Sketch of proof : ρ ( E 0 XY H ) ρ ( E 0 U Σ V H ) = ρ ( V H E 0 U Σ) = �� �� β � XY H � 2 ∗ ≥ � XY H � 2 . = ρ 0 ∗ D. Kressner p. 17

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: E 0 u = β v with | β | = 1, � E 0 � 2 ≈ 1, E 0 ∈ T A S . (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 ) . D. Kressner p. 18

Matrices Matrix Pencils Matrix Polynomials S = complex skew -symmetric matrices Case I : Nonzero eigenvalue, r 1 = 1. � � � n 1 , � X � 2 � Y � 2 − | Y T X | 2 κ ( λ, S ) = Case IIa : Zero eigenvalue, r 1 = 1, n 1 odd. Y = X � Y H EX = 0 for all E ∈ S � α S = 0 . Case IIb : Zero eigenvalue, r 1 > 1, n 1 odd. κ ( λ, S ) = ( n 1 , √ σ 1 σ 2 ) , where σ 1 , σ 2 are the two largest singular values of XX T . Case III : Zero eigenvalue, n 1 even. κ ( λ ) = κ ( λ, S ) . D. Kressner p. 19

Matrices Matrix Pencils Matrix Polynomials Multiple Eigenvalues of Matrix Pencils D. Kressner p. 20