An application of the Bauer-Fike theorem to nonlinear eigenproblems - PowerPoint PPT Presentation

An application of the Bauer-Fike theorem to nonlinear eigenproblems Elias Jarlebring TU Braunschweig Insitut Computational Mathematics joint work with Johan Karlsson, KTH

Perturbation theorems play a very essential role in computational processes for eigenproblems. Golub, van der Vorst E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 2 / 16

Perturbation theorems play a very essential role in computational processes for eigenproblems. Golub, van der Vorst What about nonlinear eigenvalue problems ([Ruhe’78],[Mehrmann,Voss’04]) 0 = M ( λ ) x ? ( ⋆ ) E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 2 / 16

Perturbation theorems play a very essential role in computational processes for eigenproblems. Golub, van der Vorst What about nonlinear eigenvalue problems ([Ruhe’78],[Mehrmann,Voss’04]) 0 = M ( λ ) x ? ( ⋆ ) Eigenvalue perturbation: Given A 1 , A 2 ∈ R n × n ∈ σ ( A 1 ) λ 1 λ 2 ∈ σ ( A 2 ) how “large” is | λ 1 − λ 2 | ? E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 2 / 16

Perturbation theorems play a very essential role in computational processes for eigenproblems. Golub, van der Vorst What about nonlinear eigenvalue problems ([Ruhe’78],[Mehrmann,Voss’04]) λ x = G ( λ ) x ? ( ⋆ ) Nonlinear Eigenvalue perturbation: Given A 1 , A 2 ∈ R n × n ∈ σ ( G 1 ( λ 1 )) λ 1 λ 2 ∈ σ ( G 2 ( λ 2 )) how “large” is | λ 1 − λ 2 | ? E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 2 / 16

Perturbation theorems play a very essential role in computational processes for eigenproblems. Golub, van der Vorst What about nonlinear eigenvalue problems ([Ruhe’78],[Mehrmann,Voss’04]) λ x = G ( λ ) x ? ( ⋆ ) Nonlinear Eigenvalue perturbation: Given A 1 , A 2 ∈ R n × n ∈ σ ( G 1 ( λ 1 )) λ 1 λ 2 ∈ σ ( G 2 ( λ 2 )) how “large” is | λ 1 − λ 2 | ? Today: How can we apply the Bauer-Fike theorem to ( ⋆ )? E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 2 / 16

Perturbation theorems play a very essential role in computational processes for eigenproblems. Golub, van der Vorst What about nonlinear eigenvalue problems ([Ruhe’78],[Mehrmann,Voss’04]) λ x = G ( λ ) x ? ( ⋆ ) Nonlinear Eigenvalue perturbation: Given A 1 , A 2 ∈ R n × n ∈ σ ( G 1 ( λ 1 )) λ 1 λ 2 ∈ σ ( G 2 ( λ 2 )) how “large” is | λ 1 − λ 2 | ? Today: How can we apply the Bauer-Fike theorem to ( ⋆ )? Introduction: Nonlinear eigenvalue problems the Bauer-Fike Theorem Accuracy iteration E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 2 / 16

Nonlinear eigenproblems Quadratic eigenproblem: Linearization[MMMM’06], SOAR[Bai,Su’05], JD[Slejpen et al’96] � � A − λ I + λ 2 B 0 = v σ ( A + B λ 2 ) ∈ λ E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 3 / 16

Nonlinear eigenproblems Quadratic eigenproblem: Linearization[MMMM’06], SOAR[Bai,Su’05], JD[Slejpen et al’96] � � A − λ I + λ 2 B 0 = v σ ( A + B λ 2 ) ∈ λ Example: � 1 � 2 � � 1 − 1 A = , B = 1 2 − 1 2 3 λ ∈ σ ( A + Bλ 2 ) 2 1 Im 0 −1 −2 −3 0 0.5 1 1.5 E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 3 / 16 Re

Nonlinear eigenproblems Delay eigenvalue problem: LMS[Engelborghs, et al’99], Pseudospectral differencing[Breda, et al’05] � � A − λ I + e − λ B 0 = v σ ( A + Be − λ ) λ ∈ E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 3 / 16

Nonlinear eigenproblems Delay eigenvalue problem: LMS[Engelborghs, et al’99], Pseudospectral differencing[Breda, et al’05] � � A − λ I + e − λ B 0 = v σ ( A + Be − λ ) λ ∈ Example: � 1 � 2 � � 1 − 1 A = , B = 1 2 − 1 2 λ ) λ ∈ σ ( A + Be − 50 Im 0 −50 −6 −4 −2 0 2 4 6 E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs Re 3 / 16

The Bauer-Fike Theorem [Bauer,Fike’60] Classical formulation Theorem (BF) If A 1 = V 1 D 1 V 1 − 1 then for any λ 2 ∈ σ ( A 2 ) λ 1 ∈ σ ( A 1 ) | λ 2 − λ 1 | ≤ κ ( V 1 ) � A 1 − A 2 � . min E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 4 / 16

The Bauer-Fike Theorem [Bauer,Fike’60] Classical formulation Theorem (BF) If A 1 = V 1 D 1 V 1 − 1 then for any λ 2 ∈ σ ( A 2 ) λ 1 ∈ σ ( A 1 ) | λ 2 − λ 1 | ≤ κ ( V 1 ) � A 1 − A 2 � . min 8 σ ( A 1 ) 6 σ ( A 2 ) 4 2 Im 0 −2 −4 −6 −8 −5 0 5 Re E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 4 / 16

The Bauer-Fike Theorem [Bauer,Fike’60] Classical formulation Theorem (BF) If A 1 = V 1 D 1 V 1 − 1 then for any λ 2 ∈ σ ( A 2 ) λ 1 ∈ σ ( A 1 ) | λ 2 − λ 1 | ≤ κ ( V 1 ) � A 1 − A 2 � . min 8 σ ( A 1 ) 6 σ ( A 2 ) 4 κ 1 ∆ 2 Im 0 κ 1 ∆ κ 1 ∆ −2 −4 κ 1 ∆ −6 −8 −5 0 5 Re E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 4 / 16

The Bauer-Fike Theorem [Bauer,Fike’60] Classical formulation Theorem (BF) If A 2 = V 2 D 2 V 2 − 1 then for any λ 1 ∈ σ ( A 1 ) λ 2 ∈ σ ( A 2 ) | λ 2 − λ 1 | ≤ κ ( V 2 ) � A 1 − A 2 � . min 8 σ ( A 1 ) 6 σ ( A 2 ) 4 κ 2 ∆ 2 Im 0 κ 2 ∆ κ 2 ∆ −2 κ 2 ∆ −4 −6 −8 −5 0 5 Re E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 4 / 16

The Bauer-Fike Theorem [Bauer,Fike’60] Classical formulation Theorem (BF) If A 2 = V 2 D 2 V 2 − 1 then for any λ 1 ∈ σ ( A 1 ) dist( λ 1 , σ ( A 2 )) := λ 2 ∈ σ ( A 2 ) | λ 2 − λ 1 | ≤ κ ( V 2 ) � A 1 − A 2 � . min 8 σ ( A 1 ) 6 σ ( A 2 ) 4 κ 2 ∆ 2 Im 0 κ 2 ∆ κ 2 ∆ −2 κ 2 ∆ −4 −6 −8 −5 0 5 Re E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 4 / 16

The Bauer-Fike Theorem Theorem (BF - Dist) If λ 1 ∈ σ ( A 1 ) and λ 2 ∈ σ ( A 2 ) then a ) dist( λ 1 , σ ( A 2 )) ≤ κ ( V 2 ) � A 1 − A 2 � b ) dist( λ 2 , σ ( A 1 )) ≤ κ ( V 1 ) � A 1 − A 2 � where dist( λ, S ) = min s ∈ S | λ − s | . E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 5 / 16

The Bauer-Fike Theorem Theorem (BF - Dist) If λ 1 ∈ σ ( A 1 ) and λ 2 ∈ σ ( A 2 ) then a ) dist( λ 1 , σ ( A 2 )) ≤ κ ( V 2 ) � A 1 − A 2 � b ) dist( λ 2 , σ ( A 1 )) ≤ κ ( V 1 ) � A 1 − A 2 � where dist( λ, S ) = min s ∈ S | λ − s | . For the Hausdorff metric � � d H ( S 1 , S 2 ) := max s 1 ∈ S 1 dist( s 1 , S 2 ) , max max s 2 ∈ S 2 dist( s 2 , S 1 ) E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 5 / 16

The Bauer-Fike Theorem Theorem (BF - Dist) If λ 1 ∈ σ ( A 1 ) and λ 2 ∈ σ ( A 2 ) then a ) dist( λ 1 , σ ( A 2 )) ≤ κ ( V 2 ) � A 1 − A 2 � b ) dist( λ 2 , σ ( A 1 )) ≤ κ ( V 1 ) � A 1 − A 2 � where dist( λ, S ) = min s ∈ S | λ − s | . For the Hausdorff metric � � d H ( S 1 , S 2 ) := max s 1 ∈ S 1 dist( s 1 , S 2 ) , max max s 2 ∈ S 2 dist( s 2 , S 1 ) Theorem (BF - Hausdorff metric) d H ( σ ( A 1 ) , σ ( A 2 )) ≤ max( κ ( V 1 ) , κ ( V 2 )) � A 1 − A 2 � E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 5 / 16

Literature (non-exhaustive) Perturbation of polynomial eigenvalue problems: [Tisseur’00] (backward error) [Higham, Tisseur’03] [Chu,Lin’04] (Bauer-Fike) [Dedieu, Tisseur’03] [Karow,Kressner,Tisseur’06] (condition number) Nonlinear eigenvalue problems: [Hadeler ’69] (generalized Rayleigh-functionals) [Ehrmann ’65] [Cullum, Ruehli ’01] (pseudospectra) [Wagenknecht,Michiels,Green’07] (pseudospectra) E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 6 / 16

Accuracy iteration Root-finding problem: find λ ∗ s.t. λ ∗ = f ( λ ∗ ). E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 7 / 16

Accuracy iteration Root-finding problem: find λ ∗ s.t. λ ∗ = f ( λ ∗ ). Error bound: (a posteori) Given an approximation ˜ λ , bound | ˜ λ − λ ∗ | . 4 ˜ λ 3 λ ∗ 2 Im 1 0 −1 −2 0 2 4 Re E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 7 / 16

Accuracy iteration Root-finding problem: find λ ∗ s.t. λ ∗ = f ( λ ∗ ). Error bound: (a posteori) Given an approximation ˜ λ , bound | ˜ λ − λ ∗ | . Assume | ˜ λ − λ ∗ | < ∆ k , i.e., λ ∗ ∈ V (∆ k ). 4 ˜ λ 3 λ ∗ 2 Im 1 0 ∆ k −1 −2 0 2 4 Re E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 7 / 16

Accuracy iteration Root-finding problem: find λ ∗ s.t. λ ∗ = f ( λ ∗ ). Error bound: (a posteori) Given an approximation ˜ λ , bound | ˜ λ − λ ∗ | . Assume | ˜ λ − λ ∗ | < ∆ k , i.e., λ ∗ ∈ V (∆ k ). | ˜ λ − λ ∗ | ≤ 4 ˜ λ 3 λ ∗ 2 Im 1 0 ∆ k −1 −2 0 2 4 Re E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 7 / 16

Accuracy iteration Root-finding problem: find λ ∗ s.t. λ ∗ = f ( λ ∗ ). Error bound: (a posteori) Given an approximation ˜ λ , bound | ˜ λ − λ ∗ | . Assume | ˜ λ − λ ∗ | < ∆ k , i.e., λ ∗ ∈ V (∆ k ). | ˜ | ˜ λ − f (˜ λ ) | + | f (˜ λ − λ ∗ | ≤ λ ) − f ( λ ∗ ) | 4 ˜ λ 3 λ ∗ f (˜ 2 λ ) ˜ Im | λ ∗ − λ | 1 0 ∆ k −1 −2 0 2 4 Re E. Jarlebring (TU Braunschweig) Bauer-Fike and NLEVPs 7 / 16