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 A1, A2 ∈ Rn×n λ1 ∈ σ(A1) λ2 ∈ σ(A2) how “large” is |λ1 − λ2|?

• 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]) λx = G(λ)x? (⋆) Nonlinear Eigenvalue perturbation: Given A1, A2 ∈ Rn×n λ1 ∈ σ(G1(λ1)) λ2 ∈ σ(G2(λ2)) how “large” is |λ1 − λ2|?

• 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]) λx = G(λ)x? (⋆) Nonlinear Eigenvalue perturbation: Given A1, A2 ∈ Rn×n λ1 ∈ σ(G1(λ1)) λ2 ∈ σ(G2(λ2)) how “large” is |λ1 − λ2|? 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]) λx = G(λ)x? (⋆) Nonlinear Eigenvalue perturbation: Given A1, A2 ∈ Rn×n λ1 ∈ σ(G1(λ1)) λ2 ∈ σ(G2(λ2)) how “large” is |λ1 − λ2|? 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 − λI + λ2B
• v

λ ∈ σ(A + Bλ2)

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Bauer-Fike and NLEVPs 3 / 16

Nonlinear eigenproblems

Quadratic eigenproblem: Linearization[MMMM’06], SOAR[Bai,Su’05], JD[Slejpen et al’96] =

• A − λI + λ2B
• v

λ ∈ σ(A + Bλ2) Example: A = 2 1 1 2

• , B =

1 −1 −1 2

• 0.5

1 1.5 −3 −2 −1 1 2 3 Re Im

λ ∈ σ(A + Bλ2)

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Bauer-Fike and NLEVPs 3 / 16

Nonlinear eigenproblems

Delay eigenvalue problem: LMS[Engelborghs, et al’99], Pseudospectral differencing[Breda, et al’05] =

• A − λI + e−λB
• v

λ ∈ σ(A + Be−λ)

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Bauer-Fike and NLEVPs 3 / 16

Nonlinear eigenproblems

Delay eigenvalue problem: LMS[Engelborghs, et al’99], Pseudospectral differencing[Breda, et al’05] =

• A − λI + e−λB
• v

λ ∈ σ(A + Be−λ) Example: A = 2 1 1 2

• , B =

1 −1 −1 2

• −6

−4 −2 2 4 6 −50 50 Re Im

λ ∈ σ(A + Be−

λ )

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Bauer-Fike and NLEVPs 3 / 16

The Bauer-Fike Theorem [Bauer,Fike’60]

Classical formulation Theorem (BF) If A1 = V1D1V1−1 then for any λ2 ∈ σ(A2) min

λ1∈σ(A1) |λ2 − λ1| ≤ κ(V1)A1 − A2.

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Bauer-Fike and NLEVPs 4 / 16

The Bauer-Fike Theorem [Bauer,Fike’60]

Classical formulation Theorem (BF) If A1 = V1D1V1−1 then for any λ2 ∈ σ(A2) min

λ1∈σ(A1) |λ2 − λ1| ≤ κ(V1)A1 − A2.

−5 5 −8 −6 −4 −2 2 4 6 8 Re Im

σ(A1) σ(A2)

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Bauer-Fike and NLEVPs 4 / 16

The Bauer-Fike Theorem [Bauer,Fike’60]

Classical formulation Theorem (BF) If A1 = V1D1V1−1 then for any λ2 ∈ σ(A2) min

λ1∈σ(A1) |λ2 − λ1| ≤ κ(V1)A1 − A2.

−5 5 −8 −6 −4 −2 2 4 6 8 Re Im

κ1∆ κ1∆ κ1∆ κ1∆ σ(A1) σ(A2)

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Bauer-Fike and NLEVPs 4 / 16

The Bauer-Fike Theorem [Bauer,Fike’60]

Classical formulation Theorem (BF) If A2 = V2D2V2−1 then for any λ1 ∈ σ(A1) min

λ2∈σ(A2) |λ2 − λ1| ≤ κ(V2)A1 − A2.

−5 5 −8 −6 −4 −2 2 4 6 8 Re Im

κ2∆ κ2∆ κ2∆ κ2∆ σ(A1) σ(A2)

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Bauer-Fike and NLEVPs 4 / 16

The Bauer-Fike Theorem [Bauer,Fike’60]

Classical formulation Theorem (BF) If A2 = V2D2V2−1 then for any λ1 ∈ σ(A1) dist(λ1, σ(A2)) := min

λ2∈σ(A2) |λ2 − λ1| ≤ κ(V2)A1 − A2.

−5 5 −8 −6 −4 −2 2 4 6 8 Re Im

κ2∆ κ2∆ κ2∆ κ2∆ σ(A1) σ(A2)

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Bauer-Fike and NLEVPs 4 / 16

The Bauer-Fike Theorem

Theorem (BF - Dist) If λ1 ∈ σ(A1) and λ2 ∈ σ(A2) then a) dist(λ1, σ(A2)) ≤ κ(V2)A1 − A2 b) dist(λ2, σ(A1)) ≤ κ(V1)A1 − A2 where dist(λ, S) = mins∈S |λ − s|.

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Bauer-Fike and NLEVPs 5 / 16

The Bauer-Fike Theorem

Theorem (BF - Dist) If λ1 ∈ σ(A1) and λ2 ∈ σ(A2) then a) dist(λ1, σ(A2)) ≤ κ(V2)A1 − A2 b) dist(λ2, σ(A1)) ≤ κ(V1)A1 − A2 where dist(λ, S) = mins∈S |λ − s|. For the Hausdorff metric dH(S1, S2) := max

• max

s1∈S1 dist(s1, S2), max s2∈S2 dist(s2, S1)

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Bauer-Fike and NLEVPs 5 / 16

The Bauer-Fike Theorem

Theorem (BF - Dist) If λ1 ∈ σ(A1) and λ2 ∈ σ(A2) then a) dist(λ1, σ(A2)) ≤ κ(V2)A1 − A2 b) dist(λ2, σ(A1)) ≤ κ(V1)A1 − A2 where dist(λ, S) = mins∈S |λ − s|. For the Hausdorff metric dH(S1, S2) := max

• max

s1∈S1 dist(s1, S2), max s2∈S2 dist(s2, S1)

• Theorem (BF - Hausdorff metric)

dH(σ(A1), σ(A2)) ≤ max(κ(V1), κ(V2))A1 − A2

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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)

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Bauer-Fike and NLEVPs 6 / 16

Accuracy iteration

Root-finding problem: find λ∗ s.t. λ∗ = f (λ∗).

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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∗|.

−2 2 4 −1 1 2 3 4 Re Im

˜ λ λ∗

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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).

−2 2 4 −1 1 2 3 4 Re Im

∆k ˜ λ λ∗

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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∗| ≤

−2 2 4 −1 1 2 3 4 Re Im

∆k ˜ λ λ∗

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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∗| ≤ |˜ λ − f (˜ λ)| + |f (˜ λ) − f (λ∗)|

−2 2 4 −1 1 2 3 4 Re Im

∆k |λ∗ − ˜ λ| ˜ λ λ∗ f(˜ λ)

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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∗| ≤ |˜ λ − f (˜ λ)| + |f (˜ λ) − f (λ∗)| ≤ |˜ λ − f (˜ λ)| + f ′

max|˜

λ − λ∗| ⇒ |˜ λ − λ∗| ≤ 1 1 − f ′

max

|˜ λ − f (˜ λ)|

−2 2 4 −1 1 2 3 4 Re Im

∆k |λ∗ − ˜ λ| ˜ λ λ∗ f(˜ λ)

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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∗| ≤ |˜ λ − f (˜ λ)| + |f (˜ λ) − f (λ∗)| ≤ |˜ λ − f (˜ λ)| + f ′

max|˜

λ − λ∗| ⇒ |˜ λ − λ∗| ≤ 1 1 − f ′

max

|˜ λ − f (˜ λ)| ≤ |˜ λ − f (˜ λ)| + max

λ∈V (∆k) |f (˜

λ) − f (λ)|

−2 2 4 −1 1 2 3 4 Re Im

∆k |λ∗ − ˜ λ| ˜ λ λ∗ f(˜ λ)

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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∗| ≤ |˜ λ − f (˜ λ)| + |f (˜ λ) − f (λ∗)| ≤ |˜ λ − f (˜ λ)| + f ′

max|˜

λ − λ∗| ⇒ |˜ λ − λ∗| ≤ 1 1 − f ′

max

|˜ λ − f (˜ λ)| ≤ |˜ λ − f (˜ λ)| + max

λ∈V (∆k) |f (˜

λ) − f (λ)|

• =:D(∆k)

−2 2 4 −1 1 2 3 4 Re Im

∆k |λ∗ − ˜ λ| D(∆k) ˜ λ λ∗ f(˜ λ)

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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∗| ≤ |˜ λ − f (˜ λ)| + |f (˜ λ) − f (λ∗)| ≤ |˜ λ − f (˜ λ)| + f ′

max|˜

λ − λ∗| ⇒ |˜ λ − λ∗| ≤ 1 1 − f ′

max

|˜ λ − f (˜ λ)| ≤ |˜ λ − f (˜ λ)| + max

λ∈V (∆k) |f (˜

λ) − f (λ)|

• =:D(∆k)

Accuracy Fixpoint Iteration λ∗ ∈ V (∆k) ⇒ λ∗ ∈ V (∆k+1) where ∆k+1 = ϕ(∆k) := |˜ λ − f (˜ λ)| + D(∆k)

−2 2 4 −1 1 2 3 4 Re Im

∆k |λ∗ − ˜ λ| D(∆k) ∆k+1 ˜ λ λ∗ f(˜ λ)

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Bauer-Fike and NLEVPs 7 / 16

Example λ∗ = 0.1λ2

∗ + 1,

˜ λ = 1;

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Bauer-Fike and NLEVPs 8 / 16

Example λ∗ = 0.1λ2

∗ + 1,

˜ λ = 1; ∆k+1 = |˜ λ − f (˜ λ)| + D(∆k) = 0.1 + D(∆k) D(∆k) = max

λ∈V (∆k) |f (λ) − f (˜

λ))| = 0.2∆k + 0.1∆2

k

∆k+1 = φ(∆k) = 0.1 + 0.2∆k + 0.1∆2

k

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Bauer-Fike and NLEVPs 8 / 16

Example λ∗ = 0.1λ2

∗ + 1,

˜ λ = 1; ∆k+1 = |˜ λ − f (˜ λ)| + D(∆k) = 0.1 + D(∆k) D(∆k) = max

λ∈V (∆k) |f (λ) − f (˜

λ))| = 0.2∆k + 0.1∆2

k

∆k+1 = φ(∆k) = 0.1 + 0.2∆k + 0.1∆2

k

∆0 2 ∆1 = φ(∆0) 0.9 ∆2 = φ(∆1) 0.361 ∆3 = φ(∆2) 0.185 . . . . . . ∆∗ 4 − √ 42 − 1 ≈ 0.127

−2 2 4 6 8 10 −4 −2 2 4 Re Im

˜ λ λ∗

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Bauer-Fike and NLEVPs 8 / 16

Example λ∗ = 0.1λ2

∗ + 1,

˜ λ = 1; ∆k+1 = |˜ λ − f (˜ λ)| + D(∆k) = 0.1 + D(∆k) D(∆k) = max

λ∈V (∆k) |f (λ) − f (˜

λ))| = 0.2∆k + 0.1∆2

k

∆k+1 = φ(∆k) = 0.1 + 0.2∆k + 0.1∆2

k

∆0 2 ∆1 = φ(∆0) 0.9 ∆2 = φ(∆1) 0.361 ∆3 = φ(∆2) 0.185 . . . . . . ∆∗ 4 − √ 42 − 1 ≈ 0.127

−2 2 4 6 8 10 −4 −2 2 4 Re Im

∆0 = 2 ˜ λ λ∗

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Bauer-Fike and NLEVPs 8 / 16

Example λ∗ = 0.1λ2

∗ + 1,

˜ λ = 1; ∆k+1 = |˜ λ − f (˜ λ)| + D(∆k) = 0.1 + D(∆k) D(∆k) = max

λ∈V (∆k) |f (λ) − f (˜

λ))| = 0.2∆k + 0.1∆2

k

∆k+1 = φ(∆k) = 0.1 + 0.2∆k + 0.1∆2

k

∆0 2 ∆1 = φ(∆0) 0.9 ∆2 = φ(∆1) 0.361 ∆3 = φ(∆2) 0.185 . . . . . . ∆∗ 4 − √ 42 − 1 ≈ 0.127

−2 2 4 6 8 10 −4 −2 2 4 Re Im

∆0 = 2 ∆1 = 0.9 ˜ λ λ∗

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Bauer-Fike and NLEVPs 8 / 16

Example λ∗ = 0.1λ2

∗ + 1,

˜ λ = 1; ∆k+1 = |˜ λ − f (˜ λ)| + D(∆k) = 0.1 + D(∆k) D(∆k) = max

λ∈V (∆k) |f (λ) − f (˜

λ))| = 0.2∆k + 0.1∆2

k

∆k+1 = φ(∆k) = 0.1 + 0.2∆k + 0.1∆2

k

∆0 2 ∆1 = φ(∆0) 0.9 ∆2 = φ(∆1) 0.361 ∆3 = φ(∆2) 0.185 . . . . . . ∆∗ 4 − √ 42 − 1 ≈ 0.127

−2 2 4 6 8 10 −4 −2 2 4 Re Im

∆0 = 2 ∆1 = 0.9 ˜ λ λ∗

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Bauer-Fike and NLEVPs 8 / 16

Example λ∗ = 0.1λ2

∗ + 1,

˜ λ = 1; ∆k+1 = |˜ λ − f (˜ λ)| + D(∆k) = 0.1 + D(∆k) D(∆k) = max

λ∈V (∆k) |f (λ) − f (˜

λ))| = 0.2∆k + 0.1∆2

k

∆k+1 = φ(∆k) = 0.1 + 0.2∆k + 0.1∆2

k

∆0 2 ∆1 = φ(∆0) 0.9 ∆2 = φ(∆1) 0.361 ∆3 = φ(∆2) 0.185 . . . . . . ∆∗ 4 − √ 42 − 1 ≈ 0.127

−2 2 4 6 8 10 −4 −2 2 4 Re Im

∆0 = 2 ∆1 = 0.9 ˜ λ λ∗

Error of ˜ λ less than ∆0 = 2 ⇒ error less than ∆∗ = 0.127.

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Bauer-Fike and NLEVPs 8 / 16

Main results: A BF-NLVP comparison theorem Accuracy iteration with constant comparison Accuracy iteration with linear comparison Accuracy iteration with linear normalized comparison

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Bauer-Fike and NLEVPs 9 / 16

Theorem (BF-NLEVP - General comparison) Let λ1 ∈ σ(G1(λ1)), λ2 ∈ σ(G2(λ2)) and G1(λ) − G2(λ) ≤ δ, G ′

1(λ) ≤ ε1, G ′ 2(λ) ≤ ε2.

for all λ ∈ V . Then, a) |λ1 − λ2| ≤ κ2 1 − κ2ε2 δ b) |λ1 − λ2| ≤ κ1 1 − κ1ε1 δ

Theorem (BF-NLEVP - General comparison) Let λ1 ∈ σ(G1(λ1)), λ2 ∈ σ(G2(λ2)) and G1(λ) − G2(λ) ≤ δ, G ′

1(λ) ≤ ε1, G ′ 2(λ) ≤ ε2.

for all λ ∈ V . Then, a) λ2 = argmin

λ∈σ(G2(λ2))

|λ − λ1| ⇒ |λ1 − λ2| ≤ κ2 1 − κ2ε2 δ b) λ1 = argmin

λ∈σ(G1(λ1))

|λ − λ2| ⇒ |λ1 − λ2| ≤ κ1 1 − κ1ε1 δ

Theorem (BF-NLEVP - General comparison) Let λ1 ∈ σ(G1(λ1)), λ2 ∈ σ(G2(λ2)) and G1(λ) − G2(λ) ≤ δ, G ′

1(λ) ≤ ε1, G ′ 2(λ) ≤ ε2.

for all λ ∈ V . Then, a) λ2 = argmin

λ∈σ(G2(λ2))

|λ − λ1| ⇒ |λ1 − λ2| ≤ κ2 1 − κ2ε2 δ b) λ1 = argmin

λ∈σ(G1(λ1))

|λ − λ2| ⇒ |λ1 − λ2| ≤ κ1 1 − κ1ε1 δ

• Pf. Sketch.

|λ1 − λ2|

BF

≤ κ1G1(λ1) − G2(λ2) ≤ κ1 (G1(λ1) − G1(λ2) + G2(λ2) − G1(λ2)) ≤ κ1 (ε1|λ1 − λ2| + δ) ⇒ |λ1 − λ2| ≤ κ1 1 − κ1ε1 δ

Accuracy iteration (constant comparison) Given ˜ λ, how well does it approximate λ1 ∈ σ(G(λ1))?

−2 −1 1 2 3 4 −1 1 2 3 4 Re Im

˜ λ λ1

Accuracy iteration (constant comparison) Given ˜ λ, how well does it approximate λ1 ∈ σ(G(λ1))? Assume λ1 ∈ V (∆k).

−2 −1 1 2 3 4 −1 1 2 3 4 Re Im

∆k ˜ λ λ1

Accuracy iteration (constant comparison) Given ˜ λ, how well does it approximate λ1 ∈ σ(G(λ1))? Assume λ1 ∈ V (∆k). Define λ2 ∈ σ(G(˜ λ)).

−2 −1 1 2 3 4 −1 1 2 3 4 Re Im

∆k |λ1 − ˜ λ| ˜ λ λ1 λ2

Accuracy iteration (constant comparison) Given ˜ λ, how well does it approximate λ1 ∈ σ(G(λ1))? Assume λ1 ∈ V (∆k). Define λ2 ∈ σ(G(˜ λ)). Use BF-NLEVP to bound |λ1−λ2| < D(∆k).

−2 −1 1 2 3 4 −1 1 2 3 4 Re Im

∆k |λ1 − ˜ λ| D(∆k) = κ max ˜ λ λ1 λ2

Accuracy iteration (constant comparison) Given ˜ λ, how well does it approximate λ1 ∈ σ(G(λ1))? Assume λ1 ∈ V (∆k). Define λ2 ∈ σ(G(˜ λ)). Use BF-NLEVP to bound |λ1−λ2| < D(∆k). Triangle inequality |λ1−˜ λ| ≤ |˜ λ−λ2|+D(∆k)

−2 −1 1 2 3 4 −1 1 2 3 4 Re Im

∆k |λ1 − ˜ λ| D(∆k) = κ max ∆k+1 ˜ λ λ1 λ2

Accuracy iteration (constant comparison) Given ˜ λ, how well does it approximate λ1 ∈ σ(G(λ1))? Assume λ1 ∈ V (∆k). Define λ2 ∈ σ(G(˜ λ)). Use BF-NLEVP to bound |λ1−λ2| < D(∆k). Triangle inequality |λ1−˜ λ| ≤ |˜ λ−λ2|+D(∆k)=: ∆k+1

−2 −1 1 2 3 4 −1 1 2 3 4 Re Im

∆k |λ1 − ˜ λ| D(∆k) = κ max ∆k+1 ˜ λ λ1 λ2

Accuracy iteration (constant comparison) Given ˜ λ, how well does it approximate λ1 ∈ σ(G(λ1))? Assume λ1 ∈ V (∆k). Define λ2 ∈ σ(G(˜ λ)). Use BF-NLEVP to bound |λ1−λ2| < D(∆k). Triangle inequality |λ1−˜ λ| ≤ |˜ λ−λ2|+D(∆k)=: ∆k+1 Iteration If λ1 ∈ V (∆k) ⇒ λ1 ∈ V (∆k+1) where ∆k+1 = |˜ λ − λ2| + κ2 max

λ∈V (∆k) G(λ) − G(˜

λ)

Cubic toy example

A0 = B B B B B B B B B @ 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 1 C C C C C C C C C A , A1 = B B B B B B B B B @ 0.02 1 C C C C C C C C C A ,

λ∗ ∈ σ(A0 + A1λ3

∗)

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Bauer-Fike and NLEVPs 12 / 16

Cubic toy example

A0 = B B B B B B B B B @ 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 1 C C C C C C C C C A , A1 = B B B B B B B B B @ 0.02 1 C C C C C C C C C A ,

λ∗ ∈ σ(A0 + A1λ3

∗)

−10 −5 5 10 −2 −1.5 −1 −0.5 0.5 1 1.5 2 Re Im

λ∗ σ(A0)

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Bauer-Fike and NLEVPs 12 / 16

Cubic toy example

A0 = B B B B B B B B B @ 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 1 C C C C C C C C C A , A1 = B B B B B B B B B @ 0.02 1 C C C C C C C C C A ,

λ∗ ∈ σ(A0 + A1λ3

∗)

Given approximation ˜ λ we compute λ2 ∈ σ(A0 + A1˜ λ3) ∆k+1 = |λ2 − ˜ λ| + κ2A1 max

λ∈V (∆k) |λ3 − ˜

λ3| = = |λ2 − ˜ λ| + κ2A1

• (|˜

λ| + ∆k)3 − |˜ λ|3

• E. Jarlebring (TU Braunschweig)

Bauer-Fike and NLEVPs 12 / 16

Cubic toy example

A0 = B B B B B B B B B @ 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 1 C C C C C C C C C A , A1 = B B B B B B B B B @ 0.02 1 C C C C C C C C C A ,

λ∗ ∈ σ(A0 + A1λ3

∗)

Given approximation ˜ λ we compute λ2 ∈ σ(A0 + A1˜ λ3) ∆k+1 = |λ2 − ˜ λ| + κ2A1 max

λ∈V (∆k) |λ3 − ˜

λ3| = = |λ2 − ˜ λ| + κ2A1

• (|˜

λ| + ∆k)3 − |˜ λ|3 λ∗ ˜ λ |λ∗ − ˜ λ| ∆(0)

∗

0.2721 ± 0.2792i 0.2722 ± 0.2787i 5.82e-04 6.02e-04 0.1447 ± 0.9138i 0.1478 ± 0.9135i 3.09e-03 3.89e-03 0.0596 ± 1.4842i 0.0640 ± 1.4848i 4.49e-03 9.44e-03 0.0140 ± 1.8660i 0.0160 ± 1.8664i 1.99e-03 1.20e-02 6.5989

• 7.5798
• E. Jarlebring (TU Braunschweig)

Bauer-Fike and NLEVPs 12 / 16

Accuracy iteration (linear comparison)

Compute one step of “Newton” (MSLP): λ2 ∈ σ(G(˜ λ) + (λ2 − ˜ λ)G ′(˜ λ)).

• E. Jarlebring (TU Braunschweig)

Bauer-Fike and NLEVPs 13 / 16

Accuracy iteration (linear comparison)

Compute one step of “Newton” (MSLP): λ2 ∈ σ(G(˜ λ) + (λ2 − ˜ λ)G ′(˜ λ)). Iteration (linear) ∆k+1 = |λ2 − ˜ λ| + κ2 1 − κ2G ′(˜ λ) max

λ∈V (∆k) G(λ) − G(˜

λ) − (λ − ˜ λ)G ′(˜ λ)

• E. Jarlebring (TU Braunschweig)

Bauer-Fike and NLEVPs 13 / 16

Accuracy iteration (linear comparison)

Compute one step of “Newton” (MSLP): λ2 ∈ σ(G(˜ λ) + (λ2 − ˜ λ)G ′(˜ λ)). Iteration (linear) ∆k+1 = |λ2 − ˜ λ| + κ2 1 − κ2G ′(˜ λ) max

λ∈V (∆k) G(λ) − G(˜

λ) − (λ − ˜ λ)G ′(˜ λ) For cubic toy example: ∆k+1 = |λ2 − ˜ λ| + κ2 1 − 3κ2|˜ λ|2A1 A1 max

λ∈V (∆k) |λ3 − ˜

λ3 − 3(λ − ˜ λ)˜ λ2|

• E. Jarlebring (TU Braunschweig)

Bauer-Fike and NLEVPs 13 / 16

Accuracy iteration (linear comparison)

Compute one step of “Newton” (MSLP): λ2 ∈ σ(G(˜ λ) + (λ2 − ˜ λ)G ′(˜ λ)). Iteration (linear) ∆k+1 = |λ2 − ˜ λ| + κ2 1 − κ2G ′(˜ λ) max

λ∈V (∆k) G(λ) − G(˜

λ) − (λ − ˜ λ)G ′(˜ λ) For cubic toy example: ∆k+1 = |λ2 − ˜ λ| + κ2 1 − 3κ2|˜ λ|2A1 A1 max

λ∈V (∆k) |λ3 − ˜

λ3 − 3(λ − ˜ λ)˜ λ2|

λ∗ ˜ λ |λ∗ − ˜ λ| ∆(0)

∗

∆(1)

∗

0.2721 ± 0.2792i 0.2722 ± 0.2787i 5.82e-04 6.02e-04 5.82e-04 0.1447 ± 0.9138i 0.1478 ± 0.9135i 3.09e-03 3.89e-03 3.10e-03 0.0596 ± 1.4842i 0.0640 ± 1.4848i 4.49e-03 9.44e-03 4.50e-03 0.0140 ± 1.8660i 0.0160 ± 1.8664i 1.99e-03 1.20e-02 1.20e-03 6.5989

• 7.5798
• E. Jarlebring (TU Braunschweig)

Bauer-Fike and NLEVPs 13 / 16

Example (DDE) λ∗ ∈ σ 2 1 1 2

• +

1 −1 −1 2

• e−λ∗

Example (DDE) λ∗ ∈ σ 2 1 1 2

• +

1 −1 −1 2

• e−λ∗
• −6

−4 −2 2 4 6 −50 50 Re Im

λ ∈ σ(A + Be−

λ )

Example (DDE) λ∗ ∈ σ 2 1 1 2

• +

1 −1 −1 2

• e−λ∗
• Pick ˜

λ as first LMS-approximation in DDE-BIFTOOL

λ∗ ? ? ? ? ? ˜ λ 1.5326 3.0246

• 0.58±4.35i
• 1.44±10.76i
• 1.89±17.11i

|λ∗ − ˜ λ| ? ? ? ? ?

Example (DDE) λ∗ ∈ σ 2 1 1 2

• +

1 −1 −1 2

• e−λ∗
• Pick ˜

λ as first LMS-approximation in DDE-BIFTOOL

λ∗ ? ? ? ? ? ˜ λ 1.5326 3.0246

• 0.58±4.35i
• 1.44±10.76i
• 1.89±17.11i

|λ∗ − ˜ λ| ? ? ? ? ? ∆(0)

∗

5.04e-05 3.50e-06 Inf Inf Inf ∆(1)

∗

8.94e-06 2.06e-07

• Inf
• Inf
• Inf

Example (DDE) λ∗ ∈ σ 2 1 1 2

• +

1 −1 −1 2

• e−λ∗
• Pick ˜

λ as first LMS-approximation in DDE-BIFTOOL

λ∗ ? ? ? ? ? ˜ λ 1.5326 3.0246

• 0.58±4.35i
• 1.44±10.76i
• 1.89±17.11i

|λ∗ − ˜ λ| ? ? ? ? ? ∆(0)

∗

5.04e-05 3.50e-06 Inf Inf Inf ∆(1)

∗

8.94e-06 2.06e-07

• Inf
• Inf
• Inf

∆(0)

∗

= |λ2 − ˜ λ| − κA1e− Re ˜

λ − W0(−κA1e− Re ˜ λ+|λ2−˜ λ|−κA1e− Re ˜

λ)

Example (DDE) λ∗ ∈ σ 2 1 1 2

• +

1 −1 −1 2

• e−λ∗
• Pick ˜

λ as first LMS-approximation in DDE-BIFTOOL

λ∗ ? ? ? ? ? ˜ λ 1.5326 3.0246

• 0.58±4.35i
• 1.44±10.76i
• 1.89±17.11i

|λ∗ − ˜ λ| ? ? ? ? ? ∆(0)

∗

5.04e-05 3.50e-06 Inf Inf Inf ∆(1)

∗

8.94e-06 2.06e-07

• Inf
• Inf
• Inf

∆(1n)

∗

8.94e-06 2.06e-07 1.40e-05 5.45e-04 3.49e-03

Example (DDE) λ∗ ∈ σ 2 1 1 2

• +

1 −1 −1 2

• e−λ∗
• Pick ˜

λ as first LMS-approximation in DDE-BIFTOOL

λ∗ ? ? ? ? ? ˜ λ 1.5326 3.0246

• 0.58±4.35i
• 1.44±10.76i
• 1.89±17.11i

|λ∗ − ˜ λ|∼ 8.93e-06 2.06e-07 1.40e-05 5.44e-04 3.48e-03 ∆(0)

∗

5.04e-05 3.50e-06 Inf Inf Inf ∆(1)

∗

8.94e-06 2.06e-07

• Inf
• Inf
• Inf

∆(1n)

∗

8.94e-06 2.06e-07 1.40e-05 5.45e-04 3.49e-03

Example (DDE) λ∗ ∈ σ 2 1 1 2

• +

1 −1 −1 2

• e−λ∗
• Pick ˜

λ as first LMS-approximation in DDE-BIFTOOL

λ∗ ? ? ? ? ? ˜ λ 1.5326 3.0246

• 0.58±4.35i
• 1.44±10.76i
• 1.89±17.11i

|λ∗ − ˜ λ|∼ 8.93e-06 2.06e-07 1.40e-05 5.44e-04 3.48e-03 ∆(0)

∗

5.04e-05 3.50e-06 Inf Inf Inf ∆(1)

∗

8.94e-06 2.06e-07

• Inf
• Inf
• Inf

∆(1n)

∗

8.94e-06 2.06e-07 1.40e-05 5.45e-04 3.49e-03

Normalization λ ∈ σ(G(λ)) ⇔ λ ∈ σ((I − A)−1(G(λ) − λA)

• =:H(λ)

)

Example (DDE) λ∗ ∈ σ 2 1 1 2

• +

1 −1 −1 2

• e−λ∗
• Pick ˜

λ as first LMS-approximation in DDE-BIFTOOL

λ∗ ? ? ? ? ? ˜ λ 1.5326 3.0246

• 0.58±4.35i
• 1.44±10.76i
• 1.89±17.11i

|λ∗ − ˜ λ|∼ 8.93e-06 2.06e-07 1.40e-05 5.44e-04 3.48e-03 ∆(0)

∗

5.04e-05 3.50e-06 Inf Inf Inf ∆(1)

∗

8.94e-06 2.06e-07

• Inf
• Inf
• Inf

∆(1n)

∗

8.94e-06 2.06e-07 1.40e-05 5.45e-04 3.49e-03

Normalization λ ∈ σ(G(λ)) ⇔ λ ∈ σ((I − A)−1(G(λ) − λA)

• =:H(λ)

) Let A = G ′(˜ λ) then H′(˜ λ) = 0 ⇒

κ2 1−κ2ε2 = κ2

Final remarks and future work

A Bauer-Fike comparison theorem for nonlinear eigenvalue problems Accuracy of solutions nonlinear eigenvalue problems with a accuracy iterations Future: Relation to method of linear/constant problems

• E. Jarlebring (TU Braunschweig)

Bauer-Fike and NLEVPs 15 / 16

Example (DDE)

λ∗ ∈ σ B B B B B B B B B B B B B @ B B B B B B B B B B B B B @ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 C C C C C C C C C C C C C A + B B B B B B B B B B B B B @ −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 C C C C C C C C C C C C C A e−λ∗ 1 C C C C C C C C C C C C C A

Example (DDE)

λ∗ ∈ σ B B B B B B B B B B B B B @ B B B B B B B B B B B B B @ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 C C C C C C C C C C C C C A + B B B B B B B B B B B B B @ −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 C C C C C C C C C C C C C A e−λ∗ 1 C C C C C C C C C C C C C A 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 Im

Example (DDE)

λ∗ ∈ σ B B B B B B B B B B B B B @ B B B B B B B B B B B B B @ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 C C C C C C C C C C C C C A + B B B B B B B B B B B B B @ −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 1 −6 1 C C C C C C C C C C C C C A e−λ∗ 1 C C C C C C C C C C C C C A

˜ λ ∆(0)

∗

∆(1)

∗

∆(1n)

∗

1.2526±0.8837i Inf

• Inf

2.62e-02 1.1967±1.0464i Inf

• Inf

1.70e-01 1.0924±1.2885i Inf

• Inf

2.80e-01 0.9536±1.5407i NaN

• Inf

Inf 0.7973±1.7675i NaN

• Inf

Inf 0.6404±1.9548i NaN

• Inf

Inf 0.4973±2.0999i NaN

• Inf

2.70e-01 0.3784±2.2058i NaN

• Inf

1.73e-01 0.2900±2.2771i Inf

• Inf

5.48e-02 0.2359±2.3179i Inf

• Inf

6.46e-02