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A projection method for the eigenvalues of Elias Jarlebring A projection method for the eigenvalues of Hot shower problem a delay-differential equation Lambert W Solution operator Infinitesimal generator 8th GAMM Workshop - Applied and numerical linear algebra A Projection method Hamburg, 11-12th September, 2008 Numerical examples Elias Jarlebring Institut Computational Mathematics Technische Universtität Braunschweig 1/20

A projection method Delay-differential equations (DDEs) for the eigenvalues of Elias Jarlebring Retarded DDE with a single delay � ˙ x ( t ) = A 0 x ( t ) + A 1 x ( t − τ ) , t ≥ 0 Σ = (1) x ( t ) = ϕ ( t ) , t ∈ [ − τ, 0 ] Hot shower problem Lambert W where A 0 , A 1 ∈ R n × n and ϕ : [ − τ, 0 ] → R n . Solution operator Infinitesimal generator A Projection method Numerical examples 2/20

A projection method Delay-differential equations (DDEs) for the eigenvalues of Elias Jarlebring Retarded DDE with a single delay � ˙ x ( t ) = A 0 x ( t ) + A 1 x ( t − τ ) , t ≥ 0 Σ = (1) x ( t ) = ϕ ( t ) , t ∈ [ − τ, 0 ] Hot shower problem Lambert W where A 0 , A 1 ∈ R n × n and ϕ : [ − τ, 0 ] → R n . Solution operator Ansatz: x ( t ) = ve st ⇒ Infinitesimal generator A Projection method Numerical examples � − sI + A 0 + A 1 e − s τ � v = 0 2/20

A projection method Delay-differential equations (DDEs) for the eigenvalues of Elias Jarlebring Retarded DDE with a single delay � ˙ x ( t ) = A 0 x ( t ) + A 1 x ( t − τ ) , t ≥ 0 Σ = (1) x ( t ) = ϕ ( t ) , t ∈ [ − τ, 0 ] Hot shower problem Lambert W where A 0 , A 1 ∈ R n × n and ϕ : [ − τ, 0 ] → R n . Solution operator Ansatz: x ( t ) = ve st ⇒ Infinitesimal generator A Projection method Numerical examples � − sI + A 0 + A 1 e − s τ � v = 0 Definition Spectrum: σ (Σ) := { s ∈ C : det ( − sI + A 0 + A 1 e − s τ ) = 0 } 2/20

A projection method The hot-shower example for the eigenvalues of • x ( t ) = Temp diff from optimal Temp Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples 3/20

A projection method The hot-shower example for the eigenvalues of • x ( t ) = Temp diff from optimal Temp Elias Jarlebring • α = Human sensitivity Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples 3/20

A projection method The hot-shower example for the eigenvalues of • x ( t ) = Temp diff from optimal Temp Elias Jarlebring • α = Human sensitivity • τ = Length of pipe Speed of water Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples 3/20

A projection method The hot-shower example for the eigenvalues of • x ( t ) = Temp diff from optimal Temp Elias Jarlebring • α = Human sensitivity • τ = Length of pipe Speed of water Delayed negative feedback: ˙ x ( t ) = − α x ( t − τ ) Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples 3/20

A projection method The hot-shower example for the eigenvalues of • x ( t ) = Temp diff from optimal Temp Elias Jarlebring • α = Human sensitivity • τ = Length of pipe Speed of water Delayed negative feedback: ˙ x ( t ) = − α x ( t − τ ) , τ = 1 Hot shower problem 4 α =1.4~ Lambert W α =2 Solution operator 2 Infinitesimal generator A Projection method 0 x Numerical examples −2 −4 0 5 10 15 t Solution x ( t ) 3/20

A projection method The hot-shower example for the eigenvalues of • x ( t ) = Temp diff from optimal Temp Elias Jarlebring • α = Human sensitivity • τ = Length of pipe Speed of water Delayed negative feedback: ˙ x ( t ) = − α x ( t − τ ) , τ = 1 50 50 Hot shower problem 4 α =1.4~ Lambert W α =2 Solution operator 2 Infinitesimal generator Imag Imag 0 0 A Projection method 0 x Numerical examples −2 −50 −50 −4 −4 −4 −2 −2 0 0 2 2 0 5 10 15 Real Real t Spectrum Solution x ( t ) σ (Σ) = { s ∈ C : − s − α e − τ s = 0 } 3/20

A projection method Computing the spectrum for the eigenvalues of Example ( [ Ebenbauer, Allgöwer ’06 ] ) Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples 4/20

A projection method Computing the spectrum for the eigenvalues of Example ( [ Ebenbauer, Allgöwer ’06 ] ) Elias Jarlebring Spectrum σ (Σ) : s ∈ C such that 0 0 1 0 1 1 − 1 13 . 5 − 1 − 5 . 9 7 . 1 − 70 . 3 A + A e − τ s A = 0 det @ − sI + − 3 − 1 − 2 2 − 1 5 @ @ − 2 − 1 − 4 2 0 6 Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples 4/20

A projection method Computing the spectrum for the eigenvalues of Example ( [ Ebenbauer, Allgöwer ’06 ] ) Elias Jarlebring Spectrum σ (Σ) : s ∈ C such that 0 0 1 0 1 1 − 1 13 . 5 − 1 − 5 . 9 7 . 1 − 70 . 3 A + A e − τ s A = 0 det @ − sI + − 3 − 1 − 2 2 − 1 5 @ @ − 2 − 1 − 4 2 0 6 Hot shower problem Lambert W τ =1 50 Solution operator Infinitesimal generator A Projection method Numerical examples Imag σ ( Σ ) 0 −50 −2 −1 0 1 Real σ ( Σ ) 4/20

Methods for σ (Σ) := { s ∈ C : det ( − sI + A 0 + A 1 e − s τ ) = 0 } A projection method for the eigenvalues of Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples 5/20

Methods for σ (Σ) := { s ∈ C : det ( − sI + A 0 + A 1 e − s τ ) = 0 } A projection method for the eigenvalues of Method 1: Lambert W Elias Jarlebring Analytic method for scalar and simultaneously triangularizable systems. Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples 5/20

Methods for σ (Σ) := { s ∈ C : det ( − sI + A 0 + A 1 e − s τ ) = 0 } A projection method for the eigenvalues of Method 1: Lambert W Elias Jarlebring Analytic method for scalar and simultaneously triangularizable systems. Method 2: Solution operator Hot shower problem Numerical method based on the discretization of the solution Lambert W operator. Solution operator Infinitesimal generator A Projection method Numerical examples 5/20

Methods for σ (Σ) := { s ∈ C : det ( − sI + A 0 + A 1 e − s τ ) = 0 } A projection method for the eigenvalues of Method 1: Lambert W Elias Jarlebring Analytic method for scalar and simultaneously triangularizable systems. Method 2: Solution operator Hot shower problem Numerical method based on the discretization of the solution Lambert W operator. Solution operator Infinitesimal generator Method 3: Infinitesimal generator A Projection method Numerical examples Numerical method based on the discretization of the infinitesimal generator. 5/20

Methods for σ (Σ) := { s ∈ C : det ( − sI + A 0 + A 1 e − s τ ) = 0 } A projection method for the eigenvalues of Method 1: Lambert W Elias Jarlebring Analytic method for scalar and simultaneously triangularizable systems. Method 2: Solution operator Hot shower problem Numerical method based on the discretization of the solution Lambert W operator. Solution operator Infinitesimal generator Method 3: Infinitesimal generator A Projection method Numerical examples Numerical method based on the discretization of the infinitesimal generator. Method 4: Subspace accelerated residual inverse iteration A projection method based on residual inverse iteration. 5/20

Methods for σ (Σ) := { s ∈ C : det ( − sI + A 0 + A 1 e − s τ ) = 0 } A projection method for the eigenvalues of Method 1: Lambert W Elias Jarlebring Analytic method for scalar and simultaneously triangularizable systems. Method 2: Solution operator Hot shower problem Numerical method based on the discretization of the solution Lambert W operator. Solution operator Infinitesimal generator Method 3: Infinitesimal generator A Projection method Numerical examples Numerical method based on the discretization of the infinitesimal generator. Method 4: Subspace accelerated residual inverse iteration A projection method based on residual inverse iteration. • Two examples indicate that Method 4 should be used for large problems. 5/20

Methods for σ (Σ) := { s ∈ C : det ( − sI + A 0 + A 1 e − s τ ) = 0 } A projection method for the eigenvalues of Method 1: Lambert W Elias Jarlebring Analytic method for scalar and simultaneously triangularizable systems. Method 2: Solution operator Hot shower problem Numerical method based on the discretization of the solution Lambert W operator. Solution operator Infinitesimal generator Method 3: Infinitesimal generator A Projection method Numerical examples Numerical method based on the discretization of the infinitesimal generator. Method 4: Subspace accelerated residual inverse iteration A projection method based on residual inverse iteration. • Two examples indicate that Method 4 should be used for large problems. • Previously in the literature n = 131, Method 4: n = 10 6 . 5/20