Structure preserving treatment of PCP-palindromic eigenvalue - PowerPoint PPT Presentation

Structure preserving treatment of PCP-palindromic eigenvalue problems Christian Schr¨ oder DFG Research Center Matheon , TU Berlin 8th GAMM Workshop Applied and Numerical Linear Algebra TU Harburg, 11. September 2008 Joint work with H. Fassbender (TU Braunschweig), N. Mackey, D.S. Mackey (Western Michigan U) C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 1 / 14

Introduction Introduction PCP linearization (briefly) PCP Schur form Application, Numerical Experiments C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 2 / 14

Introduction PCP Palindromic Eigenvalue problems ◮ Consider a regular polynomial eigenvalue problem Q ( λ ) x = ( A 0 + λ A 1 + λ 2 A 2 + · · · + λ k A k ) x = 0 . with A i ∈ C n × n given, x ∈ C n , and λ ∈ C wanted C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 2 / 14

Introduction PCP Palindromic Eigenvalue problems ◮ Consider a regular polynomial eigenvalue problem Q ( λ ) x = ( A 0 + λ A 1 + λ 2 A 2 + · · · + λ k A k ) x = 0 . with A i ∈ C n × n given, x ∈ C n , and λ ∈ C wanted ◮ Let P be a real, square, and involutory matrix, i.e., P 2 = I . ◮ Q ( λ ) is PCP palindromic, iff A i = PA k − i P ( A is the complex conjugate of A ) C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 2 / 14

Introduction PCP Palindromic Eigenvalue problems ◮ Consider a regular polynomial eigenvalue problem Q ( λ ) x = ( A 0 + λ A 1 + λ 2 A 2 + · · · + λ k A k ) x = 0 . with A i ∈ C n × n given, x ∈ C n , and λ ∈ C wanted ◮ Let P be a real, square, and involutory matrix, i.e., P 2 = I . ◮ Q ( λ ) is PCP palindromic, iff A i = PA k − i P ( A is the complex conjugate of A ) ◮ This talk is a summary of a paper ⇒ [PCP] ◮ reminicent of ∗ -palindromic problems, A i = A ∗ k − i , see [MMMM2] ◮ Application: Stability analysis of time delay equations (later) C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 2 / 14

P Introduction Eigenvalue pairing Let ( λ, x ) be an eigenpair of Q ( λ ). Then ( A 0 + λ A 1 + λ 2 A 2 + · · · + λ k A k ) x = 0 P ( A 0 + λ A 1 + λ 2 A 2 + · · · + λ k A k ) PPx = 0 ( A k + λ A k − 1 + λ 2 A k − 2 + · · · + λ k A 0 ) Px = 0 2 A k − 2 + · · · + λ k A 0 ) Px = 0 ( A k + λ A k − 1 + λ k Q (1 /λ )( Px ) = 0 λ so, (1 /λ, Px ) is also an eigenpair. C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 3 / 14

Introduction Eigenvalue pairing Let ( λ, x ) be an eigenpair of Q ( λ ). Then ( A 0 + λ A 1 + λ 2 A 2 + · · · + λ k A k ) x = 0 P ( A 0 + λ A 1 + λ 2 A 2 + · · · + λ k A k ) PPx = 0 ( A k + λ A k − 1 + λ 2 A k − 2 + · · · + λ k A 0 ) Px = 0 2 A k − 2 + · · · + λ k A 0 ) Px = 0 ( A k + λ A k − 1 + λ k Q (1 /λ )( Px ) = 0 λ so, (1 /λ, Px ) is also an eigenpair. Theorem: ([PCP]) Let Q ( λ ) = P k i =0 λ i A i , A k � = 0 be a regular PCP matrix polynomial. Then the spectrum of Q ( λ ) has the pairing ( λ, 1 /λ ) . Moreover, algebraic, gemetric and partial multiplicities of the eigenvalues in each pair are equal. Here, λ = 0 is allowed and is paired with the eigenvalue ∞ . C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 3 / 14

Introduction Eigenvalue pairing Let ( λ, x ) be an eigenpair of Q ( λ ). Then ( A 0 + λ A 1 + λ 2 A 2 + · · · + λ k A k ) x = 0 P ( A 0 + λ A 1 + λ 2 A 2 + · · · + λ k A k ) PPx = 0 ( A k + λ A k − 1 + λ 2 A k − 2 + · · · + λ k A 0 ) Px = 0 2 A k − 2 + · · · + λ k A 0 ) Px = 0 ( A k + λ A k − 1 + λ k Q (1 /λ )( Px ) = 0 λ so, (1 /λ, Px ) is also an eigenpair. Theorem: ([PCP]) Let Q ( λ ) = P k i =0 λ i A i , A k � = 0 be a regular PCP matrix polynomial. Then the spectrum of Q ( λ ) has the pairing ( λ, 1 /λ ) . Moreover, algebraic, gemetric and partial multiplicities of the eigenvalues in each pair are equal. Here, λ = 0 is allowed and is paired with the eigenvalue ∞ . Same pairing, as ∗ -palindromic polynomials [MMMM2] C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 3 / 14

Introduction A method, a problem, and a remedy ◮ Eigenvalues of PCP polynomials either ◮ come in pairs ( λ, 1 /λ ), ◮ or they are on the unit circle, i.e., | λ | = 1 (those are the interesting ones for TDSs) ◮ Standard method for polynomial EVPs: Companion form ✷ ✸ ✷ ✸ A k A k − 1 A k − 2 · · · A 0 ✻ ✼ ✻ − I 0 · · · 0 ✼ I ✻ ✼ ✻ ✼ λ + . ✻ ... ✼ ✻ ... ... ✼ . ✻ ✼ ✻ ✼ . ✹ ✺ ✹ ✺ I 0 − I 0 and QZ algorithm C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 4 / 14

Introduction A method, a problem, and a remedy ◮ Eigenvalues of PCP polynomials either ◮ come in pairs ( λ, 1 /λ ), ◮ or they are on the unit circle, i.e., | λ | = 1 (those are the interesting ones for TDSs) ◮ Standard method for polynomial EVPs: Companion form ✷ ✸ ✷ ✸ A k A k − 1 A k − 2 · · · A 0 ✻ ✼ ✻ − I 0 · · · 0 ✼ I ✻ ✼ ✻ ✼ λ + . ✻ ... ✼ ✻ ... ... ✼ . ✻ ✼ ✻ ✼ . ✹ ✺ ✹ ✺ I 0 − I 0 and QZ algorithm ◮ Problem: neither companion form nor QZ algorithm care about PCP structure ⇒ eigenvalue pairing will only be approximate (rounding errors) ◮ Remedy: structure preserving linearization and structure preserving Schur form C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 4 / 14

PCP linearization (briefly) Introduction PCP linearization (briefly) PCP Schur form Application, Numerical Experiments C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 5 / 14

PCP linearization (briefly) L 1 , L 2 , and DL (Λ = [ λ k − 1 , . . . , λ, 1] T ) Consider the pencil spaces [MMMM1,MMMM2] { L ( λ ) = λ X + Y : L ( λ ) · (Λ ⊗ I n ) = v ⊗ Q ( λ ) , v ∈ C k } L 1 ( Q ) := { L ( λ ) = λ X + Y : (Λ T ⊗ I n ) · L ( λ ) = w T ⊗ Q ( λ ) , w ∈ C k } L 2 ( Q ) := DL ( Q ) := L 1 ( Q ) ∩ L 2 ( Q ) , with v = w ◮ generalisations of companion form (member of L 1 ( Q ) with v = e 1 ) C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 5 / 14

PCP linearization (briefly) L 1 , L 2 , and DL (Λ = [ λ k − 1 , . . . , λ, 1] T ) Consider the pencil spaces [MMMM1,MMMM2] { L ( λ ) = λ X + Y : L ( λ ) · (Λ ⊗ I n ) = v ⊗ Q ( λ ) , v ∈ C k } L 1 ( Q ) := { L ( λ ) = λ X + Y : (Λ T ⊗ I n ) · L ( λ ) = w T ⊗ Q ( λ ) , w ∈ C k } L 2 ( Q ) := DL ( Q ) := L 1 ( Q ) ∩ L 2 ( Q ) , with v = w ◮ generalisations of companion form (member of L 1 ( Q ) with v = e 1 ) ◮ source for structured pencils: Theorem: ([PCP]) (Existence/Uniqueness of PCP Pencils in DL ( Q )) Suppose Q ( λ ) is a PCP-polynomial with respect to the involution P. Let F be the flip matrix and let v ∈ C k be any vector such that Fv = v, and let L ( λ ) be the unique pencil in DL ( Q ) with ansatz vector v. Then L ( λ ) is a PCP-pencil with respect to the involution ˜ P = F ⊗ P. C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 5 / 14

PCP linearization (briefly) L 1 , L 2 , and DL (Λ = [ λ k − 1 , . . . , λ, 1] T ) Consider the pencil spaces [MMMM1,MMMM2] { L ( λ ) = λ X + Y : L ( λ ) · (Λ ⊗ I n ) = v ⊗ Q ( λ ) , v ∈ C k } L 1 ( Q ) := { L ( λ ) = λ X + Y : (Λ T ⊗ I n ) · L ( λ ) = w T ⊗ Q ( λ ) , w ∈ C k } L 2 ( Q ) := DL ( Q ) := L 1 ( Q ) ∩ L 2 ( Q ) , with v = w ◮ generalisations of companion form (member of L 1 ( Q ) with v = e 1 ) ◮ source for structured pencils: Theorem: ([PCP]) (Existence/Uniqueness of PCP Pencils in DL ( Q )) Suppose Q ( λ ) is a PCP-polynomial with respect to the involution P. Let F be the flip matrix and let v ∈ C k be any vector such that Fv = v, and let L ( λ ) be the unique pencil in DL ( Q ) with ansatz vector v. Then L ( λ ) is a PCP-pencil with respect to the involution ˜ P = F ⊗ P. ◮ Eigenvalue exclusion [MMMM1]: L ( λ ) is a linearization of Q ( λ ) iff no root of the polynomial v 1 x k − 1 + v 2 x k − 2 + . . . + v 2 x + v 1 is an eigenvalue of Q ( λ ). C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 5 / 14

❶ ➊ ➍ ➊ ➍➀ ➊ ➍ ➊ ➍ ➊ ➍ ➊ ➍ ➊ ➍ ➊ ➍ PCP linearization (briefly) Example: quadratic case with A 1 = PA 1 P , P 2 = I Q ( λ ) x = λ 2 A 2 + λ A 1 + PA 2 Px = 0 , ◮ chose v = [ α, α ] T where − α/α is not an eigenvalue of Q ( λ ) C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 6 / 14

PCP linearization (briefly) Example: quadratic case with A 1 = PA 1 P , P 2 = I Q ( λ ) x = λ 2 A 2 + λ A 1 + PA 2 Px = 0 , ◮ chose v = [ α, α ] T where − α/α is not an eigenvalue of Q ( λ ) ◮ DL ( Q )-linearization is (rows resamble Q ( λ )) ❶ ➊ ➍ ➊ ➍➀ ➊ ➍ ➊ ➍ α A 2 α A 2 α A 1 − α A 2 α PA 2 P λ x 0 λ + = α A 2 α A 1 − α PA 2 P α PA 2 P α PA 2 P x 0 ◮ it is PCP, because ➊ ➍ ➊ ➍ ➊ ➍ ➊ ➍ α A 2 α A 2 P α A 1 − α A 2 α PA 2 P P = α A 2 α A 1 − α PA 2 P P α PA 2 P α PA 2 P P C. Schr¨ oder (TU Berlin) PCP-Palindromic Eigenvalue Problems GAMM LA 08 6 / 14