Palindromic polynomial evp. Linearization Trimmed linearization Palindromic pencils Numerical Methods Conclusions Nonlinear palindromic eigenvalue problems and their numerical solution Volker Mehrmann TU Berlin DFG Research Center Institut für Mathematik M ATHEON IN MEMORIAM RALPH BYERS Volker Mehrmann mehrmann@math.tu-berlin.de Nonlinear palindromic eigenvalue problems and their numerical solution
Palindromic polynomial evp. Linearization Trimmed linearization Palindromic pencils Numerical Methods Conclusions Polynomial eigenvalue problems k � A i λ i ) x = 0 , P ( λ ) x = ( i = 0 where ◮ x is a real or complex eigenvector; ◮ λ is a real or complex eigenvalue. Volker Mehrmann mehrmann@math.tu-berlin.de Nonlinear palindromic eigenvalue problems and their numerical solution
Palindromic polynomial evp. Linearization Trimmed linearization Palindromic pencils Numerical Methods Conclusions Palindromic structure Definition A nonlinear matrix function P ( λ ) is called ◮ H-palindromic if P ( λ ) = P ( λ − 1 ) H . ◮ T-palindromic if P ( λ ) = P ( λ − 1 ) T . Volker Mehrmann mehrmann@math.tu-berlin.de Nonlinear palindromic eigenvalue problems and their numerical solution
Palindromic polynomial evp. Linearization Trimmed linearization Palindromic pencils Numerical Methods Conclusions Properties of palindromic matrix polynomials. Proposition Consider a T-palindromic eigenvalue problem P ( λ ) x = 0. Then P ( λ ) x = 0 if and only if x T P ( 1 /λ ) = 0, i.e., the eigenvalues occur in pairs λ , 1 /λ . Consider a H-palindromic eigenvalue problem P ( λ ) x = 0. Then P ( λ ) x = 0 if and only if x H P ( 1 / ¯ λ ) = 0, i.e., the eigenvalues occur in pairs λ , 1 / ¯ λ . Palindromic matrix functions have symplectic spectrum, they generalize symplectic problems λ I + S , where S is a symplectic matrix. In the following the prefix T and H is dropped and we use ∗ for both. The differences are pointed pointed out when necessary. Volker Mehrmann mehrmann@math.tu-berlin.de Nonlinear palindromic eigenvalue problems and their numerical solution
Palindromic polynomial evp. Linearization Trimmed linearization Palindromic pencils Numerical Methods Conclusions Applications of palindromic matrix polynomials. Excitation of rail tracks by high speed trains Hilliges 04, Hilliges/Mehl/M. 04. Periodic surface acoustic wave filters Zaglmeyer 02. Optimal control of (high order) discrete time systems Mackey/Mackey/Mehl/M. 06, Schröder 08. Computation of the Crawford number Higham/Tisseur/Van Dooren 02. See also survey papers by Meerbergen/Tisseur 00, M./Voss 05. Essentially all problems, where symplectic/unitary matrices/pencils appear (nonlinear structures) can be formulated as palindromic problems (linear structure). Byers/Mackey/M./Xu 08 Volker Mehrmann mehrmann@math.tu-berlin.de Nonlinear palindromic eigenvalue problems and their numerical solution
Palindromic polynomial evp. Linearization Trimmed linearization Palindromic pencils Numerical Methods Conclusions Example: Discrete time optimal control Minimize ∞ � ( x ∗ j Qx j + u ∗ j Ru j ) j = 0 subject to the discrete time control problem Ex k + 1 = Ax k + Bu k , with x 0 given. Instead of the usual symplectic formulation, the necessary condition for optimality can be formulated (Schröder 08) as a discrete palindromic boundary value problem Z ∗ z k + 1 = Zz k , with 0 A B . Z = E ∗ Q S 0 S ∗ R Volker Mehrmann mehrmann@math.tu-berlin.de Nonlinear palindromic eigenvalue problems and their numerical solution
Palindromic polynomial evp. Linearization Trimmed linearization Palindromic pencils Numerical Methods Conclusions Linearization of matrix polynomials Definition: For an n × n matrix polynomial P ( λ ) , a matrix pencil L ( λ ) = λ E + A of size nk × nk is called linearization of P ( λ ) , if there exist nonsingular unimodular matrices (i.e., of constant nonzero determinant) S ( λ ) , T ( λ ) such that S ( λ ) L ( λ ) T ( λ ) = diag ( P ( λ ) , I ( k − 1 ) n ) . Volker Mehrmann mehrmann@math.tu-berlin.de Nonlinear palindromic eigenvalue problems and their numerical solution
Palindromic polynomial evp. Linearization Trimmed linearization Palindromic pencils Numerical Methods Conclusions Linearization vs. polynomial methods Direct solution of polynomial problem ◮ Newton, high order Arnoldi, Jacobi-Davidson, inverse iteration. ◮ All these essentially compute one or a few eigenvalues and associated eigenvectors. ◮ Good for large scale and general nonlinear problems. Linearization ◮ Linearization and solution of linear problem. ◮ Compute all eigenvalues and eigenvectors, invariant subspaces. ◮ Large scale and small scale problems. ◮ Potentially increased ill-conditioning. Volker Mehrmann mehrmann@math.tu-berlin.de Nonlinear palindromic eigenvalue problems and their numerical solution
Palindromic polynomial evp. Linearization Trimmed linearization Palindromic pencils Numerical Methods Conclusions Linearization and structure Example The companion linearization of the palindromic quadratic eigenvalue problem ( λ 2 A + λ B + A T ) x = 0 with B = B T is given by � � � � � − A T � � � A 0 y − B y λ = , 0 I x I 0 x which is not palindromic. ◮ Numerical methods destroy symplectic spectrum in finite arithmetic ! ◮ Perturbation theory requires structured perturbations near unit circle Ran/Rodman 1988, Bora/M. 2005. ◮ We need structure preserving linearizations. Volker Mehrmann mehrmann@math.tu-berlin.de Nonlinear palindromic eigenvalue problems and their numerical solution
Palindromic polynomial evp. Linearization Trimmed linearization Palindromic pencils Numerical Methods Conclusions Optimal Linearizations. Goal: Find class of linearizations for which: ◮ the linear pencil is easily constructed; ◮ structure preserving linearizations exist; ◮ the conditioning of the linear problem can be characterized and optimized; ◮ eigenvalues/eigenvectors of the original problem are easily read off; ◮ we have structure preserving numerical methods; ◮ a structured perturbation analysis is possible. Volker Mehrmann mehrmann@math.tu-berlin.de Nonlinear palindromic eigenvalue problems and their numerical solution
Palindromic polynomial evp. Linearization Trimmed linearization Palindromic pencils Numerical Methods Conclusions Vector spaces of potential linearizations Notation: Λ := [ λ k − 1 , λ k − 2 , . . . , λ, 1 ] T , ⊗ - Kronecker product. Definition Mackey 2 /Mehl/M. 06. For a given n × n matrix polynomial P ( λ ) of degree k define the sets: { v ⊗ P ( λ ) : v ∈ F k } , v is called right ansatz vector , V P := { w T ⊗ P ( λ ) : w ∈ F k } , w is called left ansatz vector , W P := � � L ( λ ) = λ X + Y : X , Y ∈ F kn × kn , L ( λ ) · (Λ ⊗ I n ) ∈ V P L 1 ( P ) := , � Λ T ⊗ I n � L ( λ ) = λ X + Y : X , Y ∈ F kn × kn , � � L 2 ( P ) := · L ( λ ) ∈ W P , DL ( P ) := L 1 ( P ) ∩ L 2 ( P ) . We have the freedom to choose the vector v . These are not all linearizations but they form a large class. Volker Mehrmann mehrmann@math.tu-berlin.de Nonlinear palindromic eigenvalue problems and their numerical solution
Palindromic polynomial evp. Linearization Trimmed linearization Palindromic pencils Numerical Methods Conclusions Example The first and second companion forms A k 0 · · · 0 A k − 1 A k − 2 · · · A 0 . ... . − I n 0 · · · 0 0 I n . C 1 ( λ ) := λ + . . ... ... . ... ... . . . . . . 0 0 · · · − I n 0 0 · · · 0 I n A k 0 · · · 0 A k − 1 − I n · · · 0 . . ... ... . . 0 I n . A k − 2 0 . C 2 ( λ ) := λ + . . . . ... ... ... . . . . 0 . . − I n 0 · · · 0 I n A 0 0 · · · 0 are linearizations in L 1 ( P ) , L 2 ( P ) , respectively. Volker Mehrmann mehrmann@math.tu-berlin.de Nonlinear palindromic eigenvalue problems and their numerical solution
Palindromic polynomial evp. Linearization Trimmed linearization Palindromic pencils Numerical Methods Conclusions Eigenvector Recovery Property Theorem: Mackey 2 /Mehl/M. 06. Let P ( λ ) be an n × n matrix polynomial of degree k , and let L ( λ ) be any pencil in L 1 ( P ) with ansatz vector v � = 0. Then x ∈ C n is a right eigenvector for P ( λ ) with finite eigenvalue λ ∈ C if and only if Λ ⊗ x is a right eigenvector for L ( λ ) with eigenvalue λ . If in addition P is regular, i.e. det P ( λ ) �≡ 0, and L ∈ L 1 ( P ) is a linearization, then every eigenvector of L with finite eigenvalue λ is of the form Λ ⊗ x for some eigenvector x of P . Similar results hold for L 2 ( P ) . Volker Mehrmann mehrmann@math.tu-berlin.de Nonlinear palindromic eigenvalue problems and their numerical solution
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