Structured Algorithms for Palindromic Quadratic Eigenvalue Problems : Vibration of Fast Trains Wen-Wei Lin Department of Mathematics Tsing Hua University, Taiwan A joint work with J. Qian and T.-M. Huang Jan., 2008 W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 1 / 54
Outline 1 Palindromic quadratic eigenvalue problem 2 Structured Algorithm I 3 Structured Algorithm II vs. URV-based structured Algorithm 4 T-skew-Hamiltonian Arnoldi algorithm 5 Generalized ⊤ -skew-Hamiltonian Arnoldi Algorithm 6 Numerical results 7 Conclusions W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 2 / 54
Resonances in rail tracks excited by high speed trains With new ICE trains crossing Europe at speeds of up to 300 km/h, sound and vibration levels in the trains are an important issue. Hilliges/Mehrmann/Mehl(2004) first proposed this problem on a project with company SFE GmbH in Berlin. W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 3 / 54
Finite Element Model (a). The rails are straight and infinitely long: A 3D finite element discretization of the rail with linear isoparametric tetrahedron elements(Chu/Huang/Lin/Wu, JCAM, 2007) produces an infinite-dimensional system of O.D.E.: M ¨ x + D ˙ x + Kx = F, where M, D and K are block-tridiagonal matrices. Figure: A 3D rail model. W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 4 / 54
(b). The rail sections of track between two ties are identical: The system is periodic and leads to a Palindromic QEP P ( λ ) x ≡ ( λ 2 A ⊤ 1 + λA 0 + A 1 ) x = 0 , (1) where A 0 , A 1 ∈ C n × n , and A ⊤ 0 = A 0 . ◮ The coefficient matrices A 0 and A 1 in (1) depend on some ω associated with the excitation frequency of the external force. ◮ Eq. (1) is called a palindromic QEP problem. Symplectic property If λ is an eigenvalue of P ( λ ) , then so is λ − 1 . W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 5 / 54
Numerical difficulties Need to compute all finite, nonzero eigenvalues and eigenvectors for all ω ; Lots of eigenvalues are 0 and ∞ ; Problem size 1,000–100,000; Range of eigenvalues | λ | ∈ [10 − 20 , 10 20 ] ; Problem is badly scaled; 20 10 15 10 10 10 5 10 | λ | 0 10 - 5 � 10 - 10 � 10 - 15 � 10 10 - 20 0 20 40 60 80 100 120 W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 6 / 54
Find all eigenpairs of a 2 n × 2 n linearized form �� � � �� � x � A 1 0 0 I ( M − λ L ) z ≡ − λ = 0 A ⊤ − A 0 − I 0 λx 1 by using QZ algorithm (standard approach). ⇒ The symplectic structure is not preserved which produces large numerical errors. D. Mackey/N. Mackey/Mehl/Mehrmann (SIMAX, 2006 (2)) linearize Eq. (1) to a palindromic linear pencil λZ ± Z ⊤ . Hilliges/Mehl/Mehrmann (2005) propose a hybrid method (Laub’s trick+ Jacobi-like method) to solve λZ ± Z ⊤ . ⇒ It work well if there are no EWs near ± 1 . and needs O ( n 3 log( n )) flops. W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 7 / 54
Schr ¨ o der (2001) proposed an URV based structured method for solving λZ ± Z ⊤ . ⇒ It is equivalent to the structured Algorithm I (SA I) applying to µ 2 Z ⊤ + µ 0 + Z with µ 2 = λ . Chu/Huang/L/Wu (JCAM, 2007) proposed a structure-preserving doubling alg. for solving NME: X + A ⊤ 1 X − 1 A 1 = − A 0 associated with (1). ⇒ The numerical results show much promise, but the convergence theory holds under assumption that the sequence by SDA is well-defined. W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 8 / 54
Consider the linearization: �� A 1 � � 0 �� � x � 0 I ( M − λ L ) z ≡ − λ = 0 . (2) A ⊤ − A 0 − I 0 λx 1 M and L satisfy � 0 � I n MJ M ⊤ = LJ L ⊤ with J = . (3) − I n 0 M − λ L is called a ⊤ -symplectic pencil. If ν is an eigenvalue of M − λ L , so is 1 /ν . W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 9 / 54
Structured Algorithm I (SA I) ( S + S − 1 ) -transform for ⊤ -symplectic M − λ L (L, LAA, 1987): M s ≡ MJ L ⊤ + LJ M ⊤ , L s ≡ LJ L ⊤ . (4) µ is a double eigenvalue of M s − λ L s ⇔ ν, 1 ν are eigenvalues of M − λ L , where ν, 1 ν are two roots of λ + 1 λ = µ. Theorem 2 ] ⊤ is an EV Let M − λ L be the T-symplectic pencil of (2). If z s = [ z ⊤ 1 , z ⊤ of M s − λ L s corresp. to µ = ν + 1 ν ( µ � = ± 2) , then z 1 + 1 ν z 2 and z 1 + νz 2 are EVs of P ( λ ) in (1) corresp. to ν and 1 ν , resp.. W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 10 / 54
From (2), we have � 0 � A 1 − A ⊤ � � A 0 − A 1 1 M s − λ L s = − λ A 1 − A ⊤ A ⊤ − A 0 0 1 1 �� � � − A 1 �� A ⊤ A 0 1 − A 1 0 = − λ J A 1 − A ⊤ − A ⊤ A 0 0 1 1 ≡ ( K − λ N ) J , (5) where K , N are both ⊤ -skew-Hamiltonian, i.e., ( KJ ) ⊤ = −KJ and ( NJ ) ⊤ = −NJ . Note that a matrix S is ⊤ -skew-Hamiltonian, then � S 11 � S 12 with S ⊤ 12 = − S 12 , and S ⊤ S = 21 = − S 21 . S ⊤ S 21 11 W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 11 / 54
Remark The ( S + S − 1 ) -transform M s − λ L s , in general, is a nonlinear transform, e.g., the ( S + S − 1 ) -transform of � A � � I � 0 G M − λ L ≡ − λ arising in discrete-time optimal A ⊤ − H I 0 controls produces a quadratic ( S + S − 1 ) -transform. The special form of the T-symplectic pencil (2) leads to a linear ( S + S − 1 ) -transform. This is an advantage by applying patel’s trick (Patel, LAA, 1993) to K − λ N to find all eigenpair of P ( λ ) . W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 12 / 54
If K and N are both real skew-Hamiltonian, Patel (LAA, 1993) developed an efficient structured algorithm for reducing K − µ N to a block triangular condensed form � K 11 � N 11 � � K 12 N 12 Q H ( K − µ N ) Z = − µ , K ⊤ N ⊤ 0 0 11 11 where K 11 , N 11 are upper Hessenberg and triangular, resp.. respectively, K 12 and N 12 are skew-symm., Y and Z are orthogonal matrices satisfying Z = J ⊤ QJ . Goal (Straightforward) Extend Patel’s approach to complex ⊤ -skew-Hamiltonian pencil ( K , N ) in (5). W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 13 / 54
Definition A matrix U ∈ C 2 n × 2 n is unitary ⊤ -symplectic, if UU H = I 2 n and UJ U ⊤ = J . The set of unitary ⊤ -symplectic matrices in C 2 n × 2 n is denoted by UTS 2 n . Lemma � U 1 � ¯ U 2 (i) Any matrix U ∈ UTS 2 n is of the form U = , where ¯ − U 2 U 1 U 1 , U 2 ∈ C n × n . (ii) For any matrix U ∈ UTS 2 n , if S is ⊤ -skew-Hamiltonian, then U H SU is still ⊤ -skew-Hamiltonian. W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 14 / 54
A standard Givens rotation in C 2 n × 2 n is given by I i − 1 c s ¯ G ( i, j, c, s ) = I j − i − 1 , (6) − s c ¯ I 2 n − j where c ¯ c + s ¯ s = 1 . Three types of unitary ⊤ -symplectic matrices 1 G s 1 ( i, c, s ) := G ( i, n + i, c, s ) � G ( i, j, c, s ) � 0 2 G s 2 ( i, j, c, s ) := ¯ 0 G ( i, j, c, s ) � H ( k, v ) � 0 3 H s ( k, v ) := , where ¯ 0 H ( k, v ) H ( k, v ) = I k ⊕ ( I n − k − 2 vv H v H v ) with v ∈ C n − k . W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 15 / 54
Two types of transformations in the structured algorithm 1 Using Givens ⊤ -symplectic matrices G s 1 ( i, c, s ) : K := G H N := G H s 1 K G s 1 , s 1 N G s 1 . (7) 2 Let U, V ∈ C n × n be unitary representing the application of Givens rotations. Set K := Q H N := Q H 0 K Z 0 , 0 N Z 0 , (8) where � U ⊤ � � V � 0 0 Q H 0 = , Z 0 = . (9) V ⊤ 0 0 U Remark Let ( K , N ) be ⊤ -skew-Hamiltonian pencil. Then the resulting K and N in (7) and (8) are still ⊤ -skew-Hamiltonian. Sketch W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 16 / 54
Condensed form of the structured algorithm I Let K and N be ⊤ -skew-Hamiltonian. Then K − λ N can be reduced to a block triangular structure, that is, � K 11 � � N 11 � K 12 N 12 K := Q H KZ = N := Q H NZ = , , (10) K ⊤ N ⊤ 0 0 11 11 where K 11 ∈ C n × n is upper Hessenberg, N 11 ∈ C n × n is upper triangular, and Q , Z are unitary satisfying Z = J ⊤ QJ . W.W. Lin (Tsing Hua Univ.) Palindromic QEP Jan., 2008 17 / 54
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