Perturbation of Palindromic Eigenvalue Problems

Transcription

1 Numerische Mathematik manuscript No. (will be inserted by the editor) Eric King-wah Chu Wen-Wei Lin Chern-Shuh Wang Perturbation of Palindromic Eigenvalue Problems Received: date / Revised version: date Abstract We investigate the perturbation of the palindromic eigenvalue problem for the matrix quadratic P (λ) λ 2 A T 1 + λa 0 + A 1, with A 0, A 1 C n n and A T 0 = A 0. The perturbation of palindromic eigenvalues and eigenvectors, in terms of general matrix polynomials, palindromic linearizations, (semi-schur) anti-triangular canonical forms, differentiation and Sun s implicit function approach, are discussed. Keywords Anti-triangular form, eigenvalue, eigenvector, implicit function theorem, palindromic linearization, matrix polynomial, palindromic eigenvalue problem, perturbation. Author, Journal Volume, (year) page numbers. Eric King-wah Chu School of Mathematical Sciences, Building 28, Monash University, VIC 3800, Australia. Tel.: Fax: eric.chu@sci.monash.edu.au Wen-Wei Lin Department of Mathematics, National Tsinghua University, Hsinchu 300, Taiwan. Tel.: Fax: wwlin@am.nthu.edu.tw Chern-Shuh Wang Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan. Tel.: Fax: cswang@math.ncku.edu.tw

2 2 E.K.-w. Chu, W.-W. Lin & C.-S. Wang 1 Introduction Consider the matrix quadratic P (λ) λ 2 A T 1 + λa 0 + A 1 (1) where A 0, A 1 C n n and A T 0 = A 0, and the corresponding palindromic quadratic eigenvalue problem P (λ)x = 0, x 0 (2) In general, an eigenvalue problem is described as palindromic if the spectrum contains both λ and λ 1. In other words, the eigenvalue problem of the original matrix polynomial P (λ) (in one of its equivalent forms) has a palindromic linearization [3,8] of the form λz ± Z T. (We can transform λz Z T back to the form ν( Z) + ( Z) T with ν = λ. Similarly, λ 2 A T 1 + λa 0 + A 1 and ν 2 A T 1 νa 0 + A 1 define equivalent palindromic eigenvalue problems.) It is clear that (2) is palindromic, after checking its transpose. A great foundation for the solution of palindromic eigenvalue problems has been laid by Hilliage, Mackey, Mehl and Mehrmann in [6,8,9]. There has been much recent interest in quadratic eigenvalue problems [14]. An important example of palindromic eigenvalue problems can be found in the vibration analysis of fast trains; see [6, 7] for general introductions and [5] for details. For general perturbation of eigenvalues for polynomial eigenvalue problems, see [1, 13]. On results for general matrix polynomials, see the masterpiece [3]. This paper is organized as follows. In Sections 2 5, the perturbation of palindromic eigenvalues, in terms of general matrix polynomials, palindromic linearizations, the anti-triangular canonical form [8 10] and the semi-schur anti-triangular canonical form, is discussed using the Bauer-Fike technique [1]. The perturbation result for a simple eigenvalue and its corresponding eigenvector is obtained by differentiation in Section 6. Sun s implicit function approach [12] is then applied to palindromic linearizations, general matrix quadratics and palindromic eigenvalue problems in Section 7, to obtain perturbation results for eigenvalues and the corresponding deflating subspaces. The paper is concluded in Section 8. Note that the Bauer-Fike technique allows perturbations of arbitrary size and the clustering of eigenvalues (and the corresponding deflating subspaces) may vary greatly. Consequently, it is meaningless to talk about the perturbation of eigenvectors or deflating subspaces in Bauer-Fike type perturbation theorems in Sections Bauer-Fike Theorem for General Matrix Polynomials From [1, Theorem 4.2], we have the following theorem on the perturbation of eigenvalues of a general matrix polynomial:

3 Perturbation of Palindromic Eigenvalue Problems 3 Theorem 21 Consider a regular matrix polynomial and its perturbation L(α, β) L(α, β) l B j α j β l j, j=0 l B j α j β l j, Bj B j + δb j (j = 0,, l). j=0 Let (X, T, Z) be a resolvent triple for L constructed using some finite and infinite Jordan pairs J F and J. For (α i, β i ) σ(l) and (α, β) σ( L), the spectral variation of L from L is defined as s L ( L) max (α,β) {s (α,β)}, s (α,β) min i { αβ i βα i }. Let p be the maximum dimension of the Jordan blocks in J F or J. Then for τ (τ = 1, 2, ), we have s (α,β) max{θ 1, θ 1/p 1 }, θ 1 pfκ, (3) where c 1 = 1 (for τ = 1, 2) or l (for τ = ), and Also, we have F c 1 l + 1 2, κ X Z, [δb 0,, δb l ]. s L ( L) max{θ 1, θ 1/p 1 }. (4) Note that we use the representation (α, β) for λ = α/β in the above theorem. In our palindromic case, we have l = 2 and B 0 = A T 1 = B T 2 and B 1 = A 0 = A T 0. Ultimately, the perturbation of the palindromic eigenvalues is controlled by θ 1 in (4), which is in turn dominated by the error term involving [δa 0, δa 1 ]. Note also that the perturbation in δa 0 may be nonsymmetric, pushing a pair of reciprocal palindromic eigenvalues to one which is only approximately reciprocal. For a symmetric δa 0, we only have to consider the perturbation of half of the eigenvalues, because of the palindromic structure. The unstructured perturbation result in Theorem 21, for general matrix polynomials, may not be applicable or satisfactory for palindromic eigenvalue problems. However, it serves as a reference for the structured perturbation results are in later Sections.

4 4 E.K.-w. Chu, W.-W. Lin & C.-S. Wang 3 Bauer-Fike Theorems for Palindromic Linearizations For the linearization λz + Z T, we can work from the Kronecker canonical form ( λz Z T ) Q 2 = λλ + Λ, Q 1 = [P 1, P 2 ], Q 2 = [P 2, P 1 ] where Q T 1 [ ] [ ] Λ I Λ + =, Λ I =, Λ = diag{j Λ 1,, J N } with J i being the Jordan block for λ i inside the unit circle. Applying the techniques in [1], we consider the singular matrix Q T [ 1 β(z + δz) α(z + δz) T ] Q 2 = (βλ + αλ ) [ I + (βλ + αλ ) 1 Q T 1 (βδz αδz T ] )Q 2 which implies κ 2 (βλ + αλ ) 1 βδz αδz T 1 where κ 2 Q 1 Q 2. Estimation of (βλ + αλ ) 1, with Λ ± containing various Jordan blocks, produces the following theorem. Theorem 31 Consider the palindromic linearization M βz αz T with the above Kronecker canonical form. Let Z = Z + δz, M β Z α ZT, (α i, β i ) σ(m) and (α, β) σ( M). The spectral variation of L from L is defined as s L ( L) max (α,β) {s (α,β)}, Then for any Holder norm, we have s (α,β) min i { αβ i βα i }. s (α,β) max{θ 2, θ 1/p 2 }, θ 2 c 2 κ 2 δz (5) for some p n and c 2 = 2(α 2 + β 2 ) p. Also, we have s L ( L) max{θ 2, θ 1/p 2 }. (6) We have the following three cases: (i) for small perturbations, we have p being the size of the Jordan block associated with (α i, β i ); (ii) for large perturbations, we have p = 1; and (iii) for general perturbations, we have p being the size of the largest Jordan block in Λ. Note that for the 2- or F-norm, we have θ 2 2(α 2 + β 2 ) p δz. With scaling, α 2 + β 2 = 1, s (α,β) becomes the chordal metric and θ 2 2 p δz. From [1], when we consider asymptotically small perturbations, there will be a 1-1 correspondence between the original and perturbed eigenvalues. The above perturbation bounds can be proved for a particular eigenvalue (α i, β i ) with the condition number κ 2 replacable by the product of the norms of the individual left- and right-eigenvectors. Details can be found in [1].

5 Perturbation of Palindromic Eigenvalue Problems 5 4 Perturbation of Palindromic Linearizations in Anti-Triangular Form From [10], we have the following anti-triangular canonical form: Theorem 41 Let Z + λz T be a regular n n palindromic linearization. There exists a unitary U C n n such that U T ZU = (m ij ) with m ij = 0 (i + j n + 1) (i.e., U T ZU is anti-triangular, with zero elements on the upper left corner). The palindromic eigenvalues are: m 1n m n1, m 2,n 1 m n 1,2,, m i,n i+1 m n i+1,i,, m n i+1,i m i,n i+1,, m n 1,2 m 2,n 1, m n1 m 1n Let N be the strict lower right triangular part of U T ZU. Reorganize the anti-triangular in Theorem 41 in lower triangular form: U T (Z + λz T )UP n = (D 1 + N 1 ) + λ(d 2 + N 2 ) (7) with the order-reversing permutation matrix P n = [e n, e n 1,, e 1 ], we have D 1 = diag{m 1n, m 2,n 1,, m n 1,2, m n1 } D 2 = P n D 1 P n = diag{m n1, m n 1,2,, m 2,n 1, m 1n } with N 1 = NP n and N 2 = N T P n being strictly lower triangular. Using the Schur form in (7), we can prove the following perturbation result for a palindromic linearization. Theorem 42 Consider the palindromic linearization M βz αz T with N being the strictly lower right triangular part of its anti-triangular canonical form. Let Z = Z + δz, M β Z α ZT, (α i, β i ) σ(m) and (α, β) σ( M). Assume the scaling α 2 + β 2 = 1 = αi 2 + β2 i. Then for any Holder norm, we have s (α,β) c 0 max{θ 3, θ 1/p 3 }, θ 3 c 3 δz (8) for some p n and c 0 min{2, p}, c 3 = 2 p N. Also, we have s L ( L) c 0 max{θ 3, θ 1/p 3 }. (9) Proof. Consider the singular matrix β(z + δz) α(z + δz) T = U [ β(d 1 + N 1 + U T δzup n ) α(d 2 + N 2 + U T δz T UP n ) ] P n U H = U [β(d 1 + N 1 ) α(d 2 + N 2 )] {I + [β(d 1 + N 1 ) α(d 2 + N 2 )] 1 U H (βδz αδz T )UP n } P n U H It is easy to see that [β(d 1 + N 1 ) α(d 2 + N 2 )] 1 βδz αδz T

6 6 E.K.-w. Chu, W.-W. Lin & C.-S. Wang [β(d 1 + N 1 ) α(d 2 + N 2 )] 1 U H (βδz αδz T )UP n 1 (10) Note that [β(d 1 + N 1 ) α(d 2 + N 2 )] is assumed to be nonsingular, otherwise the results in the theorem become trivial. With z min (αi,β i) βd 1 αd 2 = min (αi,β i) βα i αβ i and M [β(d 1 + N 1 ) α(d 2 + N 2 )] 1 = [ I (βd 1 αd 2 ) 1 (βn 1 αn 2 ) ] 1 (βd1 αd 2 ) 1 Using the Neumann series with D βd 1 αd 2 and Ñ βn 1 αn 2 : [ I (βd1 αd 2 ) 1 (βn 1 αn 2 ) ] 1 = (I Ñ) 1 = I + Ñ + + Ñ p 1 we obtain M Ñ 1 η 1 z 1 (1 + Ñ z Ñ p 1 z p+1 ) With x Ñ 1 z, we have the polynomial x p η(1 + x + + x p 1 ) = 0 (see the result in the Appendix), the technique in [1] bounds M via x Ñ 1 z c 0 max{η, η 1/p } Together with the properties of norms, this completes the proof. With scaling, s (α,β) equals the chordal metric. Note also that p is the integer for which Ñ k 0 (0 k < p) and Ñ p = 0. 5 For Palindromic Linearizations in Semi-Schur Anti-Triangular Form A refinement of Theorem 42, with smaller values for p, can be proved. We first refine the decomposition in Theorem 41. Theorem 51 Let Z + λz T be a regular palindromic linearization. There exists a nonsingular U, V C n n, V T ZU = anti-diag{m 1,, m r }, m j = D j + N j (11) where m j is anti-triangular with anti-diagonal block D j = anti-diag{λ j }. Proof. The proof is similar to the standard transformation of a Schur decomposition to the corresponding Jordan canonical form. It is sufficient to show that it is possible to transform the anti-triangular form (U T ZU, U T Z T U) in Theorem 41 to anti-block diagional form, so that [ ] [ ] [ ] [ ] [ ] [ ] I 0 P T U I T I Q I 0 0 T1 I Q T ZU = 0 I P T = 1 I T 2 T 12 0 I T 2 [ ] [ ] [ ] [ ] [ ] [ ] I 0 P T U I T Z T I Q I 0 0 T T U = 2 I Q T 0 I P T I T2 T T12 T = 2 T 0 I T1 T

7 Perturbation of Palindromic Eigenvalue Problems 7 assuming that the spectra of T 1 and T 2 are nonintersecting. Multiplying out the above equation produces the simultaneous linear matrix equation φ(p, Q) (T 2 Q + P T T 1, T T 1 Q + P T T T 2 ) = (T 21, T T 21) (12) which is uniquely solvable. Similar to Theorem 51 but with p being the bounded by the maximum size of m j, we can now prove the following refined version of Theorem 42: Theorem 52 Consider the palindromic linearization M βz αz T with its semi-schur anti-triangular canonical form (11). Let Z = Z + δz, M β Z α Z T, (α i, β i ) σ(m) and (α, β) σ( M). Assume the scaling α 2 +β 2 = 1 = αi 2 + β2 i. Then for any Holder norm, we have s (α,β) c 0 max{θ 4, θ 1/p 4 }, θ 4 c 4 δz (13) for κ 4 U U 1, and c 0 min{2, p}, c 4 = 2 pκ 4 max j N j. Also, we have s L ( L) c 0 max{θ 4, θ 1/p 4 }. (14) We have the following three cases: (i) for small perturbations, we have p being the size of the Schur block m j associated with (α i, β i ); (ii) for large perturbations, we have p = 1; and (iii) for general perturbations, we have p being the size of the largest Schur block m j. Similar to Theorem 31, when we consider asymptotically small perturbations, there will be a 1-1 correspondence between the original and perturbed eigenvalues. The above perturbation bounds can be proved for a particular eigenvalue (α i, β i ) with the condition number κ 4 replacable by U j V j. In addition, we can consider a group of neighbouring eigenvalues together, instead of one particular eigenvalue. This will increase p, or the size of the corresponding semi-shcur block m j, but will improve the conditioning of the linear operator φ in (12) as well as κ 4. 6 Perturbation by Differentiation Without establishing the existence of asymptotic expansions (which can be achieved using the implicit function approach in Section 7), perturbation results can be obtained by simple differentiation. Consider the palindromic eigenvalue problem P (λ, ρ)x = 0, z T x 1 = 0, P (λ, ρ) λ(ρ) 2 A T 1 (ρ) + λ(ρ)a 0 (ρ) + A 1 (ρ)

8 8 E.K.-w. Chu, W.-W. Lin & C.-S. Wang with ρ being the perturbation parameter and A 0 (0) = A 0, A 1 (0) = A 1. For a simple eigenvalue λ, differentiation produces λ ρ = yt P ρ x y T P λ x = y T P ρ x y T (2λA T 1 + A 0)x (15) and Choosing z = y(0), we have P x ρ = (λ ρ P λ + P ρ )x, z T x ρ = 0 x ρ = P (λ ρ P λ + P ρ )x The usual conclusion can be drawn the right-eigenvector x will be rotated through a big angle, even for a small perturbation, when P is large or when the separation between λ and other eigenvalues is fine. This happens, of course, when the assumption of simplicity for the eigenvalues is nearly violated. 7 Sun s Implicit Function Approach In this Section, we shall abbreviate the development of Sun s approach [2, 12], by establishing the functions to which the implicit function theorem can be applied. Detailed and subtle argument of this approach can be found in [2,12]. 7.1 Palindromic Linearizations Using the anti-triangular form in (41), Sun s approach [2] can be applied to obtain the power series of eigenvalues and deflating subspaces. Without loss of generality, we shall use an upper-triangular form in this section (with the help of the reordering operator P r ), so that Sun s approach can be followed faithfully. Assume that U in (41) is further organized to reflect the symmetry of the eigenvalue pairs {λ j, λ 1 j }, so that and U = [U 1, U 0, U 1 ] = [U 1, U 2 ] = [U 2, U 1 ] UP n = [U 1 P p, U 0 P n 2p, U 1 P p ] = [U 1 P p, U 2 P n p ] = [U 2 P n p, U 1 P p ] The anti-triangular form now becomes U T (Z λz T )UP n [ ] [ ] Λα1 Λβ1 = λ = Λ α1 Λ 0 Λ α2 0 Λ α0 λ Λ β1 Λ β0 β2 0 Λ α, 1 0 Λ β, 1

9 Perturbation of Palindromic Eigenvalue Problems 9 with Λ α1 and Λ β1 being p p (2p < n), Λ β, 1 = P p Λ α1 P p, Λ β,1 = P p Λ α, 1 P p, Λ β0 = P 2n p Λ α0 P 2n p and {U 1, U 1 } spanning the deflating subspaces of Z λz T corresponding to (Λ α1, Λ β1 ). Assume that (Λ α1, Λ β1 ) and (Λ α2, Λ β2 ) have nonitersecting spectra. Following the usual step in Sun s approach [2], we assume that Z(ρ) is dependent on a perturbation parameter ρ. We then have [ ] [ ] M U T M11 M Z(ρ)UP n = 12 U T 1 ZU 1 P p U1 T ZU 2 P n p M 21 M 22 U2 T ZU 1 P p U2 T ZU 2 P n p and L U T Z(ρ) T UP n = [ ] L11 L 12 L 21 L 22 From the (2,1)-block of [ ] [ ] Ip 0 Ip 0 (M λl) Ψ I n p Φ I n p [ ] U T 1 Z T U 1 P p U1 T Z T U 2 P n p U T 2 Z T U 1 P p U T 2 Z T U 2 P n p we construct [ ] F (Φ, Ψ, ρ) = G(Φ, Ψ, ρ) [ ΨU1 T ZU 1 P p ΨU1 T ZU 2 P n p Φ + U2 T ZU 1 P p + U2 T ZU 2 P n p Φ ΨU1 T Z T U 1 P p ΨU1 T Z T U 2 P n p Φ + U2 T Z T U 1 P p + U2 T Z T U 2 P n p Φ ] At ρ = 0, we have Φ = 0 = Ψ and F (0, 0, 0) = 0 = G(0, 0, 0). The implicit function theorem can then be applied to (F, G). Differentiation of F and G respect to Ψ and Φ at ρ = 0 yields (after stacking columns and apply Kronecker products) [ ] [ ] F Λ T = α1 I I Λ α2 G Λ T β1 I I Λ β2 (Ψ,Φ) It is easy to see that the above operator is invertible when the spectra of (Λ α1, Λ β1 ) and (Λ α2, Λ β2 ) do not intersect (and it will be ill-conditioned when the subspectra are close). Consequently, we have proved that the power series exists for Φ(ρ) and Ψ(ρ) within some small neighbourhood of ρ = 0 and Φ(ρ) = Φ ρ ρ +, Ψ(ρ) = Ψ ρ ρ + Furthermore, the deflation subspace [2] is formed by { ( [ ]) ( [ ])} I I span U, span UP Ψ(p) n Φ(p) = { span ( U 1 + U 2 Ψ(p) ), span (U 1 P p + U 2 P n p Φ(p)) }

10 10 E.K.-w. Chu, W.-W. Lin & C.-S. Wang Differentiation of M 11 U T 1 ZU 1 P p, L 11 U T 1 Z T U 1 P p also yields M 11 = U T 1 Z ρ U 1 P p, L 11 = U T 1 Z T ρ U 1 P p When p = 1 and (Λ α1, Λ β1 ) = (λ α1, λ β1 ) represents the finite eigenvalue λ 1 = λ α1 /λ β1, the above derivatives translate to λα1 λ 1 = λ β1 λ β1 λ α1 λ 2 = β1 λ α1 λ β1 λ 1 = yt 1 Z ρ x 1 λ 1 y1 T Zρ T x 1 λ β1 λ β1 y1 T ZT x 1 producing a result analogous to (15). Lastly, differentiating F and G with respect to ρ at ρ = 0 produces F ρ = Λ α2 Φ ρ Ψ ρ Λ α1 + U T 2 Z ρ (0)U 1 P p = 0 G ρ = Λ β2 Φ ρ Ψ ρ Λ β1 + U T 2 Z ρ (0) T U 1 P p = 0 The derivatives Φ ρ (0) and Ψ ρ (0) can then be retrieved from the above equations, when (Λ α1, Λ β1 ) and (Λ α2, Λ β2 ) have nonintersecting spectra. 7.2 General Matrix Quadratics We now apply Sun s approach [2] to the general matrix quadratic Q 2 (λ) = λ 2 M + λd + K similar to the development in the previous subsection. Assume the n 2n matrices X = [x j ] and Y = [y j ] contain, respectively, the right- and lefteigenvectors, with x j and y j corresponding to λ j = α j /β j. For the companion linearization [ L(λ) 0 I K D ] λ [ I M ] (16) it is easy to check that the right- and left-eigenvectors corresponding to λ j = α j /β j are respectively [ ] βj x j, [ β α j x j yj T D + α j yj T M, β j y T ] T j j Notice that by using (α j, β j ) to represent λ j, we have avoided the difficulties involving infinite eigenvalues. We can then scale the eigenvectors to satisfy the biorthogonality equations Λ α Y T MXΛ β + Λ β Y T MXΛ α + Λ β Y T DXΛ β = Λ β (17) Λ α Y T MXΛ α Λ β Y T KXΛ β = Λ α (18)

11 Perturbation of Palindromic Eigenvalue Problems 11 Here Λ α and Λ β are block-diagonal matrices, with blocks being the usual identity or Jordan matrices for the generalized eigenvalue sub-problems. Assume further that X, Y, Λ α and Λ β are organized as follows: X = [X 1, X 2 ], Y = [Y 1, Y 2 ], Λ α = diag{λ α1, Λ α2 }, Λ β = diag{λ β1, Λ β2 } where the spectra of (Λ α1, Λ β1 ) and (Λ α2, Λ β2 ) do not intersect. Following the usual steps in Sun s approach [2] in obtaining analytic power series for the eigenvalues and eigenvectors for generalized eigenvalue problems, we assume the matrices M, D and K are dependent on a perturbation parameter ρ. We first transform the linearization in (16) by pre-(post- )multiplying with its left-(right-)eigenvectors: [ ] M11 M M 12 [ Λ M 21 M β Y T D + Λ α Y T M, Λ β Y T ] [ ] [ ] 0 I XΛβ 22 K D XΛ α and L [ ] L11 L 12 [ Λ L 21 L β Y T D + Λ α Y T M, Λ β Y T ] [ ] [ ] I XΛβ 22 M XΛ α From the (2,1)-block of [ ] [ ] Ip 0 Ip 0 (M λl) Ψ I n p Φ I n p we construct F (Φ, Ψ, ρ) Ψ(Λ α1 Y1 T MX 1 Λ α1 Λ β1 Y1 T KX 1 Λ β1 ) Ψ(Λ α1 Y1 T MX 2 Λ α2 Λ β1 Y1 T KX 2 Λ β2 )Φ +(Λ α2 Y2 T MX 1 Λ α1 Λ β2 Y2 T KX 1 Λ β1 ) +(Λ α2 Y2 T MX 2 Λ α2 Λ β2 Y2 T KX 2 Λ β2 )Φ G(Φ, Ψ, ρ) Ψ(Λ β1 Y1 T DX 1 Λ β1 + Λ α1 Y1 T MX 1 Λ β1 + Λ β1 Y1 T MX 1 Λ α1 ) Ψ(Λ β1 Y1 T DX 2 Λ β2 + Λ α1 Y1 T MX 2 Λ β2 + Λ β1 Y1 T MX 2 Λ α2 )Φ +(Λ β2 Y2 T DX 1 Λ β1 + Λ α2 Y2 T MX 1 Λ β1 + Λ β2 Y2 T MX 1 Λ α1 ) +(Λ β2 Y2 T DX 2 Λ β2 + Λ α2 Y2 T MX 2 Λ β2 + Λ β2 Y2 T MX 2 Λ α2 )Φ At ρ = 0, we have Φ = 0 = Ψ and F (0, 0, 0) = 0 = G(0, 0, 0). The implicit function theorem can then be applied to (F, G). Differentiation of F and G respect to Ψ and Φ at ρ = 0 yields (after stacking columns and apply Kronecker products) [ ] [ ] F Λ T = α1 I I Λ α2 G Λ T β1 I I Λ β2 (Ψ,Φ) It is easy to see that the above operator is invertible when the spectra of (Λ α1, Λ β1 ) and (Λ α2, Λ β2 ) do not intersect (and it will be ill-conditioned when the subspectra are close). Consequently, we have proved that the power

12 12 E.K.-w. Chu, W.-W. Lin & C.-S. Wang series exists for Φ(ρ) and Ψ(ρ) within some small neighbourhood of ρ = 0 and Φ(ρ) = Φ ρ ρ +, Ψ(ρ) = Ψ ρ ρ + Furthermore, the deflation subspace [2] of the companion linearization is formed by { ( [ ]) ( [ ])} span Ỹ T I I, span X Ψ(p) Φ(p) with Ỹ = [ Λ β Y T D + Λ α Y T M, Λ β Y T ] [ ] T XΛβ, X = XΛ α Differentiation of also yields M 11 Λ α1 Y1 T MX 1 Λ α1 Λ β1 Y1 T KX 1 Λ β1 L 11 Λ β1 Y1 T DX 1 Λ β1 + Λ α1 Y1 T MX 1 Λ β1 + Λ β1 Y1 T MX 1 Λ α1 M 11 L 11 = Λ α1 Y T 1 M ρ X 1 Λ α1 Λ β1 Y T 1 K ρ X 1 Λ β1 = Λ β1y T 1 D ρ X 1 Λ β1 + Λ α1 Y T 1 M ρ X 1 Λ β1 + Λ β1 Y T 1 M ρ X 1 Λ α1 When p = 1 and (Λ α1, Λ β1 ) = (λ α1, λ β1 ) represents the finite eigenvalue λ 1 = λ α1 /λ β1, the above derivatives translate to λα1 λ 1 = λ β1 λ β1 λ α1 λ 2 = β1 λ α1 λ β1 λ 1 λ β1 λ β1 = λ2 β1 yt 1 K ρ x 1 + λ 2 α1y T 1 M ρ x 1 λ 1 (λ 2 β1 yt 1 D ρ x 1 + 2λ α1 λ β1 y T 1 M ρ x 1 ) λ 2 β1 yt 1 Dx 1 + 2λ α1 λ β1 y T 1 Mx 1 = yt 1 (λ 2 1M ρ + λ 1 D ρ + K ρ )x 1 y T 1 (2λ 1M + D)x 1 (19) producing the same formula in (15). Lastly, differentiating F and G with respect to ρ at ρ = 0 produces F ρ = Λ α2 Φ ρ Ψ ρ Λ α1 + Λ α2 Y2 T M ρ (0)X 1 Λ α1 Λ β2 Y2 T K ρ (0)X 1 Λ β1 = 0 G ρ = Λ β2 Φ ρ Ψ ρ Λ β1 + Λ α2 Λ β2 Y2 T D ρ (0)X 1 Λ β1 +Λ α2 Y2 T M ρ (0)X 1 Λ β1 + Λ β2 Y2 T M ρ (0)X 1 Λ α1 = 0 The derivatives Φ ρ (0) and Ψ ρ (0) can then be retrieved from the above equations, when (Λ α1, Λ β1 ) and (Λ α2, Λ β2 ) have nonintersecting spectra.

13 Perturbation of Palindromic Eigenvalue Problems Palindromic Case For palindromic eigenvalue problems, the perturbation results look almost identical to those for general matrix quadratics, as shown in the previous section. The only differences are M = A T 1 = K H, D = A 0 = D T, and the palindromic properties of the eigenvalues and eigenvectors. For the spectrum, we have (Λ α1, Λ β1 ) and (Λ α2, Λ β2 ) = (Λ β1, Λ α1 ) representing, respectively, eigenvalues inside and outside the unit circle, and [Y 1, Y 2 ] = [X 2, X 1 ] (as in Section 3.5.1). We shall not rewrite all the general results for palindromic quadratics, leaving the trivial substitution of symbols as an exercise. For the simple palindromic eigenvalues λ 1 and λ 1 1, (19) implies λ 1 = yt 1 (λ 2 1K T ρ + λ 1 D ρ + K ρ )x 1 y T 1 (2λ 1K T + D)x 1 λ 1 1 = yt 1 (K T ρ + λ 1 1 D ρ + λ 2 1 K ρ)x 1 y T 1 (2λ 1K T + D)x 1 Interestingly, the relative errors of the pair of reciprocal eigenvalues equal, asymptotically, to ρ λ ±1 1 λ ±1 1 = ρyt 1 (λ 1 K T ρ + D ρ + λ 1 1 K ρ)x 1 y T 1 (2λ 1K T + D)x 1 (20) and are identical except of the opposite signs. Equation (20) can easily be understood from 1 λ + δλ = 1 λ ( ) 1 + δλ 1 ( 1 δλ ) λ λ λ After applying inequalities of norms, a condition number for both λ ±1 1 can then be produced: κ y 1 2 x 1 2 y1 T (λ λ λ 1K T 1 2 (21) + D)x 1 With appropriate scaling of the eigenvectors, κ can be interpreted as the product of the norms of the left- and the right-eigenvectors, or in terms of the angle between the eigenvectors, as in other algebraic eigenvalue problems. 8 Numerical Example We shall consider the following small example to illustrate the results in the previous Section 7.3: [ ] [ ] ρ A 0 =, A = 0 1 The parameter ρ can be used to vary the conditioning of the eigenvalue problem, but will be fixed to be 0.5 in the following calculations. The matrices are perturbed randomly to the magnitude of , with δ =

14 14 E.K.-w. Chu, W.-W. Lin & C.-S. Wang Table 1 Perturbation of a palindromic eigenvalue problem i 1 2, 3 4 λ i ± i λ i ± i δλ i e e-08 ± e e-08 i r i e e-16 ± e e-08 i r i e e e-08 r (e) i e e-16 ± e-08 i e-08 r (e) i e e e-08 κ i r (κ) i e e e-08 [δa T 1, δa 0, δa 1 ] = The eigenvalues are λ i (i = 1,, 4) with λ 1 = λ 1 4 and λ 2 = λ 1 3. The numerical results are summarized in the following table, with λ i denoting the perturbed eigenvalues, δλ i λ i λ i, r i δλ i /λ i the relative error in λ i, r (e) i estimating r i using (20), κ i the individual condition numbers as in (21), and r (e) i r (κ) i κ i δ estimating r i. All calculations were carried out using MATLAB 7.1 (R14) on a Apple MacIntosh G4 Powerbook, with eps It is obvious from Table 1 that (20) provides accurate approximations to the relative errors of palindromic eigenvalues with small perturbations and the condition number κ in (21) produces tight upper bounds for the (relative) errors. Also, the fact that the relative errors for reciprocal pairs of eigenvalues are negative of each other is confirmed by the example. 9 Conclusions Bauer-Fike type perturbation results for general matrix polynomials, palindromic linearizations, the anti-triangular canonical form and the semi-schur anti-triangular canonical form are discussed, for perturbations of arbitrary size. For asymptotic perturbations, perturbation results for eigenvalues and the corresponding deflating subspaces of palindromic linearizations and (palindromic) matrix quadratics are obtained, Sun s implicit function approach. Consistent results for simple eigenvalues and the corresponding eigenvetors are obtained using simple differentiation. These results indicate, not surprisingly, that the perturbations of an eigenvalue λ and its corresponding deflating subspace S λ, respectively, are proportional to the size of the perturbation and the reciprocal of the gap between S λ and other deflating subspaces. A

15 Perturbation of Palindromic Eigenvalue Problems 15 condition number is typically the product of the norms of the left- and righteigenvectors. Appendix: Bounding the root of x m η(1 + x + + x m 1 ) Consider the polynomial P m (x) x m η(1 + x + + x m 1 ) Descartes sign rule (La Géométrie 1637) then implies that P m (x) has at most one positive real root. As P m (0) = η < 0 and P m (x) > 0 as x, any positive number x for which P m (x ) > 0 is an upper bound of the unique real positive root of P m (x). Simple inspection leads to the upper bounds x = c 0 η when η > 1 and x = c 0 η 1/m when η 1, with c 0 = min{2, p}. Consequently, c 0 = 1 when m = 1 (with x = η) and c 0 = 2 when m > 1. The details are as follows. When c 0 η > η 1 and m > 1, we have P m (c 0 η) = (c 0 η) m η (c 0η) m 1 c 0 η 1 = cm+1 0 η m+1 c m 0 η m c m 0 η m+1 + η c 0 η 1 = (cm 0 η m+1 c m 0 η m ) + (c m+1 0 η m+1 c m 0 η m+1 c m 0 η m+1 ) + η 0 c 0 η 1 as c m+1 0 η m+1 c m 0 η m+1 c m 0 η m+1 = (c 0 2)c m 0 η m+1 = 0. Thus c 0 η is an upper bound of the root x of P m when η 1. When η < 1 and m > 1, we have P m (c 0 η 1/m ) c m 0 η η(1 + c c m 0 ) = c m 0 η η cm 0 1 c 0 1 = 0 as c m+1 0 2c m 0 = 0. Thus c 0 η 1/m is an upper bound of the root x of P m when η < 1. References 1. e.k.-w. chu. Perturbation of eigenvalues for matrix polynomials via the Bauer- Fike theorems, SIAM J. Matrix Anal. Applic., 25 (2003), pp l. elsner and j.g. sun. Perturbation theorems for the generalized eigenvalue problem, Lin. Alg. Applic., 48 (1982), pp i. gohberg, p. lancaster and l. rodman. Matrix Polynomials, Academic Press, New York, g.h. golub and c.f. van loan. Matrix Computations, 3rd Ed., Johns Hopkins University Press, Baltimore, a. hilliges. Numerische Lösung von quadratischen Eigenwertproblemen mit Anwendungen in der Schienendynamik, Master Thesis, Technical University Berlin, Germany, July 2004.

16 16 E.K.-w. Chu, W.-W. Lin & C.-S. Wang 6. a. hilliges, c. mehl and v. mehrmann. On the solution of palindromic eigenvalue problems, Proceedings 4th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS), Jyväskylä, Finland, c.f. ipsen. Accurate eigenvalues for fast trains, SIAM News, 37 (2004). 8. d.s. mackey, n.mackey, c. mehl and v. mehrmann. Linearization spaces for matrix polynomials, in preparation. 9. d.s. mackey, n. mackey, c. mehl and v. mehrmann. Palindromic polynomial eigenvalue problems: good vibrations from good linearizations, in preparation. 10. d.s. mackey, n. mackey, c. mehl and v. mehrmann. Palindromic polynomial eigenvalue problems, Preprint DFG Research Center Mathematics for Key Technologies, (to appear). 11. mathworks. MATLAB User s Guide, g.w. stewart and j.g. sun. Matrix Perturbation Theory, Academic Press, New York, f. tisseur. Backward error and condition of polynomial eigenvalue problems, Lin. Alg. Applic., 309 (2000), pp f. tisseur and k. meerbergen. A survey of the quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp