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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 MATHEON 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

P(λ) x = (

k

• i=0

Aiλi)x = 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 xTP(1/λ) = 0, i.e., the eigenvalues

• ccur in pairs λ, 1/λ.

Consider a H-palindromic eigenvalue problem P(λ)x = 0. Then P(λ)x = 0 if and only if xHP(1/¯ λ) = 0, i.e., the eigenvalues

• ccur in pairs λ, 1/¯

λ. Palindromic matrix functions have symplectic spectrum, they generalize symplectic problems λI + 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

∞

• j=0

(x∗

j Qxj + u∗ j Ruj)

subject to the discrete time control problem Exk+1 = Axk + Buk, with x0 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 ∗zk+1 = Zzk, with Z =   A B E∗ Q S 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) = λE + 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 (λ2A + λB + AT)x = 0 with B = BT is given by λ

• A

I y x

• =
• −B

−AT I y 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

• ptimized;

◮ eigenvalues/eigenvectors of the original problem are easily read

• ff;

◮ 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 := [λk−1, λk−2, . . . , λ, 1]T, ⊗ - Kronecker product. Definition Mackey2/Mehl/M. 06. For a given n × n matrix polynomial P(λ) of degree k define the sets: VP := {v ⊗ P(λ) : v ∈ Fk}, v is called right ansatz vector, WP := {wT ⊗ P(λ) : w ∈ Fk}, w is called left ansatz vector, L1(P) :=

• L(λ) = λX + Y : X, Y ∈ Fkn×kn, L(λ) · (Λ ⊗ In) ∈ VP
• ,

L2(P) :=

• L(λ) = λX + Y : X, Y ∈ Fkn×kn,
• ΛT ⊗ In
• · L(λ) ∈ WP
• ,

DL(P) := L1(P) ∩ L2(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 C1(λ) := λ       Ak · · · In ... . . . . . . ... ... · · · In       +      Ak−1 Ak−2 · · · A0 −In · · · . . . ... ... . . . · · · −In      C2(λ) := λ       Ak · · · In ... . . . . . . ... ... · · · In       +       Ak−1 −In · · · Ak−2 ... . . . . . . . . . ... −In A0 · · ·       . are linearizations in L1(P), L2(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: Mackey2/Mehl/M. 06. Let P(λ) be an n × n matrix polynomial of degree k, and let L(λ) be any pencil in L1(P) with ansatz vector v = 0. Then x ∈ Cn 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 ∈ L1(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 L2(P).

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

When are these linearizations?

Lemma: Consider an n × n matrix polynomial P(λ) of degree k. Then, for v = (v1, . . . , vk)T and w = (w1, . . . , wk)T in Fk, the associated pencil satisfies L(λ) = λX + Y ∈ DL(P) if and only if v = w. Theorem: Mackey2/Mehl/M. 06. Consider an n × n matrix polynomial P(λ) of degree k. Then for given ansatz vector v = w = [v1, . . . , vk]T the associated linear pencil in DL(P) is a linearization if and only if no root of the v-polynomial p(v; x) := v1xk−1 + . . . + vk−1x + vk is an eigenvalue of P.

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

Conditioning of linearization

◮ Perturbation analysis Tisseur 00, Higham/Mackey/Tisseur 06,

Higham/Li/Tisseur 06.

◮ Computation of a simple eigenvalue ˆ

λ via the linearized eigenvalue problem is very ill-conditioned if p(v, ˆ λ) is small.

◮ Proper scaling is necessary. ◮ Open problem. Does the solution via a good, properly

scaled, structure preserving linearization produce generally better results than the direct solution of the original structured problem.

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 linearization

Lemma: Consider an n × n palindromic matrix polynomial P(λ) of degree k. Then, for a vector v = (v1, . . . , vk)T ∈ Fk the linearization L(λ) = λX + Y ∈ DL(P) is (the permutation of) a palindromic pencil, if and only if p(v; x) is palindromic, which is the case if and only if v is a palindromic vector. What are appropriate palindromic polynomials p(v; x).

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: For the palindromic polynomial P(λ)y = (λ2A + λB + AT)y = 0, B = BT all palindromic vectors have the form v = [α, α]T, α = 0 leads to a palindromic pencil κZ T + Z, Z =

• A

B − AT A A

• .

◮ This is a linearization if and only if −1 is not an eigenvalue of

P(λ).

◮ We could use as an alternative anti-palindromic linearizations

with v = [α, −α]T. Then 1 is the critical eigenvalue.

◮ What to do if −1, 1 are eigenvalues of P(λ)?

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

Trimmed linearization

Compute appropriate (structured) staircase form associated with the eigenvalues 1, −1 and the singular part directly for matrix polynomial. Similar problems for eigenvalues are 0 and ∞. Reduce/deflate chains associated with these parts until the eigenvalues 1, −1 (0, ∞) and the singular blocks have no more chains or are completely removed. Parts associated with eigenvalues 1, −1, 0, ∞ and singular parts can (at least in principle) be removed exactly. Perform (structured) linearization on the resulting trimmed matrix polynomial.

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

Theorem Byers/M./Xu 07 Staircase form for matrix tuples. Let Ai ∈ Cm,n i = 0, . . . , k. Then, the tuple (Ak, . . . , A0) is unitarily equivalent to a matrix tuple (ˆ Ak, . . . , ˆ A0) = (UAkV, . . . , UA0V), all terms ˆ Ai, i = 0, . . . , k have form

2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 A A A . . . . . . . . . A A A(i)

l

A A A . . . . . . . . . . . . A(i)

l−1

A A A . . . . . . . . . . . . . . . . . . . . . . . . . . . A(i)

1

. . . . . . . . . . . . . . . . . . . . . A(i) . . . . . . . . . . . . . . . . . . ˜ A(i)

1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A ˜ A(i)

l−1

. . . . . . . . . . . . . . . ˜ A(i)

l

. . . . . . . . . . . . 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . 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 this staircase form

◮ Each of the blocks A(i)

j

i = 0, . . . , k, j = 1, . . . , l either has the form

• Σ
• r
• ,

◮ Each of the blocks ˜

A(i)

j

i = 1, . . . , k, j = 1, . . . , l either has the form Σ

• r
• .

◮ For each j only of the A(i)

j

and ˜ A(i)

j

is nonzero.

◮ In the tuple of middle blocks (A(k)

0 , . . . , A(k) 0 ) (essentially) no k of

the coefficients have a common nullspace.

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

Theorem Byers/M./Xu 07 Staircase form for palindromic matrix

• tuples. Let P(λ) x = (k

i=0 Aiλi)x = 0 be palindromic. Then, the

tuple (Ak, . . . , A0) is unitarily congruent to a matrix tuple (ˆ Ak, . . . , ˆ A0) = (U∗AkU, . . . , U∗A0U), where all terms ˆ Ai have the form

2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 A A A . . . . . . . . . A A A(i)

l

A A A . . . . . . . . . . . . A(i)

l−1

A A A . . . . . . . . . . . . . . . . . . . . . . . . . . . A(i)

1

. . . . . . . . . . . . . . . . . . . . . A(i) . . . . . . . . . . . . . . . . . . ±A(i)

1 T

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A ±A(i)

l−1 T

. . . . . . . . . . . . . . . ±A(i)

l T

. . . . . . . . . . . . 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 , 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

Procedure for palindromic polynomials

Perform Cayley transformation Q(µ) = (1 + µ)kP( 1−µ

1+µ).

This moves ev. −1 to ∞ and 1 and to 0. Singular parts stay singular. Compute palindromic staircase form. Use transformation on original palindromic polynomial. Deflate chains associated with −1 and singular chains. Use palindromic linearization to obtain palindromic pencil λZ ∗ + Z.

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

Canonical/condensed forms for palindromic pencils

◮ Different canonical forms for real, complex H, T palindromic

• problems. Sergeichuk 87, Horn/Sergeichuk 07, Rodman 06,

Schröder 07;

◮ Palindromic generalized Schur forms for complex T-palindromic

pencils: Theorem Mackey2,Mehl,M. 07 Palindromic Schur form for complex T-palindromic pencil always exists.

◮ Palindromic Schur forms for real and complex H-palindromic

problems exist if the eigenvalues on the unit circle satisfy certain conditions (similar to the Hamiltonian/symplectic case Schröder 07, e.g. double eigenvalues on the unit circle must form a Jordan block.

◮ Palindromic staircase forms, Byers/M./Xu 07, Schröder 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

Numerical methods for palindromic pencils

◮ Application to rail problem Hilliges 04, Hilliges/Mehl/M. 04; ◮ Palindromic Jabobi and Laub trick Mackey2,Mehl,M. 07; ◮ Palindromic QR/QZ algorithm Schröder 07, Schröder/Watkins

08;

◮ Palindromic URV algorithm Schröder 07; ◮ Recursive doubling algorithm Chu/Lin/Wang/Wu, Chu,Lin, 05 ?

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 QR algorithm

Modification of Hamiltonian/symplectic QR method to compute palindromic Schur form (if it exists) Schröder 07. Single shift QR-step.

• 1. Choose shift κ = a1,n

¯ an,1 .

• 2. A − κA∗ → QR, with R anti-triangular.
• 3. A1 ← Q∗A ¯

Q. Multiple shifts, (A − κ2jA∗)−∗(A − κ2j−1A∗)(A − κ2j−2A∗)−∗ · · · (A − κ1A∗) Technical details on the usual tricks to speed up convergence, to deflate converged parts, etc, Schröder 07.

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

Analysis of palindromic QR Schröder 07

◮ Algorithm works only on one matrix. ◮ As in the Hamiltonian/symplectic case, no palindromic

Hessenberg reduction in general available.

◮ Except when a palindromic Hessenberg form is available,the

complexity is O(n4).

◮ Method is strongly backwards stable. ◮ Convergence analysis as for full QR. ◮ Method only useful for very small, difficult problems. It should be

used if a palindromic Schur exists, and eigenvalues are near unit circle.

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 URV decomposition

Extension of method by Benner/Mehrmann/Xu 99 from Hamiltonian case to palindromic case. Schröder 07 Compute unitary transformation matrices U, V so that U∗AV = R = [ri,j], U∗(A−A∗)U = T = [ti,j], V ∗(A−A∗)V = P = [pi,j] are all anti-triangular. Then eigenvalues can be read off as 1 + 2µ2

i ±

• 1 + 4µ2

i

µ2

i

, µ2

i = rijrji

pjitji , where i = 1, 2, . . . , n/2 and j = n + 1 − i.

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 URV algorithm

• 1. Compute unitary transformation matrices U, V so that for

U∗AV = R = [ri,j], U∗(A−A∗)U = T = [ti,j], V ∗(A−A∗)V = P = [pi,j] A is anti-Hessenberg and T, P are anti-triangular.

• 2. Apply period QR/QZ algorithm to parts of the formal product

TRP.

• 3. Compute eigenvalues and some eigenvectors from formal

product.

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

Analysis of palindromic URV Schröder 07

◮ Algorithm works with three matrices. ◮ The palindromic URV decomposition always exist. ◮ Algorithm is numerically backward stable to compute

eigenvalues in O(n3).

◮ Faster than QZ. ◮ Method is strongly backwards stable for a double size problem. ◮ Convergence analysis as for standard periodic QR. ◮ Method does not compute palindromic anti-triangular form.

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

Computation of palindromic anti-triangular form

◮ The method of Chu/Liu/M. 07, Watkins 07 of extracting the

Hamiltonian Schur form from the Hamiltonian URV decomposition can be modified and generalized to compute generalized palindromic anti-Schur form Schröder 08.

◮ Palindromic ’Laub Trick’ Mackey2,Mehl,M. 07.

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 ’Laub Trick’

Theorem Mackey2,Mehl,M. 07 Let LZ(λ) = λZ ∗ + Z be a regular palindromic pencil and let the columns of W1 ∈ Cn×m span an m-dimensional deflating subspace associated with the spectrum Λ1 ⊆ C, i.e., there exists V1 ∈ Cn×m, of rank m, and X11, Y11 ∈ Cm×m such that (λZ ∗ + Z)W1 = V1(λX11 + Y11), where σ(λX11 + Y11) = Λ1. If Λ1 is reciprocal-free then W1 is Z-isotropic, i.e., W ∗

1 ZW1 = 0.

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

QZ+projection method

Algorithm Mackey2,Mehl,M. 07 Given Z ∈ Cn×n and m ≤ n

2.

Compute unitary U ∈ Cn×n such that M = U∗ZU =   Y T

11

U∗

1ZU1 U∗ 1ZV 1

X11 V ∗

1 ZU1 V ∗ 1 ZV 1

  with (2, 2)-block of size (n − 2m) × (n − 2m).

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

1) Use QZ algorithm with reordering to compute unitary matrices V = [V1, V2], W = [W1, W2], V1, W1 ∈ Cn×m, V2, W2 ∈ Cn×(n−m) such that V T(λZ + Z ∗)W = λ X11 X12 X22

• +

Y11 Y12 Y22

• ,

where X11, Y11 ∈ Cm×m are upper triangular and eigenvalues are

• rdered such that σ(λX11 + Y11) is reciprocal-free.

2) Compute an isometric matrix U1 ∈ Cn×(n−2m) such that the columns of U1 are orthogonal to the columns of W1 and V 1 and set U =

• W1

U1 V 1Fm

• . Then U is unitary and M := U∗ZU

is in partial anti-triangular form.

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

Analysis of QZ+proj. method Mackey2,Mehl,M. 07

Complex T problem λZ T + Z. If n is even and if LZ(λ) does not have ±1 as eigenvalues, or if n is odd and LZ(λ) does not have +1 as an eigenvalue, and has −1 as eigenvalue with algebraic multiplicity 1, then there are exactly m = ⌊ n

2⌋ eigenvalues in each of the sets

Λ1 :=

• λ ∈ C : |λ| > 1 or (|λ| = 1 and Im(λ) > 0)
• ,

Λ2 :=

• λ ∈ C : |λ| < 1 or (|λ| = 1 and Im(λ) < 0)
• ,

Perturbation analysis analogous to Hamiltonian case Byers/Kressner 06 Method is not appropriate if eigenvalues are close to unit circle.

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

A hybrid method Mackey2,Mehl,M. 07

◮ Use QZ+projection or URV method to obtain partial palindromic

Schur form, where the middle block contains all the eigenvalues that are near to the critical region (near unit circle in real or complex H case) or near ±1 in the complex T case).

◮ Use expensive, but structurally backwards stable method such

as palindromic QR, to deal with (hopefully small) part associated with near critical eigenvalues.

◮ Use one sweep of palindromic Jacobi method to remove ’dirt’

above anti-diagonal.

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

Numerical example, Complex T problem λZ T + Z

100 × 100 matrix Z with 10 ev. near 1. Before (left) and after (right) solving the 10 × 10 subproblem.

20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100

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

Partial Schur form followed by Jacobi method

Partial Schur form followed by one sweep of palindromic Jacobi. The matrix is shown before (left) and after (right) solving the remaining 10 × 10 problem.

20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100

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 Schur, followed by palindromic Jacobi

Partial Schur form, then palindromic QR for small problem followed by

• ne sweep of Jacobi. The matrix is shown before (left) and after

(right) solving the remaining 10 × 10 problem once again.

20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100

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

A summary of palindromic methods

◮ Palindromic QR/QZ: On4, strongly backward stable, only for

small problems.

◮ Palindromic Jacobi: On3 per sweep, sometimes not convergent,

• nly for cleaning up .

◮ Palindromic URV: On3, strongly backward stable for double size

problem, implementation on the way, current version not ideal if the problem has near multiple eigenvalues

◮ Palindromic Laub trick: On3, only if there are no eigenvalues

near unit circle.

◮ Hybrid method: On3 if the number of eigenvalues near unit circle

is small.

◮ A refinement and clean up step is good for all the algorithms to

deal with deviations from palindromic anti-Schur form.

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

Conclusions and future work.

◮ Palindromic polynomial eigenvalue problems are important in

many applications.

◮ Structured linearization methods are available. ◮ Structured staircase forms are available. ◮ New trimmed linearization techniques are available. ◮ Structure preserving numerical methods for palindromic pencils

have been constructed.

◮ Hybrid methods can be used to deal with hard problems. ◮ Numerical examples indicate the success of hybrid methods.

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

Thank you very much for your attention. information, papers, codes etc http://www.math.tu-berlin.de/˜mehrmann

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

References

• R. Byers, V. Mehrmann and H. Xu. A structured staircase algorithm

for skew-symmetric/symmetric pencils, ETNA, 2007.

• R. Byers, V. Mehrmann and H. Xu. Staircase forms and trimmed

linearization for structured matrix polynomials. PREPRINT, MATHEON, url: http://www.matheon.de/ 2007. D.S. Mackey, N. Mackey, C. Mehl and V. Mehrmann. Vector spaces of linearizations for matrix polynomials, SIMAX 2007. D.S. Mackey, N. Mackey, C. Mehl and V. Mehrmann. Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations, SIMAX 2007. D.S. Mackey, N. Mackey, C. Mehl and V. Mehrmann. Numerical methods for palindromic eigenvalue problems: Computing the anti-triangular Schur form, PREPRINT, MATHEON, url: http://www.matheon.de/, 2007.

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

• V. Mehrmann and H. Voss: Nonlinear Eigenvalue Problems: A

Challenge for Modern Eigenvalue Methods. GAMM Mitteilungen, 2005.

• C. Schröder: A QR-like algorithm for the palindromic eigenvalue
• problem. PREPRINT, MATHEON, url: http://www.matheon.de/, 2007.
• C. Schröder: URV decomposition based structured methods for

palindromic and even eigenvalue problems. PREPRINT, MATHEON, url: http://www.matheon.de/, 2007.

• C. Schröder: Palindromic and even eigenvalue problems. Analysis

and numerical methods. Dissertation, TU Berlin, 2008.

Volker Mehrmann mehrmann@math.tu-berlin.de Nonlinear palindromic eigenvalue problems and their numerical solution