Nation Chiao Tung UniversityDepartment of Applied Mathematics College of Science

Palindromic Quadratization and Structure-Preserving Algorithm for Palindromic Matrix Polynomials

Wei-Shuo Su 蘇偉碩

2013/12/81

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• Introduction and motivation• Linearization and Quadratization• Backward error influence and balancing • Numerical results

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We investigate the solution of a particular rational eigenvalue problem that arises in an industrial project studying the vibration of rail tracks under the excitation arising from high speed trains.

This eigenvalue problem has the form

where and with

Taiwan High Speed Rail

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𝐾𝑞+𝐷�̇�+𝑀 �̈�= 𝑓 (𝑡 )

The displacements of two boundary cross sections of the modeled rail are assumed to have a ratio λ, which is dependent on the excitation frequency of the external force.From the virtual work principle and strain-stress relationship, the governing equation for the displacement vector q involving viscous damping can be formulated by

where K,D, and M from the finite element discretizationon a uniform mesh satisfy the given boundary conditions.

The external force is assumed to be periodic.

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From the spectral modal analysis, we consider , where ω is the frequency of the external force and is the corresponding eigenmode.Consequently, we get the palindromic QEP

(𝜆2~𝐴1⊺+𝜆~𝐴0+

~𝐴1

❑ )~𝑥=0

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The “Linearization" is a typical and frequently used technique to solve the (PQEP) in which the problem is reformulated into a linear one which doubles the order of the system. We select suitable matrices and the vector and transform (PQEP) into the (GEP)

( 𝐴− 𝜆𝐵 )𝜑=0

satisfying the relationℰ (𝜆 ) ( 𝐴− 𝜆𝐵 )ℱ (𝜆 )=[𝒫(𝜆) 0

0 𝐼𝑛]In this case,

det = cdetshows that the eigenvalues are the same.

constant determinants

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The first companion form and

and the second companion form

and

Drawbacks:1. Doubles the size of the problem dimension2. Palindromic and symplectic may be lost

Symplectic: and both exist

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In the , W.-W. Lin propose some structure-preserving method to deal with ⊤-palindromic quadratic pencil.

1. Structure-preserving algorithms for palindromic quadratic eigenvalue problems arising from vibration on fast trains

And the numerical result is quite excellent.

Based on the spirit, we want to use the same algorithm to solve palindromic matrix eigenvalue problems.

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We consider the palindromic matric polynomials of degree

𝒫 (𝜆 )≡∑𝑘=0

𝑑−1

𝜆2𝑑−𝑘𝐴𝑑−𝑘⋆ +𝜆𝑑𝐴0+𝜀∑

𝑘=1

𝑑

𝜆𝑑−𝑘 𝐴𝑘

where and or .It also satisfies 𝒫 (𝜆 )=𝜀 𝜆2𝑑𝒫( 1𝜆 )

⋆

The eigenvalues appear in the pairs of the form .

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Definition: (Quadratization) is a quadratization of if there are matrix rational functions and with nonzero and constant determinants satisfying the two-sided factorization

ℰ (𝜆 ) 𝒬 (𝜆 )ℱ ( 𝜆 )=[𝒫(𝜆) 00 𝐼𝑞−𝑣 ]

P-Quadratization: and have the same structure property.

It also implies that detdet for some nonzero constant c, and and have the same eigenvalue.

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Theorem: (eigenvector relation) is an eigenvector of if and only if is an eigenvector of .

Theorem: can be -quadratized into a -palindromic quadratic pencil of the form𝒬 (𝜆 )≡ 𝜆2𝒜1

⋆+𝜆𝒜0+𝜀𝒜1

with . We show degree 4

(we can see the paper for generalize )

,

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H-even PEP

T-even PEP

H-odd PEP

T-odd PEP

H-palindromic

PEPT-palindromic

PEP

H-anti-palindromic

PEP

T-anti-palindromic PEP

H-palindromic

QEPT-palindromic

QEP

H-anti-palindromic

QEP

T-palindromic GEP

H-palindromic QEP

T-palindromic QEP

Solving flow chart

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Proposition:Given an H-anti-PQEP: , with Then is an eigenpair of the H-anti-PQEP if and only if is an eigenpair of the H-PQEP:

Let , where

is called a -even if and

is called a -odd if and

By the Cayley transformation, it can be transformed to a -PPEP 𝜆  → 𝜆− 𝑖

𝜆+𝑖

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H-even PEP

T-even PEP

H-odd PEP

T-odd PEP

H-palindromic

PEPT-palindromic

PEP

H-anti-palindromic

PEP

T-anti-palindromic PEP

H-palindromic

QEPT-palindromic

QEP

H-anti-palindromic

QEP

T-palindromic GEP

H-palindromic QEP

T-palindromic QEP

Linearizationwhere

Solving flow chart

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We consider the -PQEP𝒬 (𝜆 )𝑥≡ (𝜆2𝐴1⋆+𝜆𝐴0+𝐴1 )𝑥=0

In order the preserve the symplectic structure, we get the special linearization

(ℳ− 𝜆ℒ )𝑧≡([ 𝐴1 0−𝐴0 − 𝐼 ]−𝜆 [ 0 𝐼

𝐴1⋆ 0]) [𝑥𝑦 ]=0

where and it also satisfies,

So that the matrix pair has eigenvalues and .The pencil are called -symplectic

H-PQEP Linear

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The -transform of an -symplectic matrix pair is defined by

, It can be verify that and If is an eigenvalue of , so is .

Theorem:Suppose is an eigenvector of corresponding to . If or, then or is an eigenpair of , respectively.

(𝒮+𝒮−1)

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TheoremSuppose that is an eigenvector of corresponding to the eigenvalue , and denote with . Let be a root of the quadratic equation . Then

(i) At least one of vector and is nonzero(ii) If , then is an eigenvector of corresponding to (iii) If , then is an eigenvector of corresponding to .

relation

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,

However, Patel’s algorithm can only process in the real case. So we extend to a real matrix pair by

,

where and

It can be verify that if is an eigenvalue of , then and are eigenvalues of

Substituting the H-skew-Hamiltonian and can be write as

simplify case

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The pair can be reduced to block upper triangular forms

,

where are orthogonal satisfying , and are upper Hessenberg and upper triangular, respectively.

Theorem (i) If is an eigenpair of , then are eigenvectors of (ii) If is an eigenvector of corresponding to the eigenvalue ,

then is an eigenvector of

(𝒦2 ,𝒩2 )(~𝒦2 ,

~𝒩2)Patel’s Alg

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H-PQEP Linear (𝒮+𝒮−1)

simplify case

(𝒦2 ,𝒩2 )(~𝒦2 ,

~𝒩2)Patel’s Alg

Solve and follow the conversion relationship

SPA for H-PQEP flow chart

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Algorithm: SPA for H-PQEPInput: with and Output: All eigenvalues and eigenvectors of 1. Form the matrix pair 2. Reduce to block upper triangular forms3. Compute eigenpairs of by the QZ algorithm4. Compute , 5. Compute the eigenpair of by

6. Compute and by solving ; Compute and 7. If then it is an eigenvector of corresponding to ; If , then

it is an eigenvalue of corresponding to 𝑧 𝑗=𝒥𝑇 (𝑥1 𝑗−𝑦 2 𝑗+𝑖 (𝑥2 𝑗+ 𝑦1 𝑗 ))≡ [𝑧 𝑗1

𝑇 ,𝑧 𝑗2𝑇 ]𝑇

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We consider 𝒫 (𝜆 )=𝜆4 𝐴2𝐻+𝜆3 𝐴1

𝐻+𝜆2 𝐴0+𝜆 𝐴1+𝐴2

From the P-Quadration we mentioned

which satisfies𝑄 (𝜆 )=𝜆2𝒜1

𝐻+𝜆𝒜0+𝒜1

If is an eigenpair of , then is an eigenpair of with where

𝑧 2=1

𝑑1 𝜆0(𝜆02 𝐴2

𝐻 𝑧1+𝑧 1)

,

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In order to balance the entries of coefficient matrices in , we define a complex diagonal matrix

𝐷≡diag (2𝛼1 ,2𝛼2+𝑖 𝛽2 ,…,2𝛼𝑛+𝑖 𝛽𝑛)Such that in the new palindromic matrix polynomial

𝐷𝒫 (𝜆 )𝐷⋆

whose each entries are close to one as much as possible.That is

2𝛼 𝑗+𝑖 𝛽 𝑗 𝐴𝑘 ( 𝑗 , ℓ )2𝛼ℓ−𝑖 𝛽ℓ ≈1

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The parameters can be determined by solving the least square problems

,

Then, the parameters and are determined by

,

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Then we set the balancing coefficient

𝑑𝑖(𝑠 )≔√𝜌(𝑠 ) max

0≤𝑘≤𝑑 {∑𝑗=1𝑛

∑ℓ=1

𝑛

|𝐴𝑘 ( 𝑗 , ℓ )|/𝑛2}𝜌(𝑠 )=max {𝛿𝑖

( 𝑠) ; 𝑖=1 ,…,𝑚}

From the row balancing of , we first set𝜂𝑖

(𝑠 )=max {1 ,max {‖𝐴2𝑚−𝑘+𝑠− 1‖1:𝑘=0,1 ,…,2𝑖−3+𝑠}}

We take to be

𝛿𝑖(𝑠 )=√𝜂 𝑖

( 𝑠)(∑𝑗=1𝑛

∑ℓ=1

𝑛

|𝐴2𝑚−2 𝑖+1( 𝑗 , ℓ) /𝑛2|)

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We compare the computational cost for PQ_SPA and PL_SPA

We define the associated relative residual by RRes≡RRes (𝜆 ,𝑥 )≔ ‖𝒫 (𝜆 )𝑥‖2

[∑𝑘=1

𝑑

(|𝜆|𝑑+𝑘+|𝜆|𝑑−𝑘 )‖𝐴𝑘‖2+|𝜆|𝑑‖𝐴0‖2]‖𝑥‖2

The eigenvalues of H-PPEP appear in reciprocal pairs We define the reciprocities of the computed eigenvalues by

,

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Example 2.1 Consider the H-PPEP with and

complex matrices and normal distribution with zero mean and standard deviation

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Example 2.2 Consider the H-PPEP with and and and being defined as

,

where , and

{𝜑𝑖(𝑘 )=4 𝑖+𝑘− ℓ ,𝜑ℓ+𝑖

(𝑘)=4 𝑖−𝑘(𝑖=1 ,…, ℓ)

𝜑𝑛(𝑘)=4

𝑛2−𝑘if 𝑛 is odd with if n is even or

(Large

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Example 2.3 Consider the H-PPEP with and and being defined as

𝐴2=𝐵13 ∙𝑑𝑖𝑎𝑔 {𝜑1(3) ,…,𝜑𝑛

(3 ) } ∙𝐵23∈ℂ(𝑛×𝑛)

where , and is defined in ex2.2 with

(Large

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*REFERENCE*Chu, E. K.-W. and Hwang, T.-M. and Lin, W.-W. and Wu, C.-T.

Vibration of fast trains, palindromic eigenvalue problems and structure-preserving doubling algorithms, J. Comput. Appl. Math. 2008, 219:237--252*Huang, T.-M. and Lin, W.-W. and Qian, J. Structure-preserving

algorithms for palindromic quadratic eigenvalue problems arising from vibration on fast trains, SIAM J. Matrix Anal. Appl. 2009,30:1566-1592.*Mackey, D.S. and Mackey, N. and Mehl, C. and Mehrmann, V. Vector

Spaces of Linearizations for Matrix Polynomials, SIAM J. Matrix Anal. Appl. 2006, 28: 971--1004.*Li, R.-L. and Lin, W.-W. and Wang, C.-S. Structured Backward Error

for Palindromic Polynomial eigenvalue problems. Numerische Mathematik, 2010, 116(1): 95-122.*Huang, T.-M. and Lin, W.-W. and Su, W.-S. Palindromic

Quadratization and Structure-Preserving Algorithm for Palindromic Matrix Polynomials of Even Degree, Numerische Mathematik 2011, 118(4): 713-735.

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If is success in life, then equal plus and . Work is ; is play; and is keeping your mouth shut.

是成功， 是工作，是娛樂， 是沉默。亞伯特、愛因斯坦