Palindromic Quadratization and Structure-Preserving Algorithm for Palindromic Matrix Polynomials

1. Palindromic Quadratization and Structure-Preserving Algorithm for Palindromic Matrix Polynomials Wei-Shuo Su Nation ChiaoTung University Department of Applied Mathematics College of Science 蘇偉碩 2013/12/8

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

3. 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

4. The external force is assumed to be periodic. 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 qinvolving viscous damping can be formulated by where K,D, and M from the finite element discretization on a uniform mesh satisfy the given boundary conditions.

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

6. 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) satisfying the relation constant determinants In this case, det = cdet shows that the eigenvalues are the same.

7. The first companion form and and the second companion form and Drawbacks: Doubles the size of the problem dimension Palindromic and symplectic may be lost Symplectic: and both exist

8. In the , W.-W. Lin propose some structure-preserving method to deal with ⊤-palindromic quadratic pencil. And the numerical result is quite excellent. Based on the spirit, we want to use the same algorithm to solve palindromic matrix eigenvalue problems. 1. Structure-preserving algorithms for palindromic quadratic eigenvalue problems arising from vibration on fast trains

9. We consider the palindromic matric polynomials of degree where and or . It also satisfies The eigenvalues appear in the pairs of the form .

10. Definition:(Quadratization) is a quadratization of if there are matrix rational functions and with nonzero and constant determinants satisfying the two-sided factorization P-Quadratization: and have the same structure property. It also implies that detdetfor some nonzero constant c, and and have the same eigenvalue.

11. 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 form with . We show degree 4 , (we can see the paper for generalize )

12. Solving flow chart  H-even PEP H-palindromic QEP H-palindromic PEP  T-palindromic QEP T-even PEP T-palindromic PEP  H-anti-palindromic QEP H-palindromic QEP H-odd PEP H-anti-palindromic PEP T-odd PEP T-palindromic GEP T-palindromic QEP T-anti-palindromic PEP

13. 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

14. Solving flow chart  H-even PEP H-palindromic QEP H-palindromic PEP  T-palindromic QEP T-even PEP T-palindromic PEP   H-anti-palindromic QEP H-palindromic QEP H-odd PEP H-anti-palindromic PEP  T-odd PEP T-palindromic GEP T-palindromic QEP T-anti-palindromic PEP where Linearization

15. H-PQEP Linear We consider the -PQEP In order the preserve the symplectic structure, we get the special linearization where and it also satisfies , So that the matrix pair has eigenvalues and . The pencil are called -symplectic

16. 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.

17. relation Theorem Suppose 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 .

18. simplify case Substituting the H-skew-Hamiltonian and can be write as , 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

19. Theorem Patel’s Alg If is an eigenpair of , then are eigenvectors of If is an eigenvector of corresponding to the eigenvalue , then is an eigenvector of The pair can be reduced to block upper triangular forms , where are orthogonal satisfying , and are upper Hessenberg and upper triangular, respectively.

20. SPA for H-PQEP flow chart H-PQEP Linear simplify case Patel’s Alg Solve and follow the conversion relationship

21. Algorithm: SPA for H-PQEP Input:with and Output: All eigenvalues and eigenvectors of Form the matrix pair Reduce to block upper triangular forms Compute eigenpairsof by the QZ algorithm Compute , Compute the eigenpairof by Compute and by solving ; Compute and If then it is an eigenvector of corresponding to ; If , then it is an eigenvalue of corresponding to

22. We consider From the P-Quadration we mentioned , which satisfies If is an eigenpair of , then is an eigenpair of with where

23. In order to balance the entries of coefficient matrices in , we define a complex diagonal matrix Such that in the new palindromic matrix polynomial whose each entries are close to one as much as possible. That is

24. The parameters can be determined by solving the least square problems , Then, the parameters and are determined by ,

25. From the row balancing of , we first set We take to be Then we set the balancing coefficient

26. We compare the computational cost for PQ_SPA and PL_SPA We define the associated relative residual by The eigenvalues of H-PPEP appear in reciprocal pairs We define the reciprocities of the computed eigenvalues by ,

27. complex matrices and normal distribution with zero mean and standard deviation Example 2.1 Consider the H-PPEP with and

28. Example 2.2 Consider the H-PPEP with and and and being defined as , (Large where , and with if n is even or

29. (Large Example 2.3 Consider the H-PPEP with and and being defined as where , and is defined in ex2.2 with

30. 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. NumerischeMathematik, 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, NumerischeMathematik 2011, 118(4): 713-735.

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