a linearization method for polynomial eigenvalue problems using a contour integral junko asakura,...

A Linearization method for Polynomial Eigenvalue Problems using a contour integral

Junko Asakura, Tetsuya Sakurai, Hiroto Tadano Department of Computer Science, University of Tsukuba T

sutomu IkegamiGrid Technology Research Center, AIST

Kinji KimuraDepartment of Applied Mathematics and Physics, Kyoto University

Nonlinear

June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik

Outline

• Background• Linearization method for PEPs using a contour integral • Extension to analytic functions • Numerical Examples• Conclusions


Background


Polynomial Eigenvalue Problems

• Oscillation analysis with damping• Stability problems in uid dynamicsfl• 3D-Schrödinger equation etc

F(z) x = 0F(z) = zlAl + zl-1Al-1 + ･･･ + zA1 + A0

Ak

Applications:

Eigenvalues in a specified domain are required in some applications


Projection method for generalized eigenvalue problems

using a contour integral

[1] Sakurai, T., Sugiura, H., A projection method for generalized eigenvalue problems. J. Comput. Appl. Math. 159( 2003)119-128

Sakurai-Sugiura(SS) method [1]

Ax = Bx


Linearization method for polynomial eigenvalue problems using a contour integral


Sakurai-Sugiura method： a positively oriented closed Jordan curve

： eigenpairs of the matrix pencil(A, B) in Γ (j=1,..., m)

(j, uj)

The eigenvalues of the pencil (H< , Hm) are given by 1, …, m.m

v : an arbitrary nonzero vector


Modification of the moments k for PEPs

： a positively oriented closed Jordan curve

： eigenpairs of the matrix pencil(A, B) in Γ (j=1,..., m)

(j, uj)


Modification of the moments k for PEPs

F(z) = zlAl + zl-1Al-1 + ･･･ + zA1 + A0

Ak


The Main Theorem

The eigenvalues of the pencil

are given by 1, …, m

F(z): a regular polynomial matrix 1, …, m: simple eigenvalues of F(z) in


The Smith Form F(z) : n × n regular matrix polynomial

P(z)F(z)Q(z) = D(z)where

D(z) =

di ： monic scalar polynomials s.t. di is divisible by di-1

P(z), Q(z) ： n×n matrix polynomials with constant nonzero determinants

F(z) admits the representation

.


,,

F(z): a regular polynomial matrix 1, …, m: simple eigenvalues of F(z) in

P(z)F(z)Q(z) = D(z): The Smith Form of F(z)


Linearization method for polynomial eigenvalue problems

using a contour integralPolynomial Eigenvalue Problem

F(z)x = 0F(z) = zlAl + zl-1Al-1 + ･･･ + zA1 + A0

Generalized Eigenvalue ProblemH< x = Hmx

H< = [i+j-2]i, j=1, Hm = [i+j-1]i, j=1

mm

mm


Extension into Analytic Functions

fij: an analytic function in , i, j= 1, …, n

F(z) x = 0


(1) Interchange two rows(2) Add to some row another row multiplied by an analytic function inside and on the given domain(3) Multiply a row by a nonzero complex number

together with the three corresponding operations on columns.

Elementary transformations


the Smith Form for Nonlinear Eigenvalue Problem

F(z) : n × n regular matrix

P(z)F(z)Q(z) = D(z)where

D(z) =

di: analytic function inside and on such that di is divisible by di-1, i=1, …, n-1

P(z), Q(z) ： n×n matrix with constant nonzero determinants

F(z) admits the representation


Block version of the Sakurai-Sugiura method

,

Block SS method[2]

[2] T. Ikegami, T. Sakurai, U. Nagashima, A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura method (submitted)

： a positively oriented closed Jordan curve： eigenpairs of the matrix polynomial F(z) in Γ (j=1,..., m) (j, uj)

V : a regular matrix


Computation of Mk

j := + exp(2i/N(j+1/2)), j = 0, …, N-1

k = 0, …, 2m-1

Approximate the integral of k via N-point trapezoidal rule:

,

V , det(V) ≠ 0


Computation of the eigenvectors of F(z)

qn(j) = jSxj, j≠ 0

The eigenvectors of F(z) are computed by

where

xj: eigenvectors of the pencil (H<, Hm)m

S = [s0, …, sk], k=0, …, m-1


Algorithm: Block SS methodInput: F(z), V , N, M, , Output: 1, …, K, qn(1), …, qn(K)

• Set j ← + exp(2i/N(j+1/2)), j = 0, …, N-1

• Compute VHF(j)-1V, j = 0,…, N-1

• Compute Mk, k = 0, …, 2m-1• Construct Hankel matrices • Compute the eigenvalues 1, …, K of

• Compute qn(1), …, qn(K)

• Set j = + j, j = 1, ..., K


Numerical Examples


Numerical ExamplesTest Problems• Example1: Quadratic Eigenvalue Problem • Example2: Eigenvalue Problem for a Matrix whose elements are Analytic Functions• Example3: Quartic Eigenvalue Problem

Test Environment • MacBook Core2Duo 2.0GHz• Memory 2.0Gbytes• MATLAB 7.4.0


Example1

Eigenvalues:

1/3, 1/2, 1, i, -i, ∞

Test Matrix:

Γ= ei| 0≦≦2 } γ = 0, L = 1

Parameters:

5 eigenvalues lie in

×

××××

Re

Im eigenvalue


Results of Example1

k residual

1 0.333333333333717 1.05e-13 1.78e-14

2 0.499999999999529 8.24e-14 1.41e-14

3 1.000000000000120 9.10e-15 1.53e-14

4 1.000000000000009

i 1.02e-15 1.94e-14

5-1.000000000000009

i 1.02e-15 1.49e-14

： result, ： exact


Example2

Test matrix:

Eigenvalues:

0, /2, -/2, , -log7(≒1.9459) ≦z≦)

Γ= ei| 0≦≦2 } γ = 0, 3.2L = 2

Parameters:

Equivalent to


Results of Example2

k residual

1 -3.1415926535897891 1.27e-15 7.10e-11

2 -1.5707963267942768 3.95e-13 4.27e-11

3 0.0000000000006607 6.61e-13 5.87e-10

4 1.5707963267612979 2.14e-11 1.76e-09

5 1.9459101513382451 1.17e-09 8.64e-08

6 3.1415926535890546 2.35e-13 6.52e-09

： result, ： exact


Example3

Test Matrix: Quartic Matrix Polynomial “butterfly” in NLEVP[3]

[3] T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder, and F. Tisseur, NLEVP: A Collection of Nonlinear Eigenvalue Problems, MIMS EPrint 2008.40 (2008)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5

F(z) = 4A4+3A3+2A2+A1+A0

Ai , i = 0, 1, 2, 3, 4


Example3

Parameters:

Γ= ei| 0≦≦2 } γ = 1-i,

L = 24

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5

→A total of 13 eigenvalues lie in


Results of Example3

-1.4

-1.2

-1

-0.8

-0.6

0.6 0.8 1 1.2 1.4

+: results of “polyeig” o: results of the proposed method

max residual of eigenvalues calculated by the proposed method: 7.40e-12

→


Conclusions


Conclusions

Summary of Our Study• We proposed a linearization method for PEPs using a contour integral.• We extended the proposed method to nonlinear eigenvalue problems.

Future Study• Precise theoretical observation of the extension to nonli

near eigenvalue problems• Estimation of suitable parameters

a linearization method for polynomial eigenvalue problems using a contour integral junko asakura,...

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