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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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
Background
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
Linearization method for polynomial eigenvalue problems using a contour integral
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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)
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
Modification of the moments k for PEPs
F(z) = zlAl + zl-1Al-1 + ・・・ + zA1 + A0
Ak
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
.
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
,,
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)
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
Extension into Analytic Functions
fij: an analytic function in , i, j= 1, …, n
F(z) x = 0
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
(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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
Numerical Examples
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
Example2
Test matrix:
Eigenvalues:
0, /2, -/2, , -log7(≒1.9459) ≦z≦)
Γ= ei| 0≦≦2 } γ = 0, 3.2L = 2
Parameters:
Equivalent to
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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
→
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
Conclusions
June 10, 2008 A linearization method for PEPs IWASEP7, Dubrovnik
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