A Linearization method f or Polynomial Eigenvalue Problems using a contour integra l Junko Asakura, Tetsuya Sakurai, Hir oto Tadano Department of Computer Science, University of Tsu kuba Tsutomu Ikegami Grid Technology Research Center, AIST Kinji Kimura Department of Applied Mathematics and Physics, Kyoto University Nonlinear
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A Linearization method for Polynomial Eigenvalue Problems using a contour integral Junko Asakura, Tetsuya Sakurai, Hiroto Tadano Department of Computer.
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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