A Linearization method for Polynomial Nonlinear Eigenvalue Problems
A Linearization method for Polynomial Nonlinear Eigenvalue Problems using a contour integral Junko Asakura, Tetsuya Sakurai, Hiroto Tadano Department of Computer Science, University of Tsukuba Tsutomu Ikegami Grid Technology Research Center, AIST Kinji Kimura June 10, 2008 Department A linearization method forand PEPs of Applied Mathematics Physics, Kyoto University IWASEP 7, 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 IWASEP 7, Dubrovnik
Background June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Polynomial Eigenvalue Problems F(z) x = 0 F(z) = zl. Al + zl-1 Al-1 + ・・・ + z. A 1 + A 0 Ak Applications: • Oscillation analysis with damping • Stability problems in fluid dynamics • 3 D-Schrödinger equation etc Eigenvalues in a specified domain are required in some applications June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Projection method for generalized eigenvalue problems using a contour integral Sakurai-Sugiura(SS) method [1] Ax = Bx [1] Sakurai, T. , Sugiura, H. , A projection method for generalized eigenvalue problems. J. Comput. Appl. Math. 159( 2003)119 -128 June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Linearization method for polynomial eigenvalue problems using a contour integral June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Sakurai-Sugiura method :a positively oriented closed Jordan curve ( j, uj) :eigenpairs of the matrix pencil(A, B) in Γ (j=1, . . . , m) v : an arbitrary nonzero vector The eigenvalues of the pencil (Hm< , Hm) are given by 1, …, m. June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Modification of the moments k for PEPs :a positively oriented closed Jordan curve ( j, uj) :eigenpairs of the matrix pencil(A, B) in Γ (j=1, . . . , m) June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Modification of the moments k for PEPs F(z) = zl. Al + zl-1 Al-1 + ・・・ + z. A 1 + A 0 Ak June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
The Main Theorem F(z): a regular polynomial matrix 1, …, m: simple eigenvalues of F(z) in The eigenvalues of the pencil are given by 1, …, m June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
The Smith Form F(z) : n × n regular matrix polynomial F(z) admits the representation 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 June 10, 2008 A linearization method for PEPs IWASEP 7, 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 IWASEP 7, Dubrovnik ,
Linearization method for polynomial eigenvalue problems using a contour integral Polynomial Eigenvalue Problem F(z)x = 0 F(z) = zl. Al + zl-1 Al-1 + ・・・ + z. A 1 + A 0 Generalized Eigenvalue Problem H<m x = Hmx m m Hm< = [ i+j-2]i, j=1, Hm = [ i+j-1]i, j=1 June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Extension into Analytic Functions F(z) x = 0 fij: an analytic function in , June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik i, j= 1, …, n
Elementary transformations (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. June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
the Smith Form for Nonlinear Eigenvalue Problem F(z) : n × n regular matrix F(z) admits the representation 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 June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Block version of the Sakurai-Sugiura method Block SS method[2] :a positively oriented closed Jordan curve ( j, uj) :eigenpairs of the matrix polynomial F(z) in Γ (j=1, . . . , m) V : a regular matrix , [2] T. Ikegami, T. Sakurai, U. Nagashima, A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura method (submitted) June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Computation of Mk Approximate the integral of k via N-point trapezoidal rule: , k = 0, …, 2 m-1 V , det(V) ≠ 0 j : = + exp(2 i/N(j+1/2)), j = 0, …, N-1 June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Computation of the eigenvectors of F(z) xj: eigenvectors of the pencil (H<m, Hm) The eigenvectors of F(z) are computed by qn( j) = j. Sxj, j≠ 0 where S = [s 0, …, sk], k=0, …, m-1 June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Algorithm: Block SS method Input: F(z), V , N, M, , Output: 1, …, K, qn( 1), …, qn( K) • • Set j ← + exp(2 i/N(j+1/2)), j = 0, …, N-1 Compute VHF( j)-1 V, j = 0, …, N-1 Compute Mk, k = 0, …, 2 m-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 IWASEP 7, Dubrovnik
Numerical Examples June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Numerical Examples Test Problems • Example 1: Quadratic Eigenvalue Problem • Example 2: Eigenvalue Problem for a Matrix whose elements are Analytic Functions • Example 3: Quartic Eigenvalue Problem Test Environment • Mac. Book Core 2 Duo 2. 0 GHz • Memory 2. 0 Gbytes • MATLAB 7. 4. 0 June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Example 1 Test Matrix: Eigenvalues: 1/3, 1/2, 1, i, -i, ∞ Im eigenvalue × ×× × Parameters: Re Γ�=�� 0≦ ≦ 2 } × γ = 0, 5 eigenvalues lie in L = 1 ���ei | June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Results of Example 1 k residual 1 0. 333333717 1. 05 e-13 1. 78 e-14 2 0. 4999999529 8. 24 e-14 1. 41 e-14 3 1. 000000120 9. 10 e-15 1. 53 e-14 4 1. 00000009 i 1. 02 e-15 1. 94 e-14 5 -1. 00000009 i 1. 02 e-15 1. 49 e-14 :result, June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik :exact
Equivalent to Example 2 Test matrix: Eigenvalues: 0, /2, - /2, , - ��log 7(≒ 1. 9459)� �� ≦z≦ ) Parameters: Γ�=�� ��� ei | 0≦ ≦ 2 } γ = 0, 3. 2 L = 2 June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Results of Example 2 k residual 1 -3. 1415926535897891 1. 27 e-15 7. 10 e-11 2 -1. 5707963267942768 3. 95 e-13 4. 27 e-11 3 0. 0000006607 6. 61 e-13 5. 87 e-10 4 1. 5707963267612979 2. 14 e-11 1. 76 e-09 5 1. 9459101513382451 1. 17 e-09 8. 64 e-08 6 3. 1415926535890546 2. 35 e-13 6. 52 e-09 :result, June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik :exact
Example 3 Test Matrix: Quartic Matrix Polynomial “butterfly” in NLEVP[3] F(z) = 4 A 4+ 3 A 3+ 2 A 2+ A 1+A 0 Ai , i = 0, 1, 2, 3, 4 [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) June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Example 3 Parameters: Γ�=�� ��� ei | 0≦ ≦ 2 } γ = 1 -i, L = 24 A total of 13 eigenvalues lie in June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik →
Results of Example 3 +: results of “polyeig” o: results of the proposed method max residual of eigenvalues calculated by the proposed method: 7. 40 e-12 → June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
Conclusions June 10, 2008 A linearization method for PEPs IWASEP 7, 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 nonlinear eigenvalue problems • Estimation of suitable parameters June 10, 2008 A linearization method for PEPs IWASEP 7, Dubrovnik
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