structure-preserving krylov subspace methods for ...structure-preserving krylov subspace methods for...
TRANSCRIPT
![Page 1: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/1.jpg)
Structure-preserving Krylovsubspace methods for Hamiltonianand symplectic eigenvalue problems
David S. Watkins
Department of Mathematics
Washington State University
March 2007 – p.1
![Page 2: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/2.jpg)
Definitions
matrices in R2n×2n
J =
[0 I
−I 0
]
S is symplectic if ST JS = J (Lie group)
H is Hamiltonian if (JH)T = JH (Lie algebra)
B is skew Hamiltonian if (JB)T = −JB
(Jordan algebra)
Matrices with these structures arise in various applications.
Sometimes they are large and sparse.
March 2007 – p.2
![Page 3: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/3.jpg)
Definitionsmatrices in R
2n×2n
J =
[0 I
−I 0
]
S is symplectic if ST JS = J (Lie group)
H is Hamiltonian if (JH)T = JH (Lie algebra)
B is skew Hamiltonian if (JB)T = −JB
(Jordan algebra)
Matrices with these structures arise in various applications.
Sometimes they are large and sparse.
March 2007 – p.2
![Page 4: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/4.jpg)
Definitionsmatrices in R
2n×2n
J =
[0 I
−I 0
]
S is symplectic if ST JS = J (Lie group)
H is Hamiltonian if (JH)T = JH (Lie algebra)
B is skew Hamiltonian if (JB)T = −JB
(Jordan algebra)
Matrices with these structures arise in various applications.
Sometimes they are large and sparse.
March 2007 – p.2
![Page 5: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/5.jpg)
Definitionsmatrices in R
2n×2n
J =
[0 I
−I 0
]
S is symplectic if ST JS = J
(Lie group)
H is Hamiltonian if (JH)T = JH (Lie algebra)
B is skew Hamiltonian if (JB)T = −JB
(Jordan algebra)
Matrices with these structures arise in various applications.
Sometimes they are large and sparse.
March 2007 – p.2
![Page 6: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/6.jpg)
Definitionsmatrices in R
2n×2n
J =
[0 I
−I 0
]
S is symplectic if ST JS = J (Lie group)
H is Hamiltonian if (JH)T = JH (Lie algebra)
B is skew Hamiltonian if (JB)T = −JB
(Jordan algebra)
Matrices with these structures arise in various applications.
Sometimes they are large and sparse.
March 2007 – p.2
![Page 7: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/7.jpg)
Definitionsmatrices in R
2n×2n
J =
[0 I
−I 0
]
S is symplectic if ST JS = J (Lie group)
H is Hamiltonian if (JH)T = JH
(Lie algebra)
B is skew Hamiltonian if (JB)T = −JB
(Jordan algebra)
Matrices with these structures arise in various applications.
Sometimes they are large and sparse.
March 2007 – p.2
![Page 8: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/8.jpg)
Definitionsmatrices in R
2n×2n
J =
[0 I
−I 0
]
S is symplectic if ST JS = J (Lie group)
H is Hamiltonian if (JH)T = JH (Lie algebra)
B is skew Hamiltonian if (JB)T = −JB
(Jordan algebra)
Matrices with these structures arise in various applications.
Sometimes they are large and sparse.
March 2007 – p.2
![Page 9: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/9.jpg)
Definitionsmatrices in R
2n×2n
J =
[0 I
−I 0
]
S is symplectic if ST JS = J (Lie group)
H is Hamiltonian if (JH)T = JH (Lie algebra)
B is skew Hamiltonian if (JB)T = −JB
(Jordan algebra)
Matrices with these structures arise in various applications.
Sometimes they are large and sparse.
March 2007 – p.2
![Page 10: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/10.jpg)
Definitionsmatrices in R
2n×2n
J =
[0 I
−I 0
]
S is symplectic if ST JS = J (Lie group)
H is Hamiltonian if (JH)T = JH (Lie algebra)
B is skew Hamiltonian if (JB)T = −JB
(Jordan algebra)
Matrices with these structures arise in various applications.
Sometimes they are large and sparse.
March 2007 – p.2
![Page 11: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/11.jpg)
Definitionsmatrices in R
2n×2n
J =
[0 I
−I 0
]
S is symplectic if ST JS = J (Lie group)
H is Hamiltonian if (JH)T = JH (Lie algebra)
B is skew Hamiltonian if (JB)T = −JB
(Jordan algebra)
Matrices with these structures arise in various applications.
Sometimes they are large and sparse.
March 2007 – p.2
![Page 12: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/12.jpg)
Definitionsmatrices in R
2n×2n
J =
[0 I
−I 0
]
S is symplectic if ST JS = J (Lie group)
H is Hamiltonian if (JH)T = JH (Lie algebra)
B is skew Hamiltonian if (JB)T = −JB
(Jordan algebra)
Matrices with these structures arise in various applications.
Sometimes they are large and sparse.
March 2007 – p.2
![Page 13: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/13.jpg)
Objective
Produce structure-preserving Krylov subspace methods forsymplectic, Hamiltonian, and skew-Hamiltonian eigenvalueproblems. Done!
Freund / Mehrmann (unpublished)
Benner / Fassbender (1997,1998)
Benner / Fassbender / W (1988,1999)
Mehrmann / W (2001)
W (2003)
March 2007 – p.3
![Page 14: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/14.jpg)
Objective
Produce structure-preserving Krylov subspace methods forsymplectic, Hamiltonian, and skew-Hamiltonian eigenvalueproblems.
Done!
Freund / Mehrmann (unpublished)
Benner / Fassbender (1997,1998)
Benner / Fassbender / W (1988,1999)
Mehrmann / W (2001)
W (2003)
March 2007 – p.3
![Page 15: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/15.jpg)
Objective
Produce structure-preserving Krylov subspace methods forsymplectic, Hamiltonian, and skew-Hamiltonian eigenvalueproblems. Done!
Freund / Mehrmann (unpublished)
Benner / Fassbender (1997,1998)
Benner / Fassbender / W (1988,1999)
Mehrmann / W (2001)
W (2003)
March 2007 – p.3
![Page 16: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/16.jpg)
Objective
Produce structure-preserving Krylov subspace methods forsymplectic, Hamiltonian, and skew-Hamiltonian eigenvalueproblems. Done!
Freund / Mehrmann (unpublished)
Benner / Fassbender (1997,1998)
Benner / Fassbender / W (1988,1999)
Mehrmann / W (2001)
W (2003)
March 2007 – p.3
![Page 17: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/17.jpg)
Objective
Produce structure-preserving Krylov subspace methods forsymplectic, Hamiltonian, and skew-Hamiltonian eigenvalueproblems. Done!
Freund / Mehrmann (unpublished)
Benner / Fassbender (1997,1998)
Benner / Fassbender / W (1988,1999)
Mehrmann / W (2001)
W (2003)
March 2007 – p.3
![Page 18: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/18.jpg)
Objective
Produce structure-preserving Krylov subspace methods forsymplectic, Hamiltonian, and skew-Hamiltonian eigenvalueproblems. Done!
Freund / Mehrmann (unpublished)
Benner / Fassbender (1997,1998)
Benner / Fassbender / W (1988,1999)
Mehrmann / W (2001)
W (2003)
March 2007 – p.3
![Page 19: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/19.jpg)
Objective
Produce structure-preserving Krylov subspace methods forsymplectic, Hamiltonian, and skew-Hamiltonian eigenvalueproblems. Done!
Freund / Mehrmann (unpublished)
Benner / Fassbender (1997,1998)
Benner / Fassbender / W (1988,1999)
Mehrmann / W (2001)
W (2003)
March 2007 – p.3
![Page 20: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/20.jpg)
Objective
Produce structure-preserving Krylov subspace methods forsymplectic, Hamiltonian, and skew-Hamiltonian eigenvalueproblems. Done!
Freund / Mehrmann (unpublished)
Benner / Fassbender (1997,1998)
Benner / Fassbender / W (1988,1999)
Mehrmann / W (2001)
W (2003)
March 2007 – p.3
![Page 21: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/21.jpg)
Today’s Objective
Show how easy it is!
unsymmetric Lanczos process
skew-Hamiltonian structure preserved automatically
H2 = B
S + S−1 = B
David S. Watkins, The Matrix Eigenvalue Problem:GR and Krylov Subspace Methods, SIAM, to appear.
March 2007 – p.4
![Page 22: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/22.jpg)
Today’s Objective
Show how easy it is!
unsymmetric Lanczos process
skew-Hamiltonian structure preserved automatically
H2 = B
S + S−1 = B
David S. Watkins, The Matrix Eigenvalue Problem:GR and Krylov Subspace Methods, SIAM, to appear.
March 2007 – p.4
![Page 23: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/23.jpg)
Today’s Objective
Show how easy it is!
unsymmetric Lanczos process
skew-Hamiltonian structure preserved automatically
H2 = B
S + S−1 = B
David S. Watkins, The Matrix Eigenvalue Problem:GR and Krylov Subspace Methods, SIAM, to appear.
March 2007 – p.4
![Page 24: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/24.jpg)
Today’s Objective
Show how easy it is!
unsymmetric Lanczos process
skew-Hamiltonian structure preserved automatically
H2 = B
S + S−1 = B
David S. Watkins, The Matrix Eigenvalue Problem:GR and Krylov Subspace Methods, SIAM, to appear.
March 2007 – p.4
![Page 25: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/25.jpg)
Today’s Objective
Show how easy it is!
unsymmetric Lanczos process
skew-Hamiltonian structure preserved automatically
H2 = B
S + S−1 = B
David S. Watkins, The Matrix Eigenvalue Problem:GR and Krylov Subspace Methods, SIAM, to appear.
March 2007 – p.4
![Page 26: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/26.jpg)
Today’s Objective
Show how easy it is!
unsymmetric Lanczos process
skew-Hamiltonian structure preserved automatically
H2 = B
S + S−1 = B
David S. Watkins, The Matrix Eigenvalue Problem:GR and Krylov Subspace Methods, SIAM, to appear.
March 2007 – p.4
![Page 27: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/27.jpg)
Today’s Objective
Show how easy it is!
unsymmetric Lanczos process
skew-Hamiltonian structure preserved automatically
H2 = B
S + S−1 = B
David S. Watkins, The Matrix Eigenvalue Problem:GR and Krylov Subspace Methods, SIAM, to appear.
March 2007 – p.4
![Page 28: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/28.jpg)
Unsymmetric Lanczos Process(1950)
uj+1βj = Auj − ujαj − uj−1γj−1
wj+1γj = ATwj − wjαj − wj−1βj−1
Start with 〈u1, w1〉 = 1.
sequences are biorthornomal: 〈uj , wk〉 = δjk
omitting simple formulas for the coefficients
|γj | = |βj |
March 2007 – p.5
![Page 29: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/29.jpg)
Unsymmetric Lanczos Process(1950)
uj+1βj = Auj − ujαj − uj−1γj−1
wj+1γj = ATwj − wjαj − wj−1βj−1
Start with 〈u1, w1〉 = 1.
sequences are biorthornomal: 〈uj , wk〉 = δjk
omitting simple formulas for the coefficients
|γj | = |βj |
March 2007 – p.5
![Page 30: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/30.jpg)
Unsymmetric Lanczos Process(1950)
uj+1βj = Auj − ujαj − uj−1γj−1
wj+1γj = ATwj − wjαj − wj−1βj−1
Start with 〈u1, w1〉 = 1.
sequences are biorthornomal: 〈uj , wk〉 = δjk
omitting simple formulas for the coefficients
|γj | = |βj |
March 2007 – p.5
![Page 31: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/31.jpg)
Unsymmetric Lanczos Process(1950)
uj+1βj = Auj − ujαj − uj−1γj−1
wj+1γj = ATwj − wjαj − wj−1βj−1
Start with 〈u1, w1〉 = 1.
sequences are biorthornomal: 〈uj , wk〉 = δjk
omitting simple formulas for the coefficients
|γj | = |βj |
March 2007 – p.5
![Page 32: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/32.jpg)
Unsymmetric Lanczos Process(1950)
uj+1βj = Auj − ujαj − uj−1γj−1
wj+1γj = ATwj − wjαj − wj−1βj−1
Start with 〈u1, w1〉 = 1.
sequences are biorthornomal: 〈uj , wk〉 = δjk
omitting simple formulas for the coefficients
|γj | = |βj |
March 2007 – p.5
![Page 33: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/33.jpg)
Unsymmetric Lanczos Process(1950)
uj+1βj = Auj − ujαj − uj−1γj−1
wj+1γj = ATwj − wjαj − wj−1βj−1
Start with 〈u1, w1〉 = 1.
sequences are biorthornomal: 〈uj , wk〉 = δjk
omitting simple formulas for the coefficients
|γj | = |βj |
March 2007 – p.5
![Page 34: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/34.jpg)
Collect the coefficients
Tj =
α1 γ1
β1 α2 γ2
β2 α3
. . .. . . . . . γj−1
βj−1 αj
Eigenvalues are estimates of eigenvalues of A.
Tj is pseudosymmetric. (|γj | = |βj |)
Tj = TjDj ,where Tj is symmetric and Dj is a signature matrix.
March 2007 – p.6
![Page 35: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/35.jpg)
Collect the coefficients
Tj =
α1 γ1
β1 α2 γ2
β2 α3
. . .. . . . . . γj−1
βj−1 αj
Eigenvalues are estimates of eigenvalues of A.
Tj is pseudosymmetric. (|γj | = |βj |)
Tj = TjDj ,where Tj is symmetric and Dj is a signature matrix.
March 2007 – p.6
![Page 36: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/36.jpg)
Collect the coefficients
Tj =
α1 γ1
β1 α2 γ2
β2 α3
. . .. . . . . . γj−1
βj−1 αj
Eigenvalues are estimates of eigenvalues of A.
Tj is pseudosymmetric. (|γj | = |βj |)
Tj = TjDj ,where Tj is symmetric and Dj is a signature matrix.
March 2007 – p.6
![Page 37: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/37.jpg)
Collect the coefficients
Tj =
α1 γ1
β1 α2 γ2
β2 α3
. . .. . . . . . γj−1
βj−1 αj
Eigenvalues are estimates of eigenvalues of A.
Tj is pseudosymmetric. (|γj | = |βj |)
Tj = TjDj ,where Tj is symmetric and Dj is a signature matrix.
March 2007 – p.6
![Page 38: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/38.jpg)
Collect the coefficients
Tj =
α1 γ1
β1 α2 γ2
β2 α3
. . .. . . . . . γj−1
βj−1 αj
Eigenvalues are estimates of eigenvalues of A.
Tj is pseudosymmetric. (|γj | = |βj |)
Tj = TjDj ,where Tj is symmetric and Dj is a signature matrix.
March 2007 – p.6
![Page 39: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/39.jpg)
Tj =
a1 b1
b1 a2 b2
b2 a3
. . .. . . . . . bj−1
bj−1 aj
Dj =
d1
d2
d3
. . .
dj
March 2007 – p.7
![Page 40: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/40.jpg)
Lanczos recurrences recast:
uj+1bjdj = Auj − ujajdj − uj−1bj−1dj
wj+1dj+1bj = ATwj − wjdjaj − wj−1dj−1bj−1,
Recurrences rewritten as matrix equations:
AUj = UjTjDj + uj+1bjdjeTj
ATWj = WjDjTj + wj+1dj+1bjeTj .
Implicit restarts: Filter using HR algorithm
Grimme / Sorensen / Van Dooren (1996)
March 2007 – p.8
![Page 41: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/41.jpg)
Lanczos recurrences recast:
uj+1bjdj = Auj − ujajdj − uj−1bj−1dj
wj+1dj+1bj = ATwj − wjdjaj − wj−1dj−1bj−1,
Recurrences rewritten as matrix equations:
AUj = UjTjDj + uj+1bjdjeTj
ATWj = WjDjTj + wj+1dj+1bjeTj .
Implicit restarts: Filter using HR algorithm
Grimme / Sorensen / Van Dooren (1996)
March 2007 – p.8
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Lanczos recurrences recast:
uj+1bjdj = Auj − ujajdj − uj−1bj−1dj
wj+1dj+1bj = ATwj − wjdjaj − wj−1dj−1bj−1,
Recurrences rewritten as matrix equations:
AUj = UjTjDj + uj+1bjdjeTj
ATWj = WjDjTj + wj+1dj+1bjeTj .
Implicit restarts: Filter using HR algorithm
Grimme / Sorensen / Van Dooren (1996)
March 2007 – p.8
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Lanczos recurrences recast:
uj+1bjdj = Auj − ujajdj − uj−1bj−1dj
wj+1dj+1bj = ATwj − wjdjaj − wj−1dj−1bj−1,
Recurrences rewritten as matrix equations:
AUj = UjTjDj + uj+1bjdjeTj
ATWj = WjDjTj + wj+1dj+1bjeTj .
Implicit restarts:
Filter using HR algorithm
Grimme / Sorensen / Van Dooren (1996)
March 2007 – p.8
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Lanczos recurrences recast:
uj+1bjdj = Auj − ujajdj − uj−1bj−1dj
wj+1dj+1bj = ATwj − wjdjaj − wj−1dj−1bj−1,
Recurrences rewritten as matrix equations:
AUj = UjTjDj + uj+1bjdjeTj
ATWj = WjDjTj + wj+1dj+1bjeTj .
Implicit restarts: Filter using HR algorithm
Grimme / Sorensen / Van Dooren (1996)
March 2007 – p.8
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Lanczos recurrences recast:
uj+1bjdj = Auj − ujajdj − uj−1bj−1dj
wj+1dj+1bj = ATwj − wjdjaj − wj−1dj−1bj−1,
Recurrences rewritten as matrix equations:
AUj = UjTjDj + uj+1bjdjeTj
ATWj = WjDjTj + wj+1dj+1bjeTj .
Implicit restarts: Filter using HR algorithm
Grimme / Sorensen / Van Dooren (1996)
March 2007 – p.8
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Preservation of Structure
A = S−1AS
S symplectic ⇒ structure preserved
Lanczos process is a partial similarity transformation.
Vectors produced are columns of transforming matrix.
Need process that produces vectors that are columns ofa symplectic matrix.
Isotropy!
March 2007 – p.9
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Preservation of StructureA = S−1AS
S symplectic ⇒ structure preserved
Lanczos process is a partial similarity transformation.
Vectors produced are columns of transforming matrix.
Need process that produces vectors that are columns ofa symplectic matrix.
Isotropy!
March 2007 – p.9
![Page 48: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/48.jpg)
Preservation of StructureA = S−1AS
S symplectic ⇒ structure preserved
Lanczos process is a partial similarity transformation.
Vectors produced are columns of transforming matrix.
Need process that produces vectors that are columns ofa symplectic matrix.
Isotropy!
March 2007 – p.9
![Page 49: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/49.jpg)
Preservation of StructureA = S−1AS
S symplectic ⇒ structure preserved
Lanczos process is a partial similarity transformation.
Vectors produced are columns of transforming matrix.
Need process that produces vectors that are columns ofa symplectic matrix.
Isotropy!
March 2007 – p.9
![Page 50: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/50.jpg)
Preservation of StructureA = S−1AS
S symplectic ⇒ structure preserved
Lanczos process is a partial similarity transformation.
Vectors produced are columns of transforming matrix.
Need process that produces vectors that are columns ofa symplectic matrix.
Isotropy!
March 2007 – p.9
![Page 51: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/51.jpg)
Preservation of StructureA = S−1AS
S symplectic ⇒ structure preserved
Lanczos process is a partial similarity transformation.
Vectors produced are columns of transforming matrix.
Need process that produces vectors that are columns ofa symplectic matrix.
Isotropy!
March 2007 – p.9
![Page 52: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/52.jpg)
Preservation of StructureA = S−1AS
S symplectic ⇒ structure preserved
Lanczos process is a partial similarity transformation.
Vectors produced are columns of transforming matrix.
Need process that produces vectors that are columns ofa symplectic matrix.
Isotropy!
March 2007 – p.9
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Isotropy
Def: U = R(U) is isotropic if xT Jy = 0 for all x, y ∈ U , i.e.
UTJU = 0
Symplectic matrix S = [ U V ] satisfies ST JS = J , i.e.
UTJU = 0, V T JV = 0, UT JV = I.
In particular, R(U), R(V ) are isotropic.
March 2007 – p.10
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Isotropy
Def: U = R(U) is isotropic if xT Jy = 0 for all x, y ∈ U , i.e.
UTJU = 0
Symplectic matrix S = [ U V ] satisfies ST JS = J , i.e.
UTJU = 0, V T JV = 0, UT JV = I.
In particular, R(U), R(V ) are isotropic.
March 2007 – p.10
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Isotropy
Def: U = R(U) is isotropic if xT Jy = 0 for all x, y ∈ U , i.e.
UTJU = 0
Symplectic matrix S = [ U V ] satisfies ST JS = J , i.e.
UTJU = 0, V T JV = 0, UT JV = I.
In particular, R(U), R(V ) are isotropic.
March 2007 – p.10
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Isotropy
Def: U = R(U) is isotropic if xT Jy = 0 for all x, y ∈ U , i.e.
UTJU = 0
Symplectic matrix S = [ U V ] satisfies ST JS = J , i.e.
UTJU = 0, V T JV = 0, UT JV = I.
In particular, R(U), R(V ) are isotropic.
March 2007 – p.10
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Isotropy
Def: U = R(U) is isotropic if xT Jy = 0 for all x, y ∈ U , i.e.
UTJU = 0
Symplectic matrix S = [ U V ] satisfies ST JS = J , i.e.
UTJU = 0, V T JV = 0, UT JV = I.
In particular, R(U), R(V ) are isotropic.
March 2007 – p.10
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Isotropy
Def: U = R(U) is isotropic if xT Jy = 0 for all x, y ∈ U , i.e.
UTJU = 0
Symplectic matrix S = [ U V ] satisfies ST JS = J , i.e.
UTJU = 0, V T JV = 0, UT JV = I.
In particular, R(U), R(V ) are isotropic.
March 2007 – p.10
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Theorem: If B is skew Hamiltonian, then every Krylovsubspace
κj(B, x) = span{x,Bx,B2x, . . . , Bj−1x
}
is isotropic.
Proof: Mehrmann / W (2001)
Corollary: Every Krylov subspace method automatically
preserves skew-Hamiltonian structure.
March 2007 – p.11
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Theorem: If B is skew Hamiltonian, then every Krylovsubspace
κj(B, x) = span{x,Bx,B2x, . . . , Bj−1x
}
is isotropic.
Proof: Mehrmann / W (2001)
Corollary: Every Krylov subspace method automatically
preserves skew-Hamiltonian structure.
March 2007 – p.11
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Theorem: If B is skew Hamiltonian, then every Krylovsubspace
κj(B, x) = span{x,Bx,B2x, . . . , Bj−1x
}
is isotropic.
Proof: Mehrmann / W (2001)
Corollary: Every Krylov subspace method automatically
preserves skew-Hamiltonian structure.
March 2007 – p.11
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Skew-Hamiltonian Lanczos Process
uj+1bjdj = Buj − ujajdj − uj−1bj−1dj
wj+1dj+1bj = BT wj − wjdjaj − wj−1dj−1bj−1
UTj JUj = 0, W T
j JWj = 0, UTj Wj = I
Let vk = −Jwk (So Wj = JVj)
(JB)T = −JB ⇒ −JBT = −BJ
March 2007 – p.12
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Skew-Hamiltonian Lanczos Processuj+1bjdj = Buj − ujajdj − uj−1bj−1dj
wj+1dj+1bj = BT wj − wjdjaj − wj−1dj−1bj−1
UTj JUj = 0, W T
j JWj = 0, UTj Wj = I
Let vk = −Jwk (So Wj = JVj)
(JB)T = −JB ⇒ −JBT = −BJ
March 2007 – p.12
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Skew-Hamiltonian Lanczos Processuj+1bjdj = Buj − ujajdj − uj−1bj−1dj
wj+1dj+1bj = BT wj − wjdjaj − wj−1dj−1bj−1
UTj JUj = 0, W T
j JWj = 0, UTj Wj = I
Let vk = −Jwk (So Wj = JVj)
(JB)T = −JB ⇒ −JBT = −BJ
March 2007 – p.12
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Skew-Hamiltonian Lanczos Processuj+1bjdj = Buj − ujajdj − uj−1bj−1dj
wj+1dj+1bj = BT wj − wjdjaj − wj−1dj−1bj−1
UTj JUj = 0, W T
j JWj = 0, UTj Wj = I
Let vk = −Jwk (So Wj = JVj)
(JB)T = −JB ⇒ −JBT = −BJ
March 2007 – p.12
![Page 66: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/66.jpg)
Skew-Hamiltonian Lanczos Processuj+1bjdj = Buj − ujajdj − uj−1bj−1dj
wj+1dj+1bj = BT wj − wjdjaj − wj−1dj−1bj−1
UTj JUj = 0, W T
j JWj = 0, UTj Wj = I
Let vk = −Jwk (So Wj = JVj)
(JB)T = −JB ⇒ −JBT = −BJ
March 2007 – p.12
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Skew-Hamiltonian Lanczos Processuj+1bjdj = Buj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = Bvj − vjdjaj − vj−1dj−1bj−1
Start with uT1 Jv1 = 1.
UTj JUj = 0, V T
j JVj = 0, UTj JVj = I
These are the columns of a symplectic matrix.
March 2007 – p.13
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Skew-Hamiltonian Lanczos Processuj+1bjdj = Buj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = Bvj − vjdjaj − vj−1dj−1bj−1
Start with uT1 Jv1 = 1.
UTj JUj = 0, V T
j JVj = 0, UTj JVj = I
These are the columns of a symplectic matrix.
March 2007 – p.13
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Skew-Hamiltonian Lanczos Processuj+1bjdj = Buj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = Bvj − vjdjaj − vj−1dj−1bj−1
Start with uT1 Jv1 = 1.
UTj JUj = 0, V T
j JVj = 0, UTj JVj = I
These are the columns of a symplectic matrix.
March 2007 – p.13
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Skew-Hamiltonian Lanczos Processuj+1bjdj = Buj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = Bvj − vjdjaj − vj−1dj−1bj−1
Start with uT1 Jv1 = 1.
UTj JUj = 0, V T
j JVj = 0, UTj JVj = I
These are the columns of a symplectic matrix.
March 2007 – p.13
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Hamiltonian Lanczos Process
H Hamiltonian ⇒ H2 skew Hamiltonian.
Apply skew-Hamiltonian Lanczos process to H2.
uj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
Let vj = Hujdj .
Multiply first equation by H and by dj .
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
March 2007 – p.14
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Hamiltonian Lanczos ProcessH Hamiltonian ⇒ H2 skew Hamiltonian.
Apply skew-Hamiltonian Lanczos process to H2.
uj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
Let vj = Hujdj .
Multiply first equation by H and by dj .
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
March 2007 – p.14
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Hamiltonian Lanczos ProcessH Hamiltonian ⇒ H2 skew Hamiltonian.
Apply skew-Hamiltonian Lanczos process to H2.
uj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
Let vj = Hujdj .
Multiply first equation by H and by dj .
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
March 2007 – p.14
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Hamiltonian Lanczos ProcessH Hamiltonian ⇒ H2 skew Hamiltonian.
Apply skew-Hamiltonian Lanczos process to H2.
uj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
Let vj = Hujdj .
Multiply first equation by H and by dj .
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
March 2007 – p.14
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Hamiltonian Lanczos ProcessH Hamiltonian ⇒ H2 skew Hamiltonian.
Apply skew-Hamiltonian Lanczos process to H2.
uj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
Let vj = Hujdj .
Multiply first equation by H and by dj .
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
March 2007 – p.14
![Page 76: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/76.jpg)
Hamiltonian Lanczos ProcessH Hamiltonian ⇒ H2 skew Hamiltonian.
Apply skew-Hamiltonian Lanczos process to H2.
uj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
Let vj = Hujdj .
Multiply first equation by H and by dj .
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
March 2007 – p.14
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Hamiltonian Lanczos ProcessH Hamiltonian ⇒ H2 skew Hamiltonian.
Apply skew-Hamiltonian Lanczos process to H2.
uj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
Let vj = Hujdj .
Multiply first equation by H and by dj .
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
March 2007 – p.14
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Hamiltonian Lanczos Processuj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
Start the process with v1 = v1 = Hu1d1.
Then vj = vj for all j.
Conclusion: The second recurrence is redundant.
Just run the first recurrence together with thesupplementary condition
vjdj = Huj .
March 2007 – p.15
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Hamiltonian Lanczos Processuj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
Start the process with v1 = v1 = Hu1d1.
Then vj = vj for all j.
Conclusion: The second recurrence is redundant.
Just run the first recurrence together with thesupplementary condition
vjdj = Huj .
March 2007 – p.15
![Page 80: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/80.jpg)
Hamiltonian Lanczos Processuj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
Start the process with v1 = v1 = Hu1d1.
Then vj = vj for all j.
Conclusion: The second recurrence is redundant.
Just run the first recurrence together with thesupplementary condition
vjdj = Huj .
March 2007 – p.15
![Page 81: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/81.jpg)
Hamiltonian Lanczos Processuj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
Start the process with v1 = v1 = Hu1d1.
Then vj = vj for all j.
Conclusion:
The second recurrence is redundant.
Just run the first recurrence together with thesupplementary condition
vjdj = Huj .
March 2007 – p.15
![Page 82: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/82.jpg)
Hamiltonian Lanczos Processuj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
Start the process with v1 = v1 = Hu1d1.
Then vj = vj for all j.
Conclusion: The second recurrence is redundant.
Just run the first recurrence together with thesupplementary condition
vjdj = Huj .
March 2007 – p.15
![Page 83: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/83.jpg)
Hamiltonian Lanczos Processuj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
Start the process with v1 = v1 = Hu1d1.
Then vj = vj for all j.
Conclusion: The second recurrence is redundant.
Just run the first recurrence
together with thesupplementary condition
vjdj = Huj .
March 2007 – p.15
![Page 84: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/84.jpg)
Hamiltonian Lanczos Processuj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = H2vj − vjdjaj − vj−1dj−1bj−1
Start the process with v1 = v1 = Hu1d1.
Then vj = vj for all j.
Conclusion: The second recurrence is redundant.
Just run the first recurrence together with thesupplementary condition
vjdj = Huj .
March 2007 – p.15
![Page 85: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/85.jpg)
Hamiltonian Lanczos Processuj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1 = Huj+1
uj+1bj = Hvj − ujaj − uj−1bj−1
vj+1dj+1 = Huj+1
Start with v1d1 = Hu1 uT1 Jv1 = 1.
This is easy to arrange.
March 2007 – p.16
![Page 86: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/86.jpg)
Hamiltonian Lanczos Processuj+1bjdj = H2uj − ujajdj − uj−1bj−1dj
vj+1dj+1 = Huj+1
uj+1bj = Hvj − ujaj − uj−1bj−1
vj+1dj+1 = Huj+1
Start with v1d1 = Hu1 uT1 Jv1 = 1.
This is easy to arrange.
March 2007 – p.16
![Page 87: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/87.jpg)
Hamiltonian Lanczos Processuj+1bjdj = Hvjdj − ujajdj − uj−1bj−1dj
vj+1dj+1 = Huj+1
uj+1bj = Hvj − ujaj − uj−1bj−1
vj+1dj+1 = Huj+1
Start with v1d1 = Hu1 uT1 Jv1 = 1.
This is easy to arrange.
March 2007 – p.16
![Page 88: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/88.jpg)
Hamiltonian Lanczos Processuj+1bjdj = Hvjdj − ujajdj − uj−1bj−1dj
vj+1dj+1 = Huj+1
uj+1bj = Hvj − ujaj − uj−1bj−1
vj+1dj+1 = Huj+1
Start with v1d1 = Hu1 uT1 Jv1 = 1.
This is easy to arrange.
March 2007 – p.16
![Page 89: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/89.jpg)
Hamiltonian Lanczos Processuj+1bj = Hvj − ujaj − uj−1bj−1
vj+1dj+1 = Huj+1
Start with v1d1 = Hu1 uT1 Jv1 = 1.
This is easy to arrange.
March 2007 – p.16
![Page 90: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/90.jpg)
Hamiltonian Lanczos Processuj+1bj = Hvj − ujaj − uj−1bj−1
vj+1dj+1 = Huj+1
Start with v1d1 = Hu1 uT1 Jv1 = 1.
This is easy to arrange.
March 2007 – p.16
![Page 91: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/91.jpg)
Hamiltonian Lanczos Processuj+1bj = Hvj − ujaj − uj−1bj−1
vj+1dj+1 = Huj+1
Start with v1d1 = Hu1 uT1 Jv1 = 1.
This is easy to arrange.
March 2007 – p.16
![Page 92: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/92.jpg)
Recurrences written as matrixproducts
uj+1bj = Hvj − ujaj − uj−1bj−1
vj+1dj+1 = Huj+1
H[
Uj Vj
]=
[Uj Vj
] [Tj
Dj
]+ uj+1bje
T2j .
Implicit restarts: Filter with the HR algorithm.
Next up: symplectic Lanczos process.
March 2007 – p.17
![Page 93: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/93.jpg)
Recurrences written as matrixproducts
uj+1bj = Hvj − ujaj − uj−1bj−1
vj+1dj+1 = Huj+1
HVj = UjTj + uj+1bjeTj , HUj = VjDj
H[
Uj Vj
]=
[Uj Vj
] [Tj
Dj
]+ uj+1bje
T2j .
Implicit restarts: Filter with the HR algorithm.
Next up: symplectic Lanczos process.
March 2007 – p.17
![Page 94: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/94.jpg)
Recurrences written as matrixproducts
uj+1bj = Hvj − ujaj − uj−1bj−1
vj+1dj+1 = Huj+1
H[
Uj Vj
]=
[Uj Vj
] [Tj
Dj
]+ uj+1bje
T2j .
Implicit restarts: Filter with the HR algorithm.
Next up: symplectic Lanczos process.
March 2007 – p.17
![Page 95: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/95.jpg)
Recurrences written as matrixproducts
uj+1bj = Hvj − ujaj − uj−1bj−1
vj+1dj+1 = Huj+1
H[
Uj Vj
]=
[Uj Vj
] [Tj
Dj
]+ uj+1bje
T2j .
Implicit restarts:
Filter with the HR algorithm.
Next up: symplectic Lanczos process.
March 2007 – p.17
![Page 96: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/96.jpg)
Recurrences written as matrixproducts
uj+1bj = Hvj − ujaj − uj−1bj−1
vj+1dj+1 = Huj+1
H[
Uj Vj
]=
[Uj Vj
] [Tj
Dj
]+ uj+1bje
T2j .
Implicit restarts: Filter with the HR algorithm.
Next up: symplectic Lanczos process.
March 2007 – p.17
![Page 97: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/97.jpg)
Recurrences written as matrixproducts
uj+1bj = Hvj − ujaj − uj−1bj−1
vj+1dj+1 = Huj+1
H[
Uj Vj
]=
[Uj Vj
] [Tj
Dj
]+ uj+1bje
T2j .
Implicit restarts: Filter with the HR algorithm.
Next up: symplectic Lanczos process.
March 2007 – p.17
![Page 98: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/98.jpg)
Symplectic Lanczos Process
S symplectic ⇒ S + S−1 skew Hamiltonian.
Apply skew-Hamiltonian Lanczos process to S + S−1.
uj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
Let vj = S−1ujdj .
Multiply first equation by S−1 and by dj.
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
March 2007 – p.18
![Page 99: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/99.jpg)
Symplectic Lanczos ProcessS symplectic ⇒ S + S−1 skew Hamiltonian.
Apply skew-Hamiltonian Lanczos process to S + S−1.
uj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
Let vj = S−1ujdj .
Multiply first equation by S−1 and by dj.
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
March 2007 – p.18
![Page 100: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/100.jpg)
Symplectic Lanczos ProcessS symplectic ⇒ S + S−1 skew Hamiltonian.
Apply skew-Hamiltonian Lanczos process to S + S−1.
uj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
Let vj = S−1ujdj .
Multiply first equation by S−1 and by dj.
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
March 2007 – p.18
![Page 101: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/101.jpg)
Symplectic Lanczos ProcessS symplectic ⇒ S + S−1 skew Hamiltonian.
Apply skew-Hamiltonian Lanczos process to S + S−1.
uj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
Let vj = S−1ujdj .
Multiply first equation by S−1 and by dj.
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
March 2007 – p.18
![Page 102: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/102.jpg)
Symplectic Lanczos ProcessS symplectic ⇒ S + S−1 skew Hamiltonian.
Apply skew-Hamiltonian Lanczos process to S + S−1.
uj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
Let vj = S−1ujdj .
Multiply first equation by S−1 and by dj.
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
March 2007 – p.18
![Page 103: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/103.jpg)
Symplectic Lanczos ProcessS symplectic ⇒ S + S−1 skew Hamiltonian.
Apply skew-Hamiltonian Lanczos process to S + S−1.
uj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
Let vj = S−1ujdj .
Multiply first equation by S−1 and by dj.
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
March 2007 – p.18
![Page 104: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/104.jpg)
Symplectic Lanczos ProcessS symplectic ⇒ S + S−1 skew Hamiltonian.
Apply skew-Hamiltonian Lanczos process to S + S−1.
uj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
Let vj = S−1ujdj .
Multiply first equation by S−1 and by dj.
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
March 2007 – p.18
![Page 105: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/105.jpg)
Symplectic Lanczos Processuj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
Start the process with v1 = v1 = S−1u1d1.
Then vj = vj for all j.
Conclusion: The second recurrence is redundant.
Just run the first recurrence together with thesupplementary condition
vjdj = S−1uj .
March 2007 – p.19
![Page 106: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/106.jpg)
Symplectic Lanczos Processuj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
Start the process with v1 = v1 = S−1u1d1.
Then vj = vj for all j.
Conclusion: The second recurrence is redundant.
Just run the first recurrence together with thesupplementary condition
vjdj = S−1uj .
March 2007 – p.19
![Page 107: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/107.jpg)
Symplectic Lanczos Processuj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
Start the process with v1 = v1 = S−1u1d1.
Then vj = vj for all j.
Conclusion: The second recurrence is redundant.
Just run the first recurrence together with thesupplementary condition
vjdj = S−1uj .
March 2007 – p.19
![Page 108: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/108.jpg)
Symplectic Lanczos Processuj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
Start the process with v1 = v1 = S−1u1d1.
Then vj = vj for all j.
Conclusion:
The second recurrence is redundant.
Just run the first recurrence together with thesupplementary condition
vjdj = S−1uj .
March 2007 – p.19
![Page 109: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/109.jpg)
Symplectic Lanczos Processuj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
Start the process with v1 = v1 = S−1u1d1.
Then vj = vj for all j.
Conclusion: The second recurrence is redundant.
Just run the first recurrence together with thesupplementary condition
vjdj = S−1uj .
March 2007 – p.19
![Page 110: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/110.jpg)
Symplectic Lanczos Processuj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
Start the process with v1 = v1 = S−1u1d1.
Then vj = vj for all j.
Conclusion: The second recurrence is redundant.
Just run the first recurrence
together with thesupplementary condition
vjdj = S−1uj .
March 2007 – p.19
![Page 111: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/111.jpg)
Symplectic Lanczos Processuj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1bj = (S + S−1)vj − vjdjaj − vj−1dj−1bj−1
Start the process with v1 = v1 = S−1u1d1.
Then vj = vj for all j.
Conclusion: The second recurrence is redundant.
Just run the first recurrence together with thesupplementary condition
vjdj = S−1uj .
March 2007 – p.19
![Page 112: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/112.jpg)
Symplectic Lanczos Processuj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1 = S−1uj+1
uj+1bj = Sujdj + vj − ujaj − uj−1bj−1
vj+1dj+1 = S−1uj+1
Start with v1d1 = S−1u1 uT1 Jv1 = 1.
This is easy to arrange.
March 2007 – p.20
![Page 113: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/113.jpg)
Symplectic Lanczos Processuj+1bjdj = (S + S−1)uj − ujajdj − uj−1bj−1dj
vj+1dj+1 = S−1uj+1
uj+1bj = Sujdj + vj − ujaj − uj−1bj−1
vj+1dj+1 = S−1uj+1
Start with v1d1 = S−1u1 uT1 Jv1 = 1.
This is easy to arrange.
March 2007 – p.20
![Page 114: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/114.jpg)
Symplectic Lanczos Processuj+1bjdj = Suj + vjdj − ujajdj − uj−1bj−1dj
vj+1dj+1 = S−1uj+1
uj+1bj = Sujdj + vj − ujaj − uj−1bj−1
vj+1dj+1 = S−1uj+1
Start with v1d1 = S−1u1 uT1 Jv1 = 1.
This is easy to arrange.
March 2007 – p.20
![Page 115: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/115.jpg)
Symplectic Lanczos Processuj+1bjdj = Suj + vjdj − ujajdj − uj−1bj−1dj
vj+1dj+1 = S−1uj+1
uj+1bj = Sujdj + vj − ujaj − uj−1bj−1
vj+1dj+1 = S−1uj+1
Start with v1d1 = S−1u1 uT1 Jv1 = 1.
This is easy to arrange.
March 2007 – p.20
![Page 116: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/116.jpg)
Symplectic Lanczos Processuj+1bj = Sujdj + vj − ujaj − uj−1bj−1
vj+1dj+1 = S−1uj+1
Start with v1d1 = S−1u1 uT1 Jv1 = 1.
This is easy to arrange.
March 2007 – p.20
![Page 117: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/117.jpg)
Symplectic Lanczos Processuj+1bj = Sujdj + vj − ujaj − uj−1bj−1
vj+1dj+1 = S−1uj+1
Start with v1d1 = S−1u1 uT1 Jv1 = 1.
This is easy to arrange.
March 2007 – p.20
![Page 118: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/118.jpg)
Symplectic Lanczos Processuj+1bj = Sujdj + vj − ujaj − uj−1bj−1
vj+1dj+1 = S−1uj+1
Start with v1d1 = S−1u1 uT1 Jv1 = 1.
This is easy to arrange.
March 2007 – p.20
![Page 119: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/119.jpg)
Recurrences written as matrixproducts
uj+1bj = Sujdj + vj − ujaj − uj−1bj−1
vj+1dj+1 = S−1uj+1
S[
Uj Vj
]=
[Uj Vj
] [TjDj Dj
−Dj 0
]+ uj+1bj+1dj+1e
Tj
Implicit restarts: Filter with the HR algorithm.
That’s it!
March 2007 – p.21
![Page 120: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/120.jpg)
Recurrences written as matrixproducts
uj+1bj = Sujdj + vj − ujaj − uj−1bj−1
vj+1dj+1 = S−1uj+1
SUj = Uj(TjDj) + uj+1bj+1dj+1eTj − VjDj , SVj = UjDj
S[
Uj Vj
]=
[Uj Vj
] [TjDj Dj
−Dj 0
]+ uj+1bj+1dj+1e
Tj
Implicit restarts: Filter with the HR algorithm.
That’s it!
March 2007 – p.21
![Page 121: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/121.jpg)
Recurrences written as matrixproducts
uj+1bj = Sujdj + vj − ujaj − uj−1bj−1
vj+1dj+1 = S−1uj+1
S[
Uj Vj
]=
[Uj Vj
] [TjDj Dj
−Dj 0
]+ uj+1bj+1dj+1e
Tj
Implicit restarts: Filter with the HR algorithm.
That’s it!
March 2007 – p.21
![Page 122: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/122.jpg)
Recurrences written as matrixproducts
uj+1bj = Sujdj + vj − ujaj − uj−1bj−1
vj+1dj+1 = S−1uj+1
S[
Uj Vj
]=
[Uj Vj
] [TjDj Dj
−Dj 0
]+ uj+1bj+1dj+1e
Tj
Implicit restarts:
Filter with the HR algorithm.
That’s it!
March 2007 – p.21
![Page 123: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/123.jpg)
Recurrences written as matrixproducts
uj+1bj = Sujdj + vj − ujaj − uj−1bj−1
vj+1dj+1 = S−1uj+1
S[
Uj Vj
]=
[Uj Vj
] [TjDj Dj
−Dj 0
]+ uj+1bj+1dj+1e
Tj
Implicit restarts: Filter with the HR algorithm.
That’s it!
March 2007 – p.21
![Page 124: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/124.jpg)
Conclusions
Since Krylov subspaces associated withskew-Hamiltonian matrices are isotropic, . . .
. . . the unsymmetric Lanczos process automaticallypreserves skew-Hamiltonian structure.
From the skew-Hamiltonian Lanczos process we easilyderive . . .
a Hamiltonian Lanczos process using H2.
a symplectic Lanczos process using S + S−1.
These algorithms are not new, but the derivations are.
Thank you for your attention.
March 2007 – p.22
![Page 125: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/125.jpg)
ConclusionsSince Krylov subspaces associated withskew-Hamiltonian matrices are isotropic, . . .
. . . the unsymmetric Lanczos process automaticallypreserves skew-Hamiltonian structure.
From the skew-Hamiltonian Lanczos process we easilyderive . . .
a Hamiltonian Lanczos process using H2.
a symplectic Lanczos process using S + S−1.
These algorithms are not new, but the derivations are.
Thank you for your attention.
March 2007 – p.22
![Page 126: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/126.jpg)
ConclusionsSince Krylov subspaces associated withskew-Hamiltonian matrices are isotropic, . . .
. . . the unsymmetric Lanczos process automaticallypreserves skew-Hamiltonian structure.
From the skew-Hamiltonian Lanczos process we easilyderive . . .
a Hamiltonian Lanczos process using H2.
a symplectic Lanczos process using S + S−1.
These algorithms are not new, but the derivations are.
Thank you for your attention.
March 2007 – p.22
![Page 127: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/127.jpg)
ConclusionsSince Krylov subspaces associated withskew-Hamiltonian matrices are isotropic, . . .
. . . the unsymmetric Lanczos process automaticallypreserves skew-Hamiltonian structure.
From the skew-Hamiltonian Lanczos process we easilyderive . . .
a Hamiltonian Lanczos process using H2.
a symplectic Lanczos process using S + S−1.
These algorithms are not new, but the derivations are.
Thank you for your attention.
March 2007 – p.22
![Page 128: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/128.jpg)
ConclusionsSince Krylov subspaces associated withskew-Hamiltonian matrices are isotropic, . . .
. . . the unsymmetric Lanczos process automaticallypreserves skew-Hamiltonian structure.
From the skew-Hamiltonian Lanczos process we easilyderive . . .
a Hamiltonian Lanczos process
using H2.
a symplectic Lanczos process using S + S−1.
These algorithms are not new, but the derivations are.
Thank you for your attention.
March 2007 – p.22
![Page 129: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/129.jpg)
ConclusionsSince Krylov subspaces associated withskew-Hamiltonian matrices are isotropic, . . .
. . . the unsymmetric Lanczos process automaticallypreserves skew-Hamiltonian structure.
From the skew-Hamiltonian Lanczos process we easilyderive . . .
a Hamiltonian Lanczos process using H2.
a symplectic Lanczos process using S + S−1.
These algorithms are not new, but the derivations are.
Thank you for your attention.
March 2007 – p.22
![Page 130: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/130.jpg)
ConclusionsSince Krylov subspaces associated withskew-Hamiltonian matrices are isotropic, . . .
. . . the unsymmetric Lanczos process automaticallypreserves skew-Hamiltonian structure.
From the skew-Hamiltonian Lanczos process we easilyderive . . .
a Hamiltonian Lanczos process using H2.
a symplectic Lanczos process
using S + S−1.
These algorithms are not new, but the derivations are.
Thank you for your attention.
March 2007 – p.22
![Page 131: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/131.jpg)
ConclusionsSince Krylov subspaces associated withskew-Hamiltonian matrices are isotropic, . . .
. . . the unsymmetric Lanczos process automaticallypreserves skew-Hamiltonian structure.
From the skew-Hamiltonian Lanczos process we easilyderive . . .
a Hamiltonian Lanczos process using H2.
a symplectic Lanczos process using S + S−1.
These algorithms are not new, but the derivations are.
Thank you for your attention.
March 2007 – p.22
![Page 132: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/132.jpg)
ConclusionsSince Krylov subspaces associated withskew-Hamiltonian matrices are isotropic, . . .
. . . the unsymmetric Lanczos process automaticallypreserves skew-Hamiltonian structure.
From the skew-Hamiltonian Lanczos process we easilyderive . . .
a Hamiltonian Lanczos process using H2.
a symplectic Lanczos process using S + S−1.
These algorithms are not new,
but the derivations are.
Thank you for your attention.
March 2007 – p.22
![Page 133: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/133.jpg)
ConclusionsSince Krylov subspaces associated withskew-Hamiltonian matrices are isotropic, . . .
. . . the unsymmetric Lanczos process automaticallypreserves skew-Hamiltonian structure.
From the skew-Hamiltonian Lanczos process we easilyderive . . .
a Hamiltonian Lanczos process using H2.
a symplectic Lanczos process using S + S−1.
These algorithms are not new, but the derivations are.
Thank you for your attention.
March 2007 – p.22
![Page 134: Structure-preserving Krylov subspace methods for ...Structure-preserving Krylov subspace methods for Hamiltonian and symplectic eigenvalue problems David S. Watkins watkins@math.wsu.edu](https://reader035.vdocuments.us/reader035/viewer/2022062603/5f039c387e708231d409e6d1/html5/thumbnails/134.jpg)
ConclusionsSince Krylov subspaces associated withskew-Hamiltonian matrices are isotropic, . . .
. . . the unsymmetric Lanczos process automaticallypreserves skew-Hamiltonian structure.
From the skew-Hamiltonian Lanczos process we easilyderive . . .
a Hamiltonian Lanczos process using H2.
a symplectic Lanczos process using S + S−1.
These algorithms are not new, but the derivations are.
Thank you for your attention.
March 2007 – p.22