Dealing with Hamiltonian Structure: Challenges and Successes David - PowerPoint PPT Presentation

Working Directly with the Pencil Hamiltonian ⇔ alternating pencil M − λN symplectic ⇔ palindromic pencil G − λG T Schröder (Ph.D. 2008) Kressner/Schröder/Watkins (2008) Cortona, September 2008 – p. 10

Working with Hamiltonian Matrices Cortona, September 2008 – p. 11

Working with Hamiltonian Matrices symplectic matrix: S T JS = J Cortona, September 2008 – p. 11

Working with Hamiltonian Matrices symplectic matrix: S T JS = J symplectic similarity transformations Cortona, September 2008 – p. 11

Working with Hamiltonian Matrices symplectic matrix: S T JS = J symplectic similarity transformations orthogonal symplectic transformations Cortona, September 2008 – p. 11

Working with Hamiltonian Matrices symplectic matrix: S T JS = J symplectic similarity transformations orthogonal symplectic transformations isotropic subspace : U T JU = 0 Cortona, September 2008 – p. 11

Working with Hamiltonian Matrices symplectic matrix: S T JS = J symplectic similarity transformations orthogonal symplectic transformations isotropic subspace : U T JU = 0 isotropy and symplectic matrices Cortona, September 2008 – p. 11

Difficulty Obtaining Hessenberg Form Cortona, September 2008 – p. 12

Difficulty Obtaining Hessenberg Form PVL form (1981) Cortona, September 2008 – p. 12

Difficulty Obtaining Hessenberg Form PVL form (1981) the desired Hessenberg form Cortona, September 2008 – p. 12

Difficulty Obtaining Hessenberg Form PVL form (1981) the desired Hessenberg form Byers (1983) Cortona, September 2008 – p. 12

Difficulty Obtaining Hessenberg Form PVL form (1981) the desired Hessenberg form Byers (1983) getting an isotropic Krylov subspace? Cortona, September 2008 – p. 12

Difficulty Obtaining Hessenberg Form PVL form (1981) the desired Hessenberg form Byers (1983) getting an isotropic Krylov subspace? Ammar/Mehrmann (1991) Cortona, September 2008 – p. 12

Difficulty Obtaining Hessenberg Form PVL form (1981) the desired Hessenberg form Byers (1983) getting an isotropic Krylov subspace? Ammar/Mehrmann (1991) new ideas needed Cortona, September 2008 – p. 12

Skew-Hamiltonian matrices . . . Cortona, September 2008 – p. 13

Skew-Hamiltonian matrices . . . . . . are easier Cortona, September 2008 – p. 13

Skew-Hamiltonian matrices . . . . . . are easier ( JK ) T = − ( JK ) skew-Hamiltonian matrix : Cortona, September 2008 – p. 13

Skew-Hamiltonian matrices . . . . . . are easier ( JK ) T = − ( JK ) skew-Hamiltonian matrix : H 2 Cortona, September 2008 – p. 13

Skew-Hamiltonian matrices . . . . . . are easier ( JK ) T = − ( JK ) skew-Hamiltonian matrix : H 2 more and bigger invariant subspaces Cortona, September 2008 – p. 13

Skew-Hamiltonian matrices . . . . . . are easier ( JK ) T = − ( JK ) skew-Hamiltonian matrix : H 2 more and bigger invariant subspaces Krylov subspaces are automatically isotropic. Cortona, September 2008 – p. 13

Skew-Hamiltonian matrices . . . . . . are easier ( JK ) T = − ( JK ) skew-Hamiltonian matrix : H 2 more and bigger invariant subspaces Krylov subspaces are automatically isotropic. reduction to Hessenberg form Cortona, September 2008 – p. 13

Skew-Hamiltonian matrices . . . . . . are easier ( JK ) T = − ( JK ) skew-Hamiltonian matrix : H 2 more and bigger invariant subspaces Krylov subspaces are automatically isotropic. reduction to Hessenberg form make use of H 2 Cortona, September 2008 – p. 13

Symplectic URV Decomposition Cortona, September 2008 – p. 14

Symplectic URV Decomposition H = UR 1 V T = V R 2 U T Cortona, September 2008 – p. 14

Symplectic URV Decomposition H = UR 1 V T = V R 2 U T � − T � S B T � � B R 1 = and R 2 = 0 T T − S T 0 Cortona, September 2008 – p. 14

Symplectic URV Decomposition H = UR 1 V T = V R 2 U T � − T � S B T � � B R 1 = and R 2 = 0 T T − S T 0 H 2 Cortona, September 2008 – p. 14

Symplectic URV Decomposition H = UR 1 V T = V R 2 U T � − T � S B T � � B R 1 = and R 2 = 0 T T − S T 0 H 2 eigenvalues of H Cortona, September 2008 – p. 14

Symplectic URV Decomposition H = UR 1 V T = V R 2 U T � − T � S B T � � B R 1 = and R 2 = 0 T T − S T 0 H 2 eigenvalues of H Benner/Mehrmann/Xu (199X) Cortona, September 2008 – p. 14

CLM Method Cortona, September 2008 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) Cortona, September 2008 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) H ← U T HU Cortona, September 2008 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) H ← U T HU H 2 has special structure. Cortona, September 2008 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) H ← U T HU H 2 has special structure. span { e 1 } invariant under H 2 Cortona, September 2008 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) H ← U T HU H 2 has special structure. span { e 1 } invariant under H 2 ⇒ span { e 1 , He 1 } invariant under H Cortona, September 2008 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) H ← U T HU H 2 has special structure. span { e 1 } invariant under H 2 ⇒ span { e 1 , He 1 } invariant under H Extract 1-D isotropic invariant subspace. Cortona, September 2008 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) H ← U T HU H 2 has special structure. span { e 1 } invariant under H 2 ⇒ span { e 1 , He 1 } invariant under H Extract 1-D isotropic invariant subspace. Build an orthogonal symplectic similarity transformation. Cortona, September 2008 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) H ← U T HU H 2 has special structure. span { e 1 } invariant under H 2 ⇒ span { e 1 , He 1 } invariant under H Extract 1-D isotropic invariant subspace. Build an orthogonal symplectic similarity transformation. Deflate. (many details skipped) Cortona, September 2008 – p. 15

Block CLM Method Cortona, September 2008 – p. 16

Block CLM Method CLM works surprisingly well. Cortona, September 2008 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Cortona, September 2008 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Cortona, September 2008 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) Cortona, September 2008 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S invariant under H 2 Cortona, September 2008 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S invariant under H 2 ⇒ span {S , H S} invariant under H Cortona, September 2008 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S invariant under H 2 ⇒ span {S , H S} invariant under H Extract k -dimensional isotropic invariant subspace. Cortona, September 2008 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S invariant under H 2 ⇒ span {S , H S} invariant under H Extract k -dimensional isotropic invariant subspace. This is Cortona, September 2008 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S invariant under H 2 ⇒ span {S , H S} invariant under H Extract k -dimensional isotropic invariant subspace. This is more robust, Cortona, September 2008 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S invariant under H 2 ⇒ span {S , H S} invariant under H Extract k -dimensional isotropic invariant subspace. This is more robust, more efficient, Cortona, September 2008 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S invariant under H 2 ⇒ span {S , H S} invariant under H Extract k -dimensional isotropic invariant subspace. This is more robust, more efficient, but we’re still working on it. Cortona, September 2008 – p. 16

Cortona, September 2008 – p. 17

Cortona, September 2008 – p. 17