Recent Progress in the Solution of the Hamiltonian Eigenvalue - PowerPoint PPT Presentation

Working with Hamiltonian Matrices symplectic matrix: S T JS = J symplectic similarity transformations Calais, May 2009 – p. 11

Working with Hamiltonian Matrices symplectic matrix: S T JS = J symplectic similarity transformations orthogonal symplectic transformations Calais, May 2009 – p. 11

Working with Hamiltonian Matrices symplectic matrix: S T JS = J symplectic similarity transformations orthogonal symplectic transformations isotropic subspace : U T JU = 0 Calais, May 2009 – 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 Calais, May 2009 – p. 11

Difficulty Obtaining Hessenberg Form Calais, May 2009 – p. 12

Difficulty Obtaining Hessenberg Form PVL form (1981) Calais, May 2009 – p. 12

Difficulty Obtaining Hessenberg Form PVL form (1981) the desired Hessenberg form Calais, May 2009 – p. 12

Difficulty Obtaining Hessenberg Form PVL form (1981) the desired Hessenberg form Byers’ Hamiltonian QR (1983) Calais, May 2009 – p. 12

Difficulty Obtaining Hessenberg Form PVL form (1981) the desired Hessenberg form Byers’ Hamiltonian QR (1983) Van Loan’s Curse Calais, May 2009 – p. 12

Difficulty Obtaining Hessenberg Form PVL form (1981) the desired Hessenberg form Byers’ Hamiltonian QR (1983) Van Loan’s Curse getting an isotropic Krylov subspace? Calais, May 2009 – p. 12

Difficulty Obtaining Hessenberg Form PVL form (1981) the desired Hessenberg form Byers’ Hamiltonian QR (1983) Van Loan’s Curse getting an isotropic Krylov subspace? Ammar/Mehrmann (1991) Calais, May 2009 – p. 12

Difficulty Obtaining Hessenberg Form PVL form (1981) the desired Hessenberg form Byers’ Hamiltonian QR (1983) Van Loan’s Curse getting an isotropic Krylov subspace? Ammar/Mehrmann (1991) new ideas needed Calais, May 2009 – p. 12

Skew-Hamiltonian matrices . . . Calais, May 2009 – p. 13

Skew-Hamiltonian matrices . . . . . . are easier Calais, May 2009 – p. 13

Skew-Hamiltonian matrices . . . . . . are easier ( JK ) T = − ( JK ) skew-Hamiltonian matrix : Calais, May 2009 – p. 13

Skew-Hamiltonian matrices . . . . . . are easier ( JK ) T = − ( JK ) skew-Hamiltonian matrix : H Hamiltonian ⇒ H 2 skew Hamiltonian Calais, May 2009 – p. 13

Skew-Hamiltonian matrices . . . . . . are easier ( JK ) T = − ( JK ) skew-Hamiltonian matrix : H Hamiltonian ⇒ H 2 skew Hamiltonian more and bigger invariant subspaces Calais, May 2009 – p. 13

Skew-Hamiltonian matrices . . . . . . are easier ( JK ) T = − ( JK ) skew-Hamiltonian matrix : H Hamiltonian ⇒ H 2 skew Hamiltonian more and bigger invariant subspaces Krylov subspaces are automatically isotropic. Calais, May 2009 – p. 13

Skew-Hamiltonian matrices . . . . . . are easier ( JK ) T = − ( JK ) skew-Hamiltonian matrix : H Hamiltonian ⇒ H 2 skew Hamiltonian more and bigger invariant subspaces Krylov subspaces are automatically isotropic. reduction to Hessenberg form Calais, May 2009 – p. 13

Skew-Hamiltonian matrices . . . . . . are easier ( JK ) T = − ( JK ) skew-Hamiltonian matrix : H Hamiltonian ⇒ H 2 skew Hamiltonian more and bigger invariant subspaces Krylov subspaces are automatically isotropic. reduction to Hessenberg form make use of H 2 Calais, May 2009 – p. 13

Symplectic URV Decomposition Calais, May 2009 – p. 14

Symplectic URV Decomposition H = UR 1 V T = V R 2 U T Calais, May 2009 – 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 Calais, May 2009 – 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 = U ( R 1 R 2 ) U T Calais, May 2009 – 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 = U ( R 1 R 2 ) U T � − ST � Z R 1 R 2 = − ( ST ) T 0 Calais, May 2009 – 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 = U ( R 1 R 2 ) U T � − ST � Z R 1 R 2 = − ( ST ) T 0 eigenvalues of H Calais, May 2009 – 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 = U ( R 1 R 2 ) U T � − ST � Z R 1 R 2 = − ( ST ) T 0 eigenvalues of H Benner/Mehrmann/Xu (1998) Calais, May 2009 – p. 14

CLM Method Calais, May 2009 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) Calais, May 2009 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) H ← U T HU Calais, May 2009 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) H ← U T HU � − ST � Z H 2 = − ( ST ) T 0 Calais, May 2009 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) H ← U T HU � − ST � Z H 2 = − ( ST ) T 0 span { e 1 } invariant under H 2 Calais, May 2009 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) H ← U T HU � − ST � Z H 2 = − ( ST ) T 0 span { e 1 } invariant under H 2 ⇒ span { e 1 , He 1 } invariant under H Calais, May 2009 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) H ← U T HU � − ST � Z H 2 = − ( ST ) T 0 span { e 1 } invariant under H 2 ⇒ span { e 1 , He 1 } invariant under H Extract 1-d isotropic invariant subspace. Calais, May 2009 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) H ← U T HU � − ST � Z H 2 = − ( ST ) T 0 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. Calais, May 2009 – p. 15

CLM Method Chu/Liu/Mehrmann (2004) H ← U T HU � − ST � Z H 2 = − ( ST ) T 0 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. (with form of H 2 preserved!) Deflate. Calais, May 2009 – p. 15

Block CLM Method Calais, May 2009 – p. 16

Block CLM Method CLM works surprisingly well. Calais, May 2009 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Calais, May 2009 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Calais, May 2009 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) Calais, May 2009 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k ) invariant under H 2 Calais, May 2009 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k ) invariant under H 2 ⇒ span {S , H S} invariant under H Calais, May 2009 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k ) invariant under H 2 ⇒ span {S , H S} invariant under H Extract k -dimensional isotropic invariant subspace. Calais, May 2009 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k ) invariant under H 2 ⇒ span {S , H S} invariant under H Extract k -dimensional isotropic invariant subspace. Build a similarity transformation that deflates 2 k eigenvalues. Calais, May 2009 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k ) invariant under H 2 ⇒ span {S , H S} invariant under H Extract k -dimensional isotropic invariant subspace. Build a similarity transformation that deflates 2 k eigenvalues. This is Calais, May 2009 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k ) invariant under H 2 ⇒ span {S , H S} invariant under H Extract k -dimensional isotropic invariant subspace. Build a similarity transformation that deflates 2 k eigenvalues. This is more robust, Calais, May 2009 – p. 16

Block CLM Method CLM works surprisingly well. difficulties with clusters Block CLM, Mehrmann/Schröder/Watkins (2008) S (of dimension k ) invariant under H 2 ⇒ span {S , H S} invariant under H Extract k -dimensional isotropic invariant subspace. Build a similarity transformation that deflates 2 k eigenvalues. This is more robust, more efficient, Calais, May 2009 – p. 16