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Dealing with Hamiltonian Structure: Challenges and Successes David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Cortona, September 2008 p. 1 Alternating Pencils Cortona, September 2008 p. 2

  1. 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

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

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

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

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

  6. 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

  7. 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

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

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

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

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

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

  13. 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

  14. 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

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

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

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

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

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

  20. 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

  21. 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

  22. 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

  23. Symplectic URV Decomposition Cortona, September 2008 – p. 14

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

  25. 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

  26. 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

  27. 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

  28. 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

  29. CLM Method Cortona, September 2008 – p. 15

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

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

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

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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

  38. Block CLM Method Cortona, September 2008 – p. 16

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

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

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

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

  43. 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

  44. 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

  45. 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

  46. 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

  47. 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

  48. 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

  49. 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

  50. Cortona, September 2008 – p. 17

  51. Cortona, September 2008 – p. 17

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