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Hamiltonian Jacobi methods: sweeping away convergence problems Christian Mehl School of Mathematics University of Birmingham United Kingdom GAMM Workshop Applied and Numerical Linear Algebra TU Hamburg-Harburg, September 11-12, 2008 Jacobi


  1. Hamiltonian Jacobi methods: sweeping away convergence problems Christian Mehl School of Mathematics University of Birmingham United Kingdom GAMM Workshop Applied and Numerical Linear Algebra TU Hamburg-Harburg, September 11-12, 2008

  2. Jacobi algorithms   ∗ · · · · · · ∗ · · · ·     · · ∗ · · ·     · · · ∗ · ·     · ∗ · · ∗ ·   · · · · · ∗ Jacobi’s algorithm for symmetric matrices: • select a pivot element in the lower triangular part;

  3. Jacobi algorithms   ∗ · · · · · · ∗ · · ◦ ·     · · ∗ · · ·     · · · ∗ · ·     · ◦ · · ∗ ·   · · · · · ∗ Jacobi’s algorithm for symmetric matrices: • select a pivot element in the lower triangular part; • diagonalize a corresponding 2 × 2 -problem with a Jacobi-rotation;

  4. Jacobi algorithms   ∗ · · · · · · ∗ · · · ·     · · ∗ · · ·     · · · ∗ · ·     · · · · ∗ ·   · · · · · ∗ Jacobi’s algorithm for symmetric matrices: • select a pivot element in the lower triangular part; • diagonalize a corresponding 2 × 2 -problem with a Jacobi-rotation; • cyclic version: do repeatedly n ( n − 1) steps, where each off-diagonal ele- 2 ment is annihilated at least once (sweep)

  5. Jacobi algorithms   ∗ · · · · · · ∗ · · · ·     �� �� · · ∗ · · ·   | a ij | 2 | a ij | 2 off ( A ) := off ( A ) :=   · · · ∗ · ·   i � = j i � = j   · · · · ∗ ·   · · · · · ∗ Jacobi’s algorithm for symmetric matrices / properties : • off ( A ) decreases monotonically; • globally convergent (...); • asymptotically quadratically convergent (...)

  6. Nonsymmetric Jacobi algorithms There are Jacobi-like algorithms for other types of eigenvalue problems: • computing the Schur form for complex matrices (Greenstadt, 1955, Eber- lein, 1962/87, Stewart 1985) • diagonalization of normal matrices (Goldstein, Hurwitz, 1959) • Hamiltonian matrices (Byers, 1986, Bunse-Gerstner Faßbender, 1997) • Generalized Schur form of regular pencils (Charlier, Van Dooren, 1989) • doubly structured matrices (Faßbender, Mackey, Mackey, 2001) • ... many more ... Drmaˇ c, Hari, Veseli´ c, ....

  7. Nonsymmetric Jacobi algorithms   ∗ ∗ ∗ ∗ ∗ ∗ · ∗ ∗ ∗ ∗ ∗     · · ∗ ∗ ∗ ∗     · · · ∗ ∗ ∗     · ∗ · · ∗ ∗   · · · · · ∗ Nonsymmetric Jacobi algorithm for general complex matrices: • select a pivot element in the lower triangular part;

  8. Nonsymmetric Jacobi algorithms   ∗ ∗ ∗ ∗ ∗ ∗ · ∗ ∗ ∗ ∗ ∗     · · ∗ ∗ ∗ ∗     · · · ∗ ∗ ∗     · ◦ · · ∗ ∗   · · · · · ∗ Nonsymmetric Jacobi algorithm for general matrices: • select a pivot element in the lower triangular part; • triangularize a corresponding 2 × 2 -problem with a Jacobi-rotation;

  9. Nonsymmetric Jacobi algorithms   ∗ ∗ ∗ ∗ ∗ ∗ · ∗ ∗ ∗ ∗ ∗     · · ∗ ∗ ∗ ∗     · · · ∗ ∗ ∗     · · · · ∗ ∗   · · · · · ∗ Nonsymmetric Jacobi algorithm for general matrices: • select a pivot element in the lower triangular part; • triangularize a corresponding 2 × 2 -problem with a Jacobi-rotation; • use a cyclic version: do repeatedly n ( n − 1) steps, where each element in 2 the lower triangular part is annihilated at least once (sweep)

  10. Nonsymmetric Jacobi algorithm: properties   ∗ ∗ ∗ ∗ ∗ ∗ · ∗ ∗ ∗ ∗ ∗     �� �� · · ∗ ∗ ∗ ∗   | a ij | 2 | a ij | 2 off ( A ) := off ( A ) :=   · · · ∗ ∗ ∗   i>j i>j   · · · · ∗ ∗   · · · · · ∗ Nonsymmetric Jacobi algorithm for general matrices / properties : • off ( A ) does NOT decrease monotonically; • global convergence in experiments (...) – proof? ; • asymptotical quadratic convergence in experiments (...) – but... ;

  11. Hamiltonian Jacobi algorithms Reminder : the Hamiltonian eigenvalue problem A matrix H ∈ R 2 n × 2 n is called Hamiltonian if � � 0 I n H T J + JH = 0 , where J = − I n 0 or, equivalently, if � A � C H = , D − A T where A, C, D are n × n and C = C T and D = D T . For Jacobi, consider the corresponding complex eigenvalue problem and re- place ( · ) T with ( · ) ∗ .

  12. Hamiltonian Jacobi algorithms Reminder : the Hamiltonian eigenvalue problem A matrix S ∈ R 2 n × 2 n is called symplectic if � � 0 I n S T JS = J, where J = . − I n 0 Similarity transformation with symplectic matrices preserve the Hamiltonian structure. In the complex case replace ( · ) T with ( · ) ∗ .

  13. Hamiltonian Jacobi algorithms Reminder : the Hamiltonian eigenvalue problem Hamiltonian Schur form : � R � C H = , 0 − R ∗ where C = C ∗ and R is upper triangular. If H ∈ C 2 n × 2 n is Hamiltonian, then there exists a unitary symplectic matrix U ∈ C 2 n × 2 n such that U ∗ HU is in Hamiltonian Schur form if H has no eigenvalues on the imaginary axis.

  14. Hamiltonian Jacobi algorithms Hamiltonian Jacobi algorithm : consider 4 × 4 subproblems instead of 2 × 2 subproblems;   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · ∗ ∗ ∗ ∗ ∗ ∗ ∗     · • ∗ ∗ ∗ ∗ ∗ ∗     · · · ∗ ∗ ∗ ∗ ∗     · · · · ∗ · · ·     · · · · ∗ ∗ · ·     · · · · ∗ ∗ ∗ ·   · · · · ∗ ∗ ∗ ∗ If a pivot element is selected...

  15. Hamiltonian Jacobi algorithms Hamiltonian Jacobi algorithm : consider 4 × 4 subproblems instead of 2 × 2 subproblems;   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · • • ∗ ∗ ∗ ∗ ∗     · • • ∗ ∗ ∗ ∗ ∗     · · · ∗ ∗ ∗ ∗ ∗     · · · · ∗ · · ·     · · · · ∗ ∗ · ·     · · · · ∗ ∗ ∗ ·   · · · · ∗ ∗ ∗ ∗ ... then the corresponding 2 × 2 subproblem is NOT Hamiltonian if the pivot element is off-diagonal.

  16. Hamiltonian Jacobi algorithms Hamiltonian Jacobi algorithm : consider 4 × 4 subproblems instead of 2 × 2 subproblems;   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · • • ∗ ∗ • • ∗     · • • ∗ ∗ • • ∗     · · · ∗ ∗ ∗ ∗ ∗     · · · · ∗ · · ·     · • • · ∗ • • ·     · • • · ∗ • • ·   · · · · ∗ ∗ ∗ ∗ The corresponding 4 × 4 subproblem is the smallest Hamiltonian sub- problem containing the off-diagonal pivot element .

  17. Hamiltonian Jacobi algorithms Hamiltonian Jacobi algorithm : consider 4 × 4 subproblems instead of 2 × 2 subproblems;   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · • • ∗ ∗ • • ∗     · ◦ • ∗ ∗ • • ∗     · · · ∗ ∗ ∗ ∗ ∗     · · · · ∗ · · ·     · ◦ ◦ · ∗ • ◦ ·     · ◦ ◦ · ∗ • • ·   · · · · ∗ ∗ ∗ ∗ Compute the Hamiltonian Schur form of the 4 × 4 subproblem and trans- form the matrix accordingly. ❀ The pivot element is annihilated.

  18. Hamiltonian Jacobi algorithms Hamiltonian Jacobi algorithm : sweep: annihilate each pivot element at least once;   • • ∗ ∗ • • ∗ ∗ ◦ • ∗ ∗ • • ∗ ∗     · · ∗ ∗ ∗ ∗ ∗ ∗     · · · ∗ ∗ ∗ ∗ ∗     ◦ ◦ · · • ◦ · ·     ◦ ◦ · · • • · ·     · · · · ∗ ∗ ∗ ·   · · · · ∗ ∗ ∗ ∗ typical row-by-column sweep

  19. Hamiltonian Jacobi algorithms Hamiltonian Jacobi algorithm : sweep: annihilate each pivot element at least once;   • ∗ • ∗ • ∗ • ∗ · ∗ ∗ ∗ ∗ ∗ ∗ ∗     ◦ ∗ • ∗ • ∗ • ∗     · · · ∗ ∗ ∗ ∗ ∗     ◦ · ◦ · • · ◦ ·     · · · · ∗ ∗ · ·     ◦ · ◦ · • · • ·   · · · · ∗ ∗ ∗ ∗ typical row-by-column sweep

  20. Hamiltonian Jacobi algorithms Hamiltonian Jacobi algorithm : sweep: annihilate each pivot element at least once;   • ∗ ∗ • • ∗ ∗ • · ∗ ∗ ∗ ∗ ∗ ∗ ∗     · · ∗ ∗ ∗ ∗ ∗ ∗     ◦ · · • • ∗ ∗ •     ◦ · · ◦ • · · ◦     · · · · ∗ ∗ · ·     · · · · ∗ ∗ ∗ ·   ◦ · · ◦ • · · • typical row-by-column sweep

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