Neutrally stable unreduced fixed points under QR iteration

1
Neutrally stable unreduced
fixed points under QR iteration

• Foundations of Computational Mathematics
• 8 August 2002
• David M. Day
• Sandia National Laboratory

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2
Outline

• QR Iteration for algebraic eigenvalue problems
• Shifts Generalized Raleigh quotient, Exceptional
• Fixed Points
• Differentiability
• Neutral Stability
• New Shift

3
QR iteration

• HQR convergence problems seldom arise
• Non-convergence due to unachievable convergence
  criteria du Croz,
• Non-convergence at fixed points Batterson and
  Smillie 1989, Day unpublished
• Domain unreduced Hessenberg matrices
• Iteration Orthogonal similarity transformation

4
Shifts

• Basic (unshifted) QR Not competitive
• Raleigh quotient shift real
• Double implicit generalized Raleigh quotient
• HQR Every 10th iteration uses an exceptional
  shift to break symmetry
• Ex Wilkinson and Reinsch, EISPACK, LAPACK all
  use different exceptional shifts
• HQR neutrally stable fixed points
  with either Ex shift

5
Fixed Points

• B QR, B Q Q B
• Sequence of shifts p(B) QR, shift polynomial
  p
• Fixed points p(B) Orthogonal scalar
• Extension of Parlett 1966 to degree(p) gt 1
• Fixed points not detectible
• Implicit shifts R not computed
• Efficiency iteration is in place
• Theory characterize p() and B p(B) in O(n)?

6
Differentiability

• Near reduced matrices flag manifold structure
• Away from reduced matrices Euclidian structure

7
Neutral Stability

• Reduced map orthogonality similarity class
• Basic QR Both the osc map and the matrix map
  have spectrum on the boundary of the unit disc at
  any unreduced fixed point
• Center manifold Dynamics depend on Riemannian
  metric
• HQR Nonconvergence Need both osc map and the
  matrix map to have spectrum on the boundary of
  the unit disc (Experimental observation)

8
HQR Work

• Coordinates on OSC Intrinsic
• Criterion of Ammar Mehrmann 91 on symplectic
  orthogonal Schur decompositions
• 4 by 4 matrices dim monic p() p(B) O
  scalar 1.
• Potential Extensions Extrinsic coords, use
  existence proofs of symp orth Schur decomp by
  Mehl, Lin, and many others.

9
New Shift

• Goal Make fixed points observable
• Batterson 1999 Generalized Raleigh quotient
  shifts min norm last row p(B)
• New max(norm first column/norm last row)
• Near Target New converges to Old
• Fixed points submatrices NW SE (positive
  subdiagonal) implies not a fixed points

10
Conclusions

• QR iteration not globally convergent
• Neutrally stable fixed points recurrent problem
• Exceptional shifts do not eliminate all fixed
  points
• Fixed points hard to detect
• New shift fixed points observable