Core-Chasing Algorithms for the Eigenvalue Problem David S. Watkins - PowerPoint PPT Presentation

Core Chasing � � � � � � � � � � � � David S. Watkins Core-Chasing Algorithms

Core Chasing � � � � � � � � � � � � � � � � � � David S. Watkins Core-Chasing Algorithms

Core Chasing � � � � � � � � � � � � David S. Watkins Core-Chasing Algorithms

Core Chasing � � � � � � � � � � � � � � � � � � David S. Watkins Core-Chasing Algorithms

Core Chasing � � � � � � � � � � � � David S. Watkins Core-Chasing Algorithms

Core Chasing � � � � � � � � � � � � � � � � David S. Watkins Core-Chasing Algorithms

Core Chasing � � � � � � � � � � � � David S. Watkins Core-Chasing Algorithms

Core Chasing � � � � � � � � � � David S. Watkins Core-Chasing Algorithms

Flop Count Cost David S. Watkins Core-Chasing Algorithms

Flop Count Cost O ( n 3 ) total flops O ( n 2 ) storage about the same as for standard Francis iteration. David S. Watkins Core-Chasing Algorithms

Advantages Are there any advantages? David S. Watkins Core-Chasing Algorithms

Advantages Are there any advantages? superior deflation procedure David S. Watkins Core-Chasing Algorithms

Advantages Are there any advantages? superior deflation procedure some structured cases David S. Watkins Core-Chasing Algorithms

Deflation Standard deflation criterion: × × × × × David S. Watkins Core-Chasing Algorithms

Deflation Standard deflation criterion: × × × × × Set a j +1 , j to zero if | a j +1 , j | < u ( | a j , j | + | a j +1 , j +1 | ) . ( u is unit roundoff .) David S. Watkins Core-Chasing Algorithms

Deflation Our deflation criterion: � � � � � � � � � �  I  − s j c j   Q j =   s j c j   I David S. Watkins Core-Chasing Algorithms

Deflation Our deflation criterion: � � � � � � � � � �  I  − s j c j   Q j =   s j c j   I Set s j to zero if | s j | < u . ( u is unit roundoff .) David S. Watkins Core-Chasing Algorithms

Deflation Both criteria are normwise backward stable. David S. Watkins Core-Chasing Algorithms

Deflation Both criteria are normwise backward stable. How does this affect eigenvalues? David S. Watkins Core-Chasing Algorithms

Deflation Both criteria are normwise backward stable. How does this affect eigenvalues? Change in λ depends on condition number κ ( λ ). David S. Watkins Core-Chasing Algorithms

Deflation Both criteria are normwise backward stable. How does this affect eigenvalues? Change in λ depends on condition number κ ( λ ). Standard result: λ is perturbed to µ , where | λ − µ | ≤ u κ ( λ ) � A � + O ( u 2 ) . David S. Watkins Core-Chasing Algorithms

Deflation Both criteria are normwise backward stable. How does this affect eigenvalues? Change in λ depends on condition number κ ( λ ). Standard result: λ is perturbed to µ , where | λ − µ | ≤ u κ ( λ ) � A � + O ( u 2 ) . This holds for both deflation criteria. David S. Watkins Core-Chasing Algorithms

Deflation But our criterion does better: David S. Watkins Core-Chasing Algorithms

Deflation But our criterion does better: Theorem (Mach and Vandebril (2014) ) | λ − µ | ≤ u κ ( λ ) | λ | + O ( u 2 ) . David S. Watkins Core-Chasing Algorithms

Deflation But our criterion does better: Theorem (Mach and Vandebril (2014) ) | λ − µ | ≤ u κ ( λ ) | λ | + O ( u 2 ) . Relative perturbation in each λ is tiny. David S. Watkins Core-Chasing Algorithms

Deflation But our criterion does better: Theorem (Mach and Vandebril (2014) ) | λ − µ | ≤ u κ ( λ ) | λ | + O ( u 2 ) . Relative perturbation in each λ is tiny. This does not hold for standard deflation criterion. David S. Watkins Core-Chasing Algorithms

Deflation Fun Example: David S. Watkins Core-Chasing Algorithms

Deflation Fun Example: � 1 � 2 A = (0 < ǫ < u ) ǫ ǫ David S. Watkins Core-Chasing Algorithms

Deflation Fun Example: � 1 � 2 A = (0 < ǫ < u ) ǫ ǫ λ 1 = 1 + 2 ǫ + O ( ǫ 2 ) λ 2 = − ǫ + O ( ǫ 2 ) David S. Watkins Core-Chasing Algorithms

Deflation Fun Example: � 1 � 2 A = (0 < ǫ < u ) ǫ ǫ λ 1 = 1 + 2 ǫ + O ( ǫ 2 ) λ 2 = − ǫ + O ( ǫ 2 ) These eigenvalues are well conditioned. David S. Watkins Core-Chasing Algorithms

Deflation Fun Example: � 1 � 2 A = (0 < ǫ < u ) ǫ ǫ λ 1 = 1 + 2 ǫ + O ( ǫ 2 ) λ 2 = − ǫ + O ( ǫ 2 ) These eigenvalues are well conditioned. Standard criterion deflates to � 1 � 2 . 0 ǫ David S. Watkins Core-Chasing Algorithms

Deflation Fun Example: � 1 � 2 A = (0 < ǫ < u ) ǫ ǫ λ 1 = 1 + 2 ǫ + O ( ǫ 2 ) λ 2 = − ǫ + O ( ǫ 2 ) These eigenvalues are well conditioned. Standard criterion deflates to � 1 � 2 . 0 ǫ Eigenvalues are µ 1 = 1 and µ 2 = ǫ . David S. Watkins Core-Chasing Algorithms

Deflation Fun Example: � 1 � 2 A = (0 < ǫ < u ) ǫ ǫ λ 1 = 1 + 2 ǫ + O ( ǫ 2 ) λ 2 = − ǫ + O ( ǫ 2 ) These eigenvalues are well conditioned. Standard criterion deflates to � 1 � 2 . 0 ǫ Eigenvalues are µ 1 = 1 and µ 2 = ǫ . Small eigenvalue is off by 200%. David S. Watkins Core-Chasing Algorithms

Deflation Example, continued: David S. Watkins Core-Chasing Algorithms

Deflation Example, continued: � 1 � 2 A = (0 < ǫ < u ) ǫ ǫ λ 1 = 1 + 2 ǫ + O ( ǫ 2 ) λ 2 = − ǫ + O ( ǫ 2 ) David S. Watkins Core-Chasing Algorithms

Deflation Example, continued: � 1 � 2 A = (0 < ǫ < u ) ǫ ǫ λ 1 = 1 + 2 ǫ + O ( ǫ 2 ) λ 2 = − ǫ + O ( ǫ 2 ) Our criterion: � 1 � � 1 − ǫ 2 � A = QR ≈ . 1 0 − ǫ ǫ David S. Watkins Core-Chasing Algorithms

Deflation Example, continued: � 1 � 2 A = (0 < ǫ < u ) ǫ ǫ λ 1 = 1 + 2 ǫ + O ( ǫ 2 ) λ 2 = − ǫ + O ( ǫ 2 ) Our criterion: � 1 � � 1 − ǫ 2 � A = QR ≈ . 1 0 − ǫ ǫ Deflates to � 1 � � 1 � 1 � � 0 2 2 = . 0 1 0 − ǫ 0 − ǫ David S. Watkins Core-Chasing Algorithms

Deflation Example, continued: � 1 � 2 A = (0 < ǫ < u ) ǫ ǫ λ 1 = 1 + 2 ǫ + O ( ǫ 2 ) λ 2 = − ǫ + O ( ǫ 2 ) Our criterion: � 1 � � 1 − ǫ 2 � A = QR ≈ . 1 0 − ǫ ǫ Deflates to � 1 � � 1 � 1 � � 0 2 2 = . 0 1 0 − ǫ 0 − ǫ Eigenvalues are µ 1 = 1 and µ 2 = − ǫ . David S. Watkins Core-Chasing Algorithms

Deflation Example, continued: � 1 � 2 A = (0 < ǫ < u ) ǫ ǫ λ 1 = 1 + 2 ǫ + O ( ǫ 2 ) λ 2 = − ǫ + O ( ǫ 2 ) Our criterion: � 1 � � 1 − ǫ 2 � A = QR ≈ . 1 0 − ǫ ǫ Deflates to � 1 � � 1 � 1 � � 0 2 2 = . 0 1 0 − ǫ 0 − ǫ Eigenvalues are µ 1 = 1 and µ 2 = − ǫ . Both eigenvalues are accurate. David S. Watkins Core-Chasing Algorithms

Exploitation of Structure Structures we can exploit David S. Watkins Core-Chasing Algorithms

Exploitation of Structure Structures we can exploit unitary David S. Watkins Core-Chasing Algorithms

Exploitation of Structure Structures we can exploit unitary companion matrix (unitary-plus-rank-one) David S. Watkins Core-Chasing Algorithms

Exploitation of Structure Structures we can exploit unitary companion matrix (unitary-plus-rank-one) unitary-plus-low-rank David S. Watkins Core-Chasing Algorithms

Unitary Case David S. Watkins Core-Chasing Algorithms

Unitary Case � � � � � A = QR = � � � � � David S. Watkins Core-Chasing Algorithms

Unitary Case � � � � � A = QR = � � � � � David S. Watkins Core-Chasing Algorithms