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

Cost David S. Watkins Core-Chasing Algorithms

Cost Most arithmetic in passing-through operation David S. Watkins Core-Chasing Algorithms

Cost Most arithmetic in passing-through operation O ( n 2 ) flops per iteration . . . O ( n 3 ) total flops . . . about the same as for standard Francis iteration. David S. Watkins Core-Chasing Algorithms

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

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

Are there any advantages? unitary case companion case (unitary-plus-rank-one) 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

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

Unitary Case � � � � � A = QR = � � � � � Cost is O ( n ) flops per iteration, David S. Watkins Core-Chasing Algorithms

Unitary Case � � � � � A = QR = � � � � � Cost is O ( n ) flops per iteration, O ( n 2 ) flops total. David S. Watkins Core-Chasing Algorithms

Unitary Case � � � � � A = QR = � � � � � Cost is O ( n ) flops per iteration, O ( n 2 ) flops total. Storage requirement is O ( n ). David S. Watkins Core-Chasing Algorithms

Unitary Case � � � � � A = QR = � � � � � Cost is O ( n ) flops per iteration, O ( n 2 ) flops total. Storage requirement is O ( n ). Gragg (1986) David S. Watkins Core-Chasing Algorithms

Unitary Case � � � � � A = QR = � � � � � Cost is O ( n ) flops per iteration, O ( n 2 ) flops total. Storage requirement is O ( n ). Gragg (1986) Ammar, Reichel, M. Stewart, Bunse-Gerstner, Elsner, He, W, . . . David S. Watkins Core-Chasing Algorithms

Companion Case David S. Watkins Core-Chasing Algorithms

Companion Case p ( x ) = x n + a n − 1 x n − 1 + a n − 2 x n − 2 + · · · + a 0 = 0 monic polynomial David S. Watkins Core-Chasing Algorithms

Companion Case p ( x ) = x n + a n − 1 x n − 1 + a n − 2 x n − 2 + · · · + a 0 = 0 monic polynomial companion matrix   0 · · · 0 − a 0 1 0 · · · 0 − a 1    . .  ... . .   A = 1 . .    ...   0 − a n − 2    1 − a n − 1 . . . get the zeros of p by computing the eigenvalues. David S. Watkins Core-Chasing Algorithms

Companion Case p ( x ) = x n + a n − 1 x n − 1 + a n − 2 x n − 2 + · · · + a 0 = 0 monic polynomial companion matrix   0 · · · 0 − a 0 1 0 · · · 0 − a 1    . .  ... . .   A = 1 . .    ...   0 − a n − 2    1 − a n − 1 . . . get the zeros of p by computing the eigenvalues. MATLAB’s roots command David S. Watkins Core-Chasing Algorithms

Cost of solving companion eigenvalue problem David S. Watkins Core-Chasing Algorithms

Cost of solving companion eigenvalue problem If structure not exploited: O ( n 2 ) storage, O ( n 3 ) flops Francis’s algorithm David S. Watkins Core-Chasing Algorithms

Cost of solving companion eigenvalue problem If structure not exploited: O ( n 2 ) storage, O ( n 3 ) flops Francis’s algorithm If structure exploited: O ( n 2 ) flops O ( n ) storage, several methods proposed data-sparse representation + Francis’s algorithm David S. Watkins Core-Chasing Algorithms

Representation of R We store the QR decomposed form. � � � � � A = QR = � � � � � David S. Watkins Core-Chasing Algorithms

Representation of R We store the QR decomposed form. � � � � � A = QR = � � � � � where   1 0 · · · − a 1 1 − a 2   R = .  ...  .  .   .  − a 0 David S. Watkins Core-Chasing Algorithms

Representation of R We store the QR decomposed form. � � � � � A = QR = � � � � � where   1 0 · · · − a 1 1 − a 2   R = .  ...  .  .   .  − a 0 This is unitary-plus-rank-one. David S. Watkins Core-Chasing Algorithms

Representation of R We store the QR decomposed form. � � � � � A = QR = � � � � � where   1 0 · · · − a 1 1 − a 2   R = .  ...  .  .   .  − a 0 This is unitary-plus-rank-one. How do we store it? David S. Watkins Core-Chasing Algorithms

Representation of R Jared covered this yesterday. David S. Watkins Core-Chasing Algorithms

Representation of R Jared covered this yesterday. Add a row and column to R . David S. Watkins Core-Chasing Algorithms

Representation of R Jared covered this yesterday. Add a row and column to R .   1 0 · · · − a 1 0 1 − a 2 0   . .  ...  ˜ . . R =  . .  .     − a 0 1   0 0 · · · 0 0 David S. Watkins Core-Chasing Algorithms

Representation of R Jared covered this yesterday. Add a row and column to R .   1 0 · · · − a 1 0 1 − a 2 0   . .  ...  ˜ . . R =  . .  .     − a 0 1   0 0 · · · 0 0 This is still unitary-plus-rank-one. David S. Watkins Core-Chasing Algorithms

Representation of R ˜ R = David S. Watkins Core-Chasing Algorithms

Representation of R ˜ 1 ( B 1 · · · B n + e 1 y T ) R = C ∗ n · · · C ∗ David S. Watkins Core-Chasing Algorithms

Representation of R ˜ 1 ( B 1 · · · B n + e 1 y T ) R = C ∗ n · · · C ∗ � � � � � � � � � � + · · · � � � � � � � � � � � � � � David S. Watkins Core-Chasing Algorithms

Representation of R ˜ 1 ( B 1 · · · B n + e 1 y T ) R = C ∗ n · · · C ∗ � � � � � � � � � � + · · · � � � � � � � � � � � � � � . . . and we don’t have to store the rank-one part! David S. Watkins Core-Chasing Algorithms

Representation of R ˜ 1 ( B 1 · · · B n + e 1 y T ) R = C ∗ n · · · C ∗ � � � � � � � � � � + · · · � � � � � � � � � � � � � � . . . and we don’t have to store the rank-one part! Storage is O ( n ). David S. Watkins Core-Chasing Algorithms

Passing a core transformation through R � � � � David S. Watkins Core-Chasing Algorithms

Passing a core transformation through R � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � C ∗ n · · · C ∗ B 1 · · · B n 1 David S. Watkins Core-Chasing Algorithms

Passing a core transformation through R � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � C ∗ n · · · C ∗ B 1 · · · B n 1 Cost: O ( n ) flops per iteration, O ( n 2 ) flops total. David S. Watkins Core-Chasing Algorithms

The rank-one part Recovering y T from the core transformations David S. Watkins Core-Chasing Algorithms