Franciss Algorithm David S. Watkins watkins@math.wsu.edu - PowerPoint PPT Presentation

Summary of Francis Iteration Pick some shifts. Compute p ( A ) e 1 . ( p determined by shifts) Build Q 0 with first column q 1 = αp ( A ) e 1 . ( A → Q − 1 0 AQ 0 ) Make a bulge. Chase the bulge. (return to Hessenberg form) Francis’s Algorithm – p. 13

Summary of Francis Iteration Pick some shifts. Compute p ( A ) e 1 . ( p determined by shifts) Build Q 0 with first column q 1 = αp ( A ) e 1 . ( A → Q − 1 0 AQ 0 ) Make a bulge. Chase the bulge. (return to Hessenberg form) ˆ A = Q − 1 AQ Francis’s Algorithm – p. 13

Quicker Summary Francis’s Algorithm – p. 14

Quicker Summary Make a bulge. Francis’s Algorithm – p. 14

Quicker Summary Make a bulge. Chase it. Francis’s Algorithm – p. 14

Remarks Francis’s Algorithm – p. 15

Remarks This is pretty simple. Francis’s Algorithm – p. 15

Remarks This is pretty simple. no QR decomposition in sight! Francis’s Algorithm – p. 15

Remarks This is pretty simple. no QR decomposition in sight! Why call it the QR algorithm? Francis’s Algorithm – p. 15

Remarks This is pretty simple. no QR decomposition in sight! Why call it the QR algorithm? Confusion! Francis’s Algorithm – p. 15

Remarks This is pretty simple. no QR decomposition in sight! Why call it the QR algorithm? Confusion! Can we think of another name? Francis’s Algorithm – p. 15

Remarks This is pretty simple. no QR decomposition in sight! Why call it the QR algorithm? Confusion! Can we think of another name? I’m calling it Francis’s Algorithm. Francis’s Algorithm – p. 15

Remarks This is pretty simple. no QR decomposition in sight! Why call it the QR algorithm? Confusion! Can we think of another name? I’m calling it Francis’s Algorithm. This is not a radical move. Francis’s Algorithm – p. 15

Question Francis’s Algorithm – p. 16

Question How should we view Francis’s algorithm? Francis’s Algorithm – p. 16

Question How should we view Francis’s algorithm? Do we have to start with the basic QR algorithm? Francis’s Algorithm – p. 16

Question How should we view Francis’s algorithm? Do we have to start with the basic QR algorithm? Couldn’t we just as well introduce Francis’s algorithm directly? Francis’s Algorithm – p. 16

Question How should we view Francis’s algorithm? Do we have to start with the basic QR algorithm? Couldn’t we just as well introduce Francis’s algorithm directly? ...bypassing the basic QR algorithm entirely? Francis’s Algorithm – p. 16

Question How should we view Francis’s algorithm? Do we have to start with the basic QR algorithm? Couldn’t we just as well introduce Francis’s algorithm directly? ...bypassing the basic QR algorithm entirely? ...and the answer is: Francis’s Algorithm – p. 16

Question How should we view Francis’s algorithm? Do we have to start with the basic QR algorithm? Couldn’t we just as well introduce Francis’s algorithm directly? ...bypassing the basic QR algorithm entirely? ...and the answer is: Why not? Francis’s Algorithm – p. 16

Question How should we view Francis’s algorithm? Do we have to start with the basic QR algorithm? Couldn’t we just as well introduce Francis’s algorithm directly? ...bypassing the basic QR algorithm entirely? ...and the answer is: Why not? This simplifies the presentation. Francis’s Algorithm – p. 16

Question How should we view Francis’s algorithm? Do we have to start with the basic QR algorithm? Couldn’t we just as well introduce Francis’s algorithm directly? ...bypassing the basic QR algorithm entirely? ...and the answer is: Why not? This simplifies the presentation. I’m putting my money where my mouth is. Francis’s Algorithm – p. 16

Francis’s Algorithm – p. 17

I’m putting my money where my mouth is ... Francis’s Algorithm – p. 17

I’m putting my money where my mouth is ... ...and saving one entire section! Francis’s Algorithm – p. 17

Pedagogical Pathway Francis’s Algorithm – p. 18

Pedagogical Pathway reduction to Hessenberg form Francis’s Algorithm – p. 18

Pedagogical Pathway reduction to Hessenberg form Francis’s algorithm Francis’s Algorithm – p. 18

Pedagogical Pathway reduction to Hessenberg form Francis’s algorithm Try it out! Francis’s Algorithm – p. 18

Pedagogical Pathway reduction to Hessenberg form Francis’s algorithm Try it out! It works great! Francis’s Algorithm – p. 18

Pedagogical Pathway reduction to Hessenberg form Francis’s algorithm Try it out! It works great! Why does it work? Francis’s Algorithm – p. 18

Ingredients of Francis’s Algorithm Francis’s Algorithm – p. 19

Ingredients of Francis’s Algorithm subspace iteration (power method) Francis’s Algorithm – p. 19

Ingredients of Francis’s Algorithm subspace iteration (power method) subspace iteration with changes of coordinate system Francis’s Algorithm – p. 19

Ingredients of Francis’s Algorithm subspace iteration (power method) subspace iteration with changes of coordinate system Krylov subspaces Francis’s Algorithm – p. 19

Ingredients of Francis’s Algorithm subspace iteration (power method) subspace iteration with changes of coordinate system Krylov subspaces (instead of the implicit- Q theorem) Francis’s Algorithm – p. 19

Ingredients of Francis’s Algorithm subspace iteration (power method) subspace iteration with changes of coordinate system Krylov subspaces (instead of the implicit- Q theorem) Krylov subspaces and subspace iteration Francis’s Algorithm – p. 19

Ingredients of Francis’s Algorithm subspace iteration (power method) subspace iteration with changes of coordinate system Krylov subspaces (instead of the implicit- Q theorem) Krylov subspaces and subspace iteration Krylov subspaces and Hessenberg form Francis’s Algorithm – p. 19

Power Method, Subspace Iteration Francis’s Algorithm – p. 20

Power Method, Subspace Iteration v , Av , A 2 v , A 3 v , ... Francis’s Algorithm – p. 20

Power Method, Subspace Iteration v , Av , A 2 v , A 3 v , ... convergence rate | λ 2 /λ 1 | Francis’s Algorithm – p. 20