francis's algorithm as a core-chasing algorithm · fundamentals of matrix computations, 3rd...
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![Page 1: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/1.jpg)
Francis’s Algorithm as a Core-Chasing Algorithm
David S. Watkins
Department of MathematicsWashington State University
PNWNAS, November 12, 2016
David S. Watkins Core-Chasing Algorithm
![Page 2: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/2.jpg)
Today’s Topic
The matrix eigenvalue problem
A ∈ Cn×n
Find the eigenvalues (. . . vectors, invariant subspaces)
Many applications
Interest dates back to the very beginning of the electroniccomputing era.
Nobody knew how to do it.
David S. Watkins Core-Chasing Algorithm
![Page 3: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/3.jpg)
Today’s Topic
The matrix eigenvalue problem
A ∈ Cn×n
Find the eigenvalues (. . . vectors, invariant subspaces)
Many applications
Interest dates back to the very beginning of the electroniccomputing era.
Nobody knew how to do it.
David S. Watkins Core-Chasing Algorithm
![Page 4: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/4.jpg)
Today’s Topic
The matrix eigenvalue problem
A ∈ Cn×n
Find the eigenvalues (. . . vectors, invariant subspaces)
Many applications
Interest dates back to the very beginning of the electroniccomputing era.
Nobody knew how to do it.
David S. Watkins Core-Chasing Algorithm
![Page 5: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/5.jpg)
Today’s Topic
The matrix eigenvalue problem
A ∈ Cn×n
Find the eigenvalues (. . . vectors, invariant subspaces)
Many applications
Interest dates back to the very beginning of the electroniccomputing era.
Nobody knew how to do it.
David S. Watkins Core-Chasing Algorithm
![Page 6: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/6.jpg)
Today’s Topic
The matrix eigenvalue problem
A ∈ Cn×n
Find the eigenvalues (. . . vectors, invariant subspaces)
Many applications
Interest dates back to the very beginning of the electroniccomputing era.
Nobody knew how to do it.
David S. Watkins Core-Chasing Algorithm
![Page 7: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/7.jpg)
Today’s Topic
The matrix eigenvalue problem
A ∈ Cn×n
Find the eigenvalues (. . . vectors, invariant subspaces)
Many applications
Interest dates back to the very beginning of the electroniccomputing era.
Nobody knew how to do it.
David S. Watkins Core-Chasing Algorithm
![Page 8: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/8.jpg)
Today’s Topic
The matrix eigenvalue problem
A ∈ Cn×n
Find the eigenvalues (. . . vectors, invariant subspaces)
Many applications
Interest dates back to the very beginning of the electroniccomputing era.
Nobody knew how to do it.
David S. Watkins Core-Chasing Algorithm
![Page 9: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/9.jpg)
John Francis
invented the winning algorithm in 1959.
commonly called: QR algorithm
more precisely: implicitly shifted QR algorithm
better yet: Francis’s algorithm, bulge-chasing algorithm.
David S. Watkins Core-Chasing Algorithm
![Page 10: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/10.jpg)
John Francis
invented the winning algorithm in 1959.
commonly called: QR algorithm
more precisely: implicitly shifted QR algorithm
better yet: Francis’s algorithm, bulge-chasing algorithm.
David S. Watkins Core-Chasing Algorithm
![Page 11: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/11.jpg)
John Francis
invented the winning algorithm in 1959.
commonly called: QR algorithm
more precisely: implicitly shifted QR algorithm
better yet: Francis’s algorithm, bulge-chasing algorithm.
David S. Watkins Core-Chasing Algorithm
![Page 12: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/12.jpg)
John Francis
invented the winning algorithm in 1959.
commonly called: QR algorithm
more precisely: implicitly shifted QR algorithm
better yet: Francis’s algorithm, bulge-chasing algorithm.
David S. Watkins Core-Chasing Algorithm
![Page 13: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/13.jpg)
John Francis
invented the winning algorithm in 1959.
commonly called: QR algorithm
more precisely: implicitly shifted QR algorithm
better yet: Francis’s algorithm,
bulge-chasing algorithm.
David S. Watkins Core-Chasing Algorithm
![Page 14: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/14.jpg)
John Francis
invented the winning algorithm in 1959.
commonly called: QR algorithm
more precisely: implicitly shifted QR algorithm
better yet: Francis’s algorithm, bulge-chasing algorithm.
David S. Watkins Core-Chasing Algorithm
![Page 15: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/15.jpg)
Francis’s algorithm (superficial description)
upper Hessenberg form
A =
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
unitary similarity transformation
direct method (O(n3) flops)
Francis: Iterate
Drive toward triangular form.
(Galois theory)
David S. Watkins Core-Chasing Algorithm
![Page 16: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/16.jpg)
Francis’s algorithm (superficial description)
upper Hessenberg form
A =
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
unitary similarity transformation
direct method (O(n3) flops)
Francis: Iterate
Drive toward triangular form.
(Galois theory)
David S. Watkins Core-Chasing Algorithm
![Page 17: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/17.jpg)
Francis’s algorithm (superficial description)
upper Hessenberg form
A =
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
unitary similarity transformation
direct method (O(n3) flops)
Francis: Iterate
Drive toward triangular form.
(Galois theory)
David S. Watkins Core-Chasing Algorithm
![Page 18: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/18.jpg)
Francis’s algorithm (superficial description)
upper Hessenberg form
A =
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
unitary similarity transformation
direct method (O(n3) flops)
Francis: Iterate
Drive toward triangular form.
(Galois theory)
David S. Watkins Core-Chasing Algorithm
![Page 19: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/19.jpg)
Francis’s algorithm (superficial description)
upper Hessenberg form
A =
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
unitary similarity transformation
direct method (O(n3) flops)
Francis: Iterate
Drive toward triangular form.
(Galois theory)
David S. Watkins Core-Chasing Algorithm
![Page 20: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/20.jpg)
Francis’s algorithm (superficial description)
Chasing the bulge ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
David S. Watkins Core-Chasing Algorithm
![Page 21: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/21.jpg)
Francis’s algorithm (superficial description)
Chasing the bulge ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗+ ∗ ∗ ∗ ∗
∗ ∗ ∗∗ ∗
David S. Watkins Core-Chasing Algorithm
![Page 22: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/22.jpg)
Francis’s algorithm (superficial description)
Chasing the bulge ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗+ ∗ ∗ ∗
∗ ∗
David S. Watkins Core-Chasing Algorithm
![Page 23: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/23.jpg)
Francis’s algorithm (superficial description)
Chasing the bulge ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗+ ∗ ∗
David S. Watkins Core-Chasing Algorithm
![Page 24: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/24.jpg)
Francis’s algorithm (superficial description)
Chasing the bulge ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
David S. Watkins Core-Chasing Algorithm
![Page 25: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/25.jpg)
Francis’s algorithm (superficial description)
Chasing the bulge ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
iteration complete!
repeated iterations ⇒ triangular form
This is the single-shift algorithm.
Double-shift algorithm chases a bigger bulge.
David S. Watkins Core-Chasing Algorithm
![Page 26: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/26.jpg)
Francis’s algorithm (superficial description)
Chasing the bulge ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
iteration complete!
repeated iterations ⇒ triangular form
This is the single-shift algorithm.
Double-shift algorithm chases a bigger bulge.
David S. Watkins Core-Chasing Algorithm
![Page 27: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/27.jpg)
Francis’s algorithm (superficial description)
Chasing the bulge ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
iteration complete!
repeated iterations ⇒ triangular form
This is the single-shift algorithm.
Double-shift algorithm chases a bigger bulge.
David S. Watkins Core-Chasing Algorithm
![Page 28: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/28.jpg)
Francis’s algorithm (superficial description)
Chasing the bulge ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
iteration complete!
repeated iterations ⇒ triangular form
This is the single-shift algorithm.
Double-shift algorithm chases a bigger bulge.
David S. Watkins Core-Chasing Algorithm
![Page 29: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/29.jpg)
Computational Cost
Computational Cost
O(n2) flops per iteration
O(n) total iterations
O(n3) total flops
David S. Watkins Core-Chasing Algorithm
![Page 30: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/30.jpg)
For details see . . .
Golub and Van Loan, Matrix Computations, 4th Ed.
Watkins, Fundamentals of Matrix Computations, 3rd Ed.
David S. Watkins Core-Chasing Algorithm
![Page 31: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/31.jpg)
For details see . . .
Golub and Van Loan, Matrix Computations, 4th Ed.
Watkins, Fundamentals of Matrix Computations, 3rd Ed.
David S. Watkins Core-Chasing Algorithm
![Page 32: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/32.jpg)
For details see . . .
Golub and Van Loan, Matrix Computations, 4th Ed.
Watkins, Fundamentals of Matrix Computations, 3rd Ed.
David S. Watkins Core-Chasing Algorithm
![Page 33: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/33.jpg)
For details see . . .
Golub and Van Loan, Matrix Computations, 4th Ed.
Watkins, Fundamentals of Matrix Computations, 3rd Ed.
David S. Watkins Core-Chasing Algorithm
![Page 34: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/34.jpg)
My History with this Topic
Understanding the QR algorithm, SIAM Rev., 1982
Fundamentals of Matrix Computations, Wiley, 1991
Some perspectives on the eigenvalue problem, 1993
QR-like algorithms—an overview of convergence theory andpractice, AMS proceedings, 1996
QR-like algorithms for eigenvalue problems, JCAM, 2000
The Matrix Eigenvalue Problem: GR and Krylov SubspaceMethods, SIAM, 2007
The QR algorithm revisited, SIAM Rev., 2008
Fundamentals of Matrix Computations, 3rd Ed., 2010
Francis’s Algorithm, Amer. Math. Monthly, 2011
. . . but we’re still not done!
David S. Watkins Core-Chasing Algorithm
![Page 35: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/35.jpg)
My History with this Topic
Understanding the QR algorithm, SIAM Rev., 1982
Fundamentals of Matrix Computations, Wiley, 1991
Some perspectives on the eigenvalue problem, 1993
QR-like algorithms—an overview of convergence theory andpractice, AMS proceedings, 1996
QR-like algorithms for eigenvalue problems, JCAM, 2000
The Matrix Eigenvalue Problem: GR and Krylov SubspaceMethods, SIAM, 2007
The QR algorithm revisited, SIAM Rev., 2008
Fundamentals of Matrix Computations, 3rd Ed., 2010
Francis’s Algorithm, Amer. Math. Monthly, 2011
. . . but we’re still not done!
David S. Watkins Core-Chasing Algorithm
![Page 36: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/36.jpg)
My History with this Topic
Understanding the QR algorithm, SIAM Rev., 1982
Fundamentals of Matrix Computations, Wiley, 1991
Some perspectives on the eigenvalue problem, 1993
QR-like algorithms—an overview of convergence theory andpractice, AMS proceedings, 1996
QR-like algorithms for eigenvalue problems, JCAM, 2000
The Matrix Eigenvalue Problem: GR and Krylov SubspaceMethods, SIAM, 2007
The QR algorithm revisited, SIAM Rev., 2008
Fundamentals of Matrix Computations, 3rd Ed., 2010
Francis’s Algorithm, Amer. Math. Monthly, 2011
. . . but we’re still not done!
David S. Watkins Core-Chasing Algorithm
![Page 37: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/37.jpg)
My History with this Topic
Understanding the QR algorithm, SIAM Rev., 1982
Fundamentals of Matrix Computations, Wiley, 1991
Some perspectives on the eigenvalue problem, 1993
QR-like algorithms—an overview of convergence theory andpractice, AMS proceedings, 1996
QR-like algorithms for eigenvalue problems, JCAM, 2000
The Matrix Eigenvalue Problem: GR and Krylov SubspaceMethods, SIAM, 2007
The QR algorithm revisited, SIAM Rev., 2008
Fundamentals of Matrix Computations, 3rd Ed., 2010
Francis’s Algorithm, Amer. Math. Monthly, 2011
. . . but we’re still not done!
David S. Watkins Core-Chasing Algorithm
![Page 38: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/38.jpg)
My History with this Topic
Understanding the QR algorithm, SIAM Rev., 1982
Fundamentals of Matrix Computations, Wiley, 1991
Some perspectives on the eigenvalue problem, 1993
QR-like algorithms—an overview of convergence theory andpractice, AMS proceedings, 1996
QR-like algorithms for eigenvalue problems, JCAM, 2000
The Matrix Eigenvalue Problem: GR and Krylov SubspaceMethods, SIAM, 2007
The QR algorithm revisited, SIAM Rev., 2008
Fundamentals of Matrix Computations, 3rd Ed., 2010
Francis’s Algorithm, Amer. Math. Monthly, 2011
. . . but we’re still not done!
David S. Watkins Core-Chasing Algorithm
![Page 39: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/39.jpg)
Our International Research Group
This is joint work with
Jared Aurentz (Oxford)
Thomas Mach (KU Leuven)
Raf Vandebril (KU Leuven)
David S. Watkins Core-Chasing Algorithm
![Page 40: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/40.jpg)
A new look at an old algorithm
Store in QR decomposed form
A = QR
Q is unitary, R is upper triangular
looks inefficient! but it’s not!
David S. Watkins Core-Chasing Algorithm
![Page 41: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/41.jpg)
A new look at an old algorithm
Store in QR decomposed form
A = QR
Q is unitary, R is upper triangular
looks inefficient! but it’s not!
David S. Watkins Core-Chasing Algorithm
![Page 42: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/42.jpg)
A new look at an old algorithm
Store in QR decomposed form
A = QR
Q is unitary, R is upper triangular
looks inefficient!
but it’s not!
David S. Watkins Core-Chasing Algorithm
![Page 43: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/43.jpg)
A new look at an old algorithm
Store in QR decomposed form
A = QR
Q is unitary, R is upper triangular
looks inefficient! but it’s not!
David S. Watkins Core-Chasing Algorithm
![Page 44: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/44.jpg)
A new look at an old algorithm
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
=
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
David S. Watkins Core-Chasing Algorithm
![Page 45: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/45.jpg)
A new look at an old algorithm
��∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
=
∗ ∗ ∗ ∗ ∗0 ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
David S. Watkins Core-Chasing Algorithm
![Page 46: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/46.jpg)
A new look at an old algorithm
��
��
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
=
∗ ∗ ∗ ∗ ∗0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗∗ ∗ ∗∗ ∗
David S. Watkins Core-Chasing Algorithm
![Page 47: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/47.jpg)
A new look at an old algorithm
��
�
��
�
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
=
∗ ∗ ∗ ∗ ∗0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗0 ∗ ∗∗ ∗
David S. Watkins Core-Chasing Algorithm
![Page 48: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/48.jpg)
A new look at an old algorithm
��
�
��
��
�
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
=
∗ ∗ ∗ ∗ ∗0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗0 ∗ ∗
0 ∗
David S. Watkins Core-Chasing Algorithm
![Page 49: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/49.jpg)
A new look at an old algorithm
��
�
��
��
�
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
=
∗ ∗ ∗ ∗ ∗0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗0 ∗ ∗
0 ∗
Def: Core Transformation
Now invert the core transformationsto move them to the other side.
David S. Watkins Core-Chasing Algorithm
![Page 50: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/50.jpg)
A new look at an old algorithm
��
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��
�
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
=
∗ ∗ ∗ ∗ ∗0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗0 ∗ ∗
0 ∗
Def: Core Transformation
Now invert the core transformationsto move them to the other side.
David S. Watkins Core-Chasing Algorithm
![Page 51: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/51.jpg)
A new look at an old algorithm
��
�
��
��
�
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
=
∗ ∗ ∗ ∗ ∗0 ∗ ∗ ∗ ∗
0 ∗ ∗ ∗0 ∗ ∗
0 ∗
Def: Core Transformation
Now invert the core transformationsto move them to the other side.
David S. Watkins Core-Chasing Algorithm
![Page 52: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/52.jpg)
A new look at an old algorithm
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
=
����
����
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
A = QR
Q =
����
����
Q requires only O(n) storage space.
David S. Watkins Core-Chasing Algorithm
![Page 53: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/53.jpg)
A new look at an old algorithm
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
=
����
����
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
A = QR
Q =
����
����
Q requires only O(n) storage space.
David S. Watkins Core-Chasing Algorithm
![Page 54: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/54.jpg)
A new look at an old algorithm
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
=
����
����
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
A = QR
Q =
����
����
Q requires only O(n) storage space.
David S. Watkins Core-Chasing Algorithm
![Page 55: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/55.jpg)
A new look at an old algorithm
Manipulating core transformations
Fusion� �� � ⇒ ��
Turnover (aka shift through, Givens swap, . . . )
� ���
�� ⇔
��
��� �
Passing a core transformation through a triangular matrix(cost O(n)) ∗ ∗ ∗ ∗∗ ∗ ∗
∗ ∗∗
�� ⇔
∗ ∗ ∗ ∗∗ ∗ ∗+ ∗ ∗
∗
⇔ ��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 56: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/56.jpg)
A new look at an old algorithm
Manipulating core transformations
Fusion� �� � ⇒ ��
Turnover (aka shift through, Givens swap, . . . )
� ���
�� ⇔
��
��� �
Passing a core transformation through a triangular matrix(cost O(n)) ∗ ∗ ∗ ∗∗ ∗ ∗
∗ ∗∗
�� ⇔
∗ ∗ ∗ ∗∗ ∗ ∗+ ∗ ∗
∗
⇔ ��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 57: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/57.jpg)
A new look at an old algorithm
Manipulating core transformations
Fusion� �� � ⇒ ��
Turnover (aka shift through, Givens swap, . . . )
� ���
�� ⇔
��
��� �
Passing a core transformation through a triangular matrix(cost O(n)) ∗ ∗ ∗ ∗∗ ∗ ∗
∗ ∗∗
�� ⇔
∗ ∗ ∗ ∗∗ ∗ ∗+ ∗ ∗
∗
⇔ ��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 58: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/58.jpg)
A new look at an old algorithm
Manipulating core transformations
Fusion� �� � ⇒ ��
Turnover (aka shift through, Givens swap, . . . )
� ���
�� ⇔
��
��� �
Passing a core transformation through a triangular matrix(cost O(n)) ∗ ∗ ∗ ∗∗ ∗ ∗
∗ ∗∗
�� ⇔
∗ ∗ ∗ ∗∗ ∗ ∗+ ∗ ∗
∗
⇔ ��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 59: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/59.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
����
��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 60: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/60.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
����
��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 61: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/61.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
� �� ���
��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
��
David S. Watkins Core-Chasing Algorithm
![Page 62: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/62.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
� �� ���
��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
��
David S. Watkins Core-Chasing Algorithm
![Page 63: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/63.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
����
��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
��
David S. Watkins Core-Chasing Algorithm
![Page 64: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/64.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
� ���
��
��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 65: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/65.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
� ���
��
��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 66: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/66.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
��
��� �
��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 67: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/67.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
��
��� �
��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 68: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/68.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
����
��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
��
David S. Watkins Core-Chasing Algorithm
![Page 69: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/69.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
��� ��
��
�
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 70: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/70.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
��� ��
��
�
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 71: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/71.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
���
��
�� �
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 72: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/72.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
���
��
�� �
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 73: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/73.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
����
��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
��
David S. Watkins Core-Chasing Algorithm
![Page 74: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/74.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
����
� �� �
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 75: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/75.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
����
� �� �
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 76: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/76.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
����
��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 77: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/77.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
����
��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 78: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/78.jpg)
A new look at an old algorithm
Francis’s algorithm on the QR decomposed form(a core chasing algorithm)
����
��
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
Done!
David S. Watkins Core-Chasing Algorithm
![Page 79: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/79.jpg)
A new look at an old algorithm
Cost
Most arithmetic in passing-through operation
O(n2) flops per iteration . . .
O(n3) total flops . . .
about the same as for standard Francis iteration.
David S. Watkins Core-Chasing Algorithm
![Page 80: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/80.jpg)
A new look at an old algorithm
Cost
Most arithmetic in passing-through operation
O(n2) flops per iteration . . .
O(n3) total flops . . .
about the same as for standard Francis iteration.
David S. Watkins Core-Chasing Algorithm
![Page 81: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/81.jpg)
A new look at an old algorithm
Cost
Most arithmetic in passing-through operation
O(n2) flops per iteration . . .
O(n3) total flops . . .
about the same as for standard Francis iteration.
David S. Watkins Core-Chasing Algorithm
![Page 82: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/82.jpg)
A new look at an old algorithm
Are there any advantages?
unitary case
companion case (unitary-plus-rank-one)
general case: efficient cache use
David S. Watkins Core-Chasing Algorithm
![Page 83: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/83.jpg)
A new look at an old algorithm
Are there any advantages?
unitary case
companion case (unitary-plus-rank-one)
general case: efficient cache use
David S. Watkins Core-Chasing Algorithm
![Page 84: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/84.jpg)
A new look at an old algorithm
Are there any advantages?
unitary case
companion case (unitary-plus-rank-one)
general case: efficient cache use
David S. Watkins Core-Chasing Algorithm
![Page 85: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/85.jpg)
A new look at an old algorithm
Are there any advantages?
unitary case
companion case (unitary-plus-rank-one)
general case: efficient cache use
David S. Watkins Core-Chasing Algorithm
![Page 86: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/86.jpg)
Unitary Case
A = QR =��
����
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 87: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/87.jpg)
Unitary Case
A = QR =��
����
∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗
David S. Watkins Core-Chasing Algorithm
![Page 88: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/88.jpg)
Unitary Case
A = QR =��
����
David S. Watkins Core-Chasing Algorithm
![Page 89: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/89.jpg)
Unitary Case
A = QR =��
����
Cost is O(n) flops per iteration, O(n2) flops total.
Storage requirement is O(n).
Gragg (1986)
Ammar, Reichel, M. Stewart, Bunse-Gerstner, Elsner, He, W,. . .
David S. Watkins Core-Chasing Algorithm
![Page 90: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/90.jpg)
Unitary Case
A = QR =��
����
Cost is O(n) flops per iteration,
O(n2) flops total.
Storage requirement is O(n).
Gragg (1986)
Ammar, Reichel, M. Stewart, Bunse-Gerstner, Elsner, He, W,. . .
David S. Watkins Core-Chasing Algorithm
![Page 91: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/91.jpg)
Unitary Case
A = QR =��
����
Cost is O(n) flops per iteration, O(n2) flops total.
Storage requirement is O(n).
Gragg (1986)
Ammar, Reichel, M. Stewart, Bunse-Gerstner, Elsner, He, W,. . .
David S. Watkins Core-Chasing Algorithm
![Page 92: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/92.jpg)
Unitary Case
A = QR =��
����
Cost is O(n) flops per iteration, O(n2) flops total.
Storage requirement is O(n).
Gragg (1986)
Ammar, Reichel, M. Stewart, Bunse-Gerstner, Elsner, He, W,. . .
David S. Watkins Core-Chasing Algorithm
![Page 93: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/93.jpg)
Unitary Case
A = QR =��
����
Cost is O(n) flops per iteration, O(n2) flops total.
Storage requirement is O(n).
Gragg (1986)
Ammar, Reichel, M. Stewart, Bunse-Gerstner, Elsner, He, W,. . .
David S. Watkins Core-Chasing Algorithm
![Page 94: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/94.jpg)
Unitary Case
A = QR =��
����
Cost is O(n) flops per iteration, O(n2) flops total.
Storage requirement is O(n).
Gragg (1986)
Ammar, Reichel, M. Stewart, Bunse-Gerstner, Elsner, He, W,. . .
David S. Watkins Core-Chasing Algorithm
![Page 95: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/95.jpg)
Companion Case
p(x) = xn + an−1xn−1 + an−2x
n−2 + · · ·+ a0 = 0
monic polynomial
companion matrix
A =
0 · · · 0 −a01 0 · · · 0 −a1
1. . .
......
. . . 0 −an−2
1 −an−1
. . . get the zeros of p by computing the eigenvalues.
MATLAB’s roots command
David S. Watkins Core-Chasing Algorithm
![Page 96: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/96.jpg)
Companion Case
p(x) = xn + an−1xn−1 + an−2x
n−2 + · · ·+ a0 = 0
monic polynomial
companion matrix
A =
0 · · · 0 −a01 0 · · · 0 −a1
1. . .
......
. . . 0 −an−2
1 −an−1
. . . get the zeros of p by computing the eigenvalues.
MATLAB’s roots command
David S. Watkins Core-Chasing Algorithm
![Page 97: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/97.jpg)
Companion Case
p(x) = xn + an−1xn−1 + an−2x
n−2 + · · ·+ a0 = 0
monic polynomial
companion matrix
A =
0 · · · 0 −a01 0 · · · 0 −a1
1. . .
......
. . . 0 −an−2
1 −an−1
. . . get the zeros of p by computing the eigenvalues.
MATLAB’s roots command
David S. Watkins Core-Chasing Algorithm
![Page 98: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/98.jpg)
Companion Case
p(x) = xn + an−1xn−1 + an−2x
n−2 + · · ·+ a0 = 0
monic polynomial
companion matrix
A =
0 · · · 0 −a01 0 · · · 0 −a1
1. . .
......
. . . 0 −an−2
1 −an−1
. . . get the zeros of p by computing the eigenvalues.
MATLAB’s roots command
David S. Watkins Core-Chasing Algorithm
![Page 99: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/99.jpg)
Cost of solving companion eigenvalue problem
If structure not exploited:
O(n2) storage, O(n3) flopsFrancis’s algorithm
If structure exploited:
O(n) storage, O(n2) flopsdata-sparse representation + Francis’s algorithmseveral methods proposed
David S. Watkins Core-Chasing Algorithm
![Page 100: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/100.jpg)
Cost of solving companion eigenvalue problem
If structure not exploited:
O(n2) storage, O(n3) flopsFrancis’s algorithm
If structure exploited:
O(n) storage, O(n2) flopsdata-sparse representation + Francis’s algorithmseveral methods proposed
David S. Watkins Core-Chasing Algorithm
![Page 101: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/101.jpg)
Cost of solving companion eigenvalue problem
If structure not exploited:
O(n2) storage, O(n3) flopsFrancis’s algorithm
If structure exploited:
O(n) storage, O(n2) flopsdata-sparse representation + Francis’s algorithm
several methods proposed
David S. Watkins Core-Chasing Algorithm
![Page 102: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/102.jpg)
Cost of solving companion eigenvalue problem
If structure not exploited:
O(n2) storage, O(n3) flopsFrancis’s algorithm
If structure exploited:
O(n) storage, O(n2) flopsdata-sparse representation + Francis’s algorithmseveral methods proposed
David S. Watkins Core-Chasing Algorithm
![Page 103: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/103.jpg)
Some of the Competitors
Chandrasekaran, Gu, Xia, Zhu (2007)
Bini, Boito, Eidelman, Gemignani, Gohberg (2010)
Boito, Eidelman, Gemignani, Gohberg (2012)
Fortran codes available
evidence of backward stability
quasiseparable generator representation
We will do something else.
Our method is faster, and we can prove backward stability.
David S. Watkins Core-Chasing Algorithm
![Page 104: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/104.jpg)
Some of the Competitors
Chandrasekaran, Gu, Xia, Zhu (2007)
Bini, Boito, Eidelman, Gemignani, Gohberg (2010)
Boito, Eidelman, Gemignani, Gohberg (2012)
Fortran codes available
evidence of backward stability
quasiseparable generator representation
We will do something else.
Our method is faster, and we can prove backward stability.
David S. Watkins Core-Chasing Algorithm
![Page 105: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/105.jpg)
Some of the Competitors
Chandrasekaran, Gu, Xia, Zhu (2007)
Bini, Boito, Eidelman, Gemignani, Gohberg (2010)
Boito, Eidelman, Gemignani, Gohberg (2012)
Fortran codes available
evidence of backward stability
quasiseparable generator representation
We will do something else.
Our method is faster, and we can prove backward stability.
David S. Watkins Core-Chasing Algorithm
![Page 106: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/106.jpg)
Some of the Competitors
Chandrasekaran, Gu, Xia, Zhu (2007)
Bini, Boito, Eidelman, Gemignani, Gohberg (2010)
Boito, Eidelman, Gemignani, Gohberg (2012)
Fortran codes available
evidence of backward stability
quasiseparable generator representation
We will do something else.
Our method is faster, and we can prove backward stability.
David S. Watkins Core-Chasing Algorithm
![Page 107: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/107.jpg)
Some of the Competitors
Chandrasekaran, Gu, Xia, Zhu (2007)
Bini, Boito, Eidelman, Gemignani, Gohberg (2010)
Boito, Eidelman, Gemignani, Gohberg (2012)
Fortran codes available
evidence of backward stability
quasiseparable generator representation
We will do something else.
Our method is faster, and we can prove backward stability.
David S. Watkins Core-Chasing Algorithm
![Page 108: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/108.jpg)
Some of the Competitors
Chandrasekaran, Gu, Xia, Zhu (2007)
Bini, Boito, Eidelman, Gemignani, Gohberg (2010)
Boito, Eidelman, Gemignani, Gohberg (2012)
Fortran codes available
evidence of backward stability
quasiseparable generator representation
We will do something else.
Our method is faster,
and we can prove backward stability.
David S. Watkins Core-Chasing Algorithm
![Page 109: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/109.jpg)
Some of the Competitors
Chandrasekaran, Gu, Xia, Zhu (2007)
Bini, Boito, Eidelman, Gemignani, Gohberg (2010)
Boito, Eidelman, Gemignani, Gohberg (2012)
Fortran codes available
evidence of backward stability
quasiseparable generator representation
We will do something else.
Our method is faster, and we can prove backward stability.
David S. Watkins Core-Chasing Algorithm
![Page 110: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/110.jpg)
Structure
Companion matrix is unitary-plus-rank-one0 · · · 0 11 0
. . ....
1 0
+
0 · · · 0 −a0 − 10 0 −a1...
......
0 · · · 0 −an−1
We exploit this structure.
David S. Watkins Core-Chasing Algorithm
![Page 111: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/111.jpg)
Structure
Companion matrix is unitary-plus-rank-one0 · · · 0 11 0
. . ....
1 0
+
0 · · · 0 −a0 − 10 0 −a1...
......
0 · · · 0 −an−1
We exploit this structure.
David S. Watkins Core-Chasing Algorithm
![Page 112: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/112.jpg)
Structure
. . . but we store the QR decomposed form
A = QR
=
0 · · · 0 11 0
. . ....
1 0
1 0 · · · −a11 −a2
. . ....−a0
=
����
. . . ��
1 0 · · · −a1
1 −a2. . .
...−a0
David S. Watkins Core-Chasing Algorithm
![Page 113: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/113.jpg)
Structure
. . . but we store the QR decomposed form
A = QR
=
0 · · · 0 11 0
. . ....
1 0
1 0 · · · −a11 −a2
. . ....−a0
=
����
. . . ��
1 0 · · · −a1
1 −a2. . .
...−a0
David S. Watkins Core-Chasing Algorithm
![Page 114: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/114.jpg)
Structure
. . . but we store the QR decomposed form
A = QR
=
0 · · · 0 11 0
. . ....
1 0
1 0 · · · −a11 −a2
. . ....−a0
=
����
. . . ��
1 0 · · · −a1
1 −a2. . .
...−a0
David S. Watkins Core-Chasing Algorithm
![Page 115: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/115.jpg)
Structure
How do we store R compactly?
R is unitary-plus-rank one.
Adjoin a row and column for wiggle room. (not obvious)
R̂ =
1 −a1 0
. . ....
...1 −an−1 0−a0 1
0 0
=
1 0 0
. . ....
...1 0 0
0 1
1 0
+
0 −a1 0
. . ....
...0 −an−1 0−a0 0
−1 0
David S. Watkins Core-Chasing Algorithm
![Page 116: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/116.jpg)
Structure
How do we store R compactly?
R is unitary-plus-rank one.
Adjoin a row and column for wiggle room. (not obvious)
R̂ =
1 −a1 0
. . ....
...1 −an−1 0−a0 1
0 0
=
1 0 0
. . ....
...1 0 0
0 1
1 0
+
0 −a1 0
. . ....
...0 −an−1 0−a0 0
−1 0
David S. Watkins Core-Chasing Algorithm
![Page 117: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/117.jpg)
Structure
How do we store R compactly?
R is unitary-plus-rank one.
Adjoin a row and column for wiggle room. (not obvious)
R̂ =
1 −a1 0
. . ....
...1 −an−1 0−a0 1
0 0
=
1 0 0
. . ....
...1 0 0
0 1
1 0
+
0 −a1 0
. . ....
...0 −an−1 0−a0 0
−1 0
David S. Watkins Core-Chasing Algorithm
![Page 118: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/118.jpg)
Structure
How do we store R compactly?
R is unitary-plus-rank one.
Adjoin a row and column for wiggle room. (not obvious)
R̂ =
1 −a1 0
. . ....
...1 −an−1 0−a0 1
0 0
=
1 0 0
. . ....
...1 0 0
0 1
1 0
+
0 −a1 0
. . ....
...0 −an−1 0−a0 0
−1 0
David S. Watkins Core-Chasing Algorithm
![Page 119: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/119.jpg)
Structure
How do we store R compactly?
R is unitary-plus-rank one.
Adjoin a row and column for wiggle room. (not obvious)
R̂ =
1 −a1 0
. . ....
...1 −an−1 0−a0 1
0 0
=
1 0 0
. . ....
...1 0 0
0 1
1 0
+
0 −a1 0
. . ....
...0 −an−1 0−a0 0
−1 0
David S. Watkins Core-Chasing Algorithm
![Page 120: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/120.jpg)
Representation of R
R = PT R̂P
R̂ = U + xyT , where
xyT =
−a1
...− an−1
−a0−1
[
0 · · · 0 1 0]
Next step: Roll up x .
David S. Watkins Core-Chasing Algorithm
![Page 121: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/121.jpg)
Representation of R
R = PT R̂P
R̂ = U + xyT , where
xyT =
−a1
...− an−1
−a0−1
[
0 · · · 0 1 0]
Next step: Roll up x .
David S. Watkins Core-Chasing Algorithm
![Page 122: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/122.jpg)
Representation of R
R = PT R̂P
R̂ = U + xyT , where
xyT =
−a1
...− an−1
−a0−1
[
0 · · · 0 1 0]
Next step: Roll up x .
David S. Watkins Core-Chasing Algorithm
![Page 123: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/123.jpg)
Representation of R
R = PT R̂P
R̂ = U + xyT , where
xyT =
−a1
...− an−1
−a0−1
[
0 · · · 0 1 0]
Next step: Roll up x .
David S. Watkins Core-Chasing Algorithm
![Page 124: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/124.jpg)
Representation of R
xxxx
=
xxxx
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Core-Chasing Algorithm
![Page 125: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/125.jpg)
Representation of R
��
xxxx
=
xxx0
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Core-Chasing Algorithm
![Page 126: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/126.jpg)
Representation of R
����
xxxx
=
xx00
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Core-Chasing Algorithm
![Page 127: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/127.jpg)
Representation of R
����
��
xxxx
=
x000
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Core-Chasing Algorithm
![Page 128: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/128.jpg)
Representation of R
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xxxx
=
x000
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Core-Chasing Algorithm
![Page 129: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/129.jpg)
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R̂ = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R̂ = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = PTC ∗(B + e1yT )P = PTC ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )P
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Core-Chasing Algorithm
![Page 130: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/130.jpg)
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R̂ = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R̂ = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = PTC ∗(B + e1yT )P = PTC ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )P
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Core-Chasing Algorithm
![Page 131: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/131.jpg)
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R̂ = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R̂ = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = PTC ∗(B + e1yT )P = PTC ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )P
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Core-Chasing Algorithm
![Page 132: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/132.jpg)
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R̂ = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R̂ = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = PTC ∗(B + e1yT )P = PTC ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )P
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Core-Chasing Algorithm
![Page 133: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/133.jpg)
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R̂ = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R̂ = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = PTC ∗(B + e1yT )P = PTC ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )P
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Core-Chasing Algorithm
![Page 134: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/134.jpg)
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R̂ = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R̂ = C ∗(B + e1yT )
B is upper Hessenberg (and unitary)
so B = B1 · · ·Bn.
R = PTC ∗(B + e1yT )P = PTC ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )P
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Core-Chasing Algorithm
![Page 135: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/135.jpg)
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R̂ = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R̂ = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = PTC ∗(B + e1yT )P = PTC ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )P
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Core-Chasing Algorithm
![Page 136: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/136.jpg)
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R̂ = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R̂ = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = PTC ∗(B + e1yT )P = PTC ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )P
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Core-Chasing Algorithm
![Page 137: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/137.jpg)
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R̂ = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R̂ = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = PTC ∗(B + e1yT )P = PTC ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )P
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Core-Chasing Algorithm
![Page 138: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/138.jpg)
Representation of R
R = PTC ∗n · · ·C ∗
1 (B1 · · ·Bn + e1yT )P
=
��
�
��
��
�
��
����
��
+ · · ·
David S. Watkins Core-Chasing Algorithm
![Page 139: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/139.jpg)
Representation of R
R = PTC ∗n · · ·C ∗
1 (B1 · · ·Bn + e1yT )P
=
��
�
��
��
�
��
����
��
+ · · ·
David S. Watkins Core-Chasing Algorithm
![Page 140: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/140.jpg)
Representation of A
Altogether we have
A = QR
=
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+ · · ·
David S. Watkins Core-Chasing Algorithm
![Page 141: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/141.jpg)
Francis Iterations
We have complex single-shift code . . .
real double-shift code.
We describe single-shift case for simplicity.
ignoring rank-one part . . .
A =
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David S. Watkins Core-Chasing Algorithm
![Page 142: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/142.jpg)
Francis Iterations
We have complex single-shift code . . .
real double-shift code.
We describe single-shift case for simplicity.
ignoring rank-one part . . .
A =
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David S. Watkins Core-Chasing Algorithm
![Page 143: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/143.jpg)
Francis Iterations
We have complex single-shift code . . .
real double-shift code.
We describe single-shift case for simplicity.
ignoring rank-one part . . .
A =
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David S. Watkins Core-Chasing Algorithm
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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The Core Chase
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![Page 171: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/171.jpg)
The Core Chase
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David S. Watkins Core-Chasing Algorithm
![Page 172: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/172.jpg)
The Core Chase
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David S. Watkins Core-Chasing Algorithm
![Page 173: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/173.jpg)
Done!
iteration complete!
Cost: 3n turnovers/iteration, so O(n) flops/iteration
Double-shift iteration is similar.
(Chase two core transformations instead of one.)
David S. Watkins Core-Chasing Algorithm
![Page 174: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/174.jpg)
Done!
iteration complete!
Cost: 3n turnovers/iteration, so O(n) flops/iteration
Double-shift iteration is similar.
(Chase two core transformations instead of one.)
David S. Watkins Core-Chasing Algorithm
![Page 175: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/175.jpg)
Done!
iteration complete!
Cost: 3n turnovers/iteration, so O(n) flops/iteration
Double-shift iteration is similar.
(Chase two core transformations instead of one.)
David S. Watkins Core-Chasing Algorithm
![Page 176: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/176.jpg)
Performance
100
101
102
103
104
105
10−5
10−4
10−3
10−2
10−1
100
101
102
103
degree
tim
e (
se
co
nd
s)
LAPACK
BEGG
AMVW
David S. Watkins Core-Chasing Algorithm
![Page 177: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/177.jpg)
Performance
At degree 1000
method time
LAPACK 7.2
BEGG 1.2
AMVW 0.2
David S. Watkins Core-Chasing Algorithm
![Page 178: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/178.jpg)
See our paper for . . .
Paper in SIAM J. Matrix Anal. Appl. has
. . . more timings,
. . . accuracy comparisons,
. . . proof of backward stability.
David S. Watkins Core-Chasing Algorithm
![Page 179: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/179.jpg)
See our paper for . . .
Paper in SIAM J. Matrix Anal. Appl. has
. . . more timings,
. . . accuracy comparisons,
. . . proof of backward stability.
David S. Watkins Core-Chasing Algorithm
![Page 180: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/180.jpg)
See our paper for . . .
Paper in SIAM J. Matrix Anal. Appl. has
. . . more timings,
. . . accuracy comparisons,
. . . proof of backward stability.
David S. Watkins Core-Chasing Algorithm
![Page 181: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/181.jpg)
See our paper for . . .
Paper in SIAM J. Matrix Anal. Appl. has
. . . more timings,
. . . accuracy comparisons,
. . . proof of backward stability.
David S. Watkins Core-Chasing Algorithm
![Page 182: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/182.jpg)
Summary
We took a new look at Francis’s algorithm
considered QR decomposed form
We demonstrated some advantages.
unitary caseunitary-plus-rank-one case (companion)efficient cache use (not demonstrated today)
Thank you for your attention.
David S. Watkins Core-Chasing Algorithm
![Page 183: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/183.jpg)
Summary
We took a new look at Francis’s algorithm
considered QR decomposed form
We demonstrated some advantages.
unitary caseunitary-plus-rank-one case (companion)efficient cache use (not demonstrated today)
Thank you for your attention.
David S. Watkins Core-Chasing Algorithm
![Page 184: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/184.jpg)
Summary
We took a new look at Francis’s algorithm
considered QR decomposed form
We demonstrated some advantages.
unitary caseunitary-plus-rank-one case (companion)efficient cache use (not demonstrated today)
Thank you for your attention.
David S. Watkins Core-Chasing Algorithm
![Page 185: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/185.jpg)
Summary
We took a new look at Francis’s algorithm
considered QR decomposed form
We demonstrated some advantages.
unitary caseunitary-plus-rank-one case (companion)efficient cache use (not demonstrated today)
Thank you for your attention.
David S. Watkins Core-Chasing Algorithm
![Page 186: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/186.jpg)
Summary
We took a new look at Francis’s algorithm
considered QR decomposed form
We demonstrated some advantages.
unitary case
unitary-plus-rank-one case (companion)efficient cache use (not demonstrated today)
Thank you for your attention.
David S. Watkins Core-Chasing Algorithm
![Page 187: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/187.jpg)
Summary
We took a new look at Francis’s algorithm
considered QR decomposed form
We demonstrated some advantages.
unitary caseunitary-plus-rank-one case (companion)
efficient cache use (not demonstrated today)
Thank you for your attention.
David S. Watkins Core-Chasing Algorithm
![Page 188: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/188.jpg)
Summary
We took a new look at Francis’s algorithm
considered QR decomposed form
We demonstrated some advantages.
unitary caseunitary-plus-rank-one case (companion)efficient cache use (not demonstrated today)
Thank you for your attention.
David S. Watkins Core-Chasing Algorithm
![Page 189: Francis's Algorithm as a Core-Chasing Algorithm · Fundamentals of Matrix Computations, 3rd Ed., 2010 Francis’s Algorithm, Amer. Math. Monthly, 2011...but we’re still not done!](https://reader034.vdocuments.net/reader034/viewer/2022050501/5f93d01c2a40b920937f3933/html5/thumbnails/189.jpg)
Summary
We took a new look at Francis’s algorithm
considered QR decomposed form
We demonstrated some advantages.
unitary caseunitary-plus-rank-one case (companion)efficient cache use (not demonstrated today)
Thank you for your attention.
David S. Watkins Core-Chasing Algorithm