![Page 1: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/1.jpg)
Francis’s AlgorithmDavid S. Watkins
Department of Mathematics
Washington State University
Francis’s Algorithm – p. 1
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Eigenvalue Problem: Av = λv
Francis’s Algorithm – p. 2
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Eigenvalue Problem: Av = λv
How to solve?
Francis’s Algorithm – p. 2
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Eigenvalue Problem: Av = λv
How to solve?
lambda = eig(A)
Francis’s Algorithm – p. 2
![Page 5: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/5.jpg)
Eigenvalue Problem: Av = λv
How to solve?
lambda = eig(A)
How doeseig do it?
Francis’s Algorithm – p. 2
![Page 6: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/6.jpg)
Eigenvalue Problem: Av = λv
How to solve?
lambda = eig(A)
How doeseig do it?
Francis’s algorithm,
Francis’s Algorithm – p. 2
![Page 7: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/7.jpg)
Eigenvalue Problem: Av = λv
How to solve?
lambda = eig(A)
How doeseig do it?
Francis’s algorithm, aka
Francis’s Algorithm – p. 2
![Page 8: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/8.jpg)
Eigenvalue Problem: Av = λv
How to solve?
lambda = eig(A)
How doeseig do it?
Francis’s algorithm, aka
the implicitly shiftedQR algorithm
Francis’s Algorithm – p. 2
![Page 9: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/9.jpg)
Eigenvalue Problem: Av = λv
How to solve?
lambda = eig(A)
How doeseig do it?
Francis’s algorithm, aka
the implicitly shiftedQR algorithm
50 years!
Francis’s Algorithm – p. 2
![Page 10: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/10.jpg)
Eigenvalue Problem: Av = λv
How to solve?
lambda = eig(A)
How doeseig do it?
Francis’s algorithm, aka
the implicitly shiftedQR algorithm
50 years!
Top Ten of the century (Dongarra and Sullivan)
Francis’s Algorithm – p. 2
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John Francis
Francis’s Algorithm – p. 3
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Who is John Francis?
Francis’s Algorithm – p. 4
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Who is John Francis?born near London in 1934
Francis’s Algorithm – p. 4
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Who is John Francis?born near London in 1934
employed in late 50’s, Pegasus computer
Francis’s Algorithm – p. 4
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Who is John Francis?born near London in 1934
employed in late 50’s, Pegasus computer
linear algebra, eigenvalue routines
Francis’s Algorithm – p. 4
![Page 16: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/16.jpg)
Who is John Francis?born near London in 1934
employed in late 50’s, Pegasus computer
linear algebra, eigenvalue routines
primitive computer
Francis’s Algorithm – p. 4
![Page 17: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/17.jpg)
Who is John Francis?born near London in 1934
employed in late 50’s, Pegasus computer
linear algebra, eigenvalue routines
primitive computer
no software
Francis’s Algorithm – p. 4
![Page 18: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/18.jpg)
Who is John Francis?born near London in 1934
employed in late 50’s, Pegasus computer
linear algebra, eigenvalue routines
primitive computer
no software
experimented with a variety of methods
Francis’s Algorithm – p. 4
![Page 19: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/19.jpg)
Who is John Francis?born near London in 1934
employed in late 50’s, Pegasus computer
linear algebra, eigenvalue routines
primitive computer
no software
experimented with a variety of methods
invented His algorithm and programmed it
Francis’s Algorithm – p. 4
![Page 20: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/20.jpg)
Who is John Francis?born near London in 1934
employed in late 50’s, Pegasus computer
linear algebra, eigenvalue routines
primitive computer
no software
experimented with a variety of methods
invented His algorithm and programmed it
moved on to other things
Francis’s Algorithm – p. 4
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Some History
Francis’s Algorithm – p. 5
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Some HistoryRutishauser (q-d 1954, LR 1958)
Francis’s Algorithm – p. 5
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Some HistoryRutishauser (q-d 1954, LR 1958)
Francis’s first paper (QR)
Francis’s Algorithm – p. 5
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Some HistoryRutishauser (q-d 1954, LR 1958)
Francis’s first paper (QR)
A − ρI = QR, RQ + ρI = A
Francis’s Algorithm – p. 5
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Some HistoryRutishauser (q-d 1954, LR 1958)
Francis’s first paper (QR)
A − ρI = QR, RQ + ρI = A repeat!
Francis’s Algorithm – p. 5
![Page 26: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/26.jpg)
Some HistoryRutishauser (q-d 1954, LR 1958)
Francis’s first paper (QR)
A − ρI = QR, RQ + ρI = A repeat!
Kublanovskaya
Francis’s Algorithm – p. 5
![Page 27: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/27.jpg)
Some HistoryRutishauser (q-d 1954, LR 1958)
Francis’s first paper (QR)
A − ρI = QR, RQ + ρI = A repeat!
Kublanovskaya
. . . but this is not “Francis’s Algorithm”
Francis’s Algorithm – p. 5
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Francis’s Algorithm
Francis’s Algorithm – p. 6
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Francis’s AlgorithmSecond paper of Francis
Francis’s Algorithm – p. 6
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Francis’s AlgorithmSecond paper of Francis
real matrices
Francis’s Algorithm – p. 6
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Francis’s AlgorithmSecond paper of Francis
real matrices with complex pairs of eigenvalues
Francis’s Algorithm – p. 6
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Francis’s AlgorithmSecond paper of Francis
real matrices with complex pairs of eigenvalues
complex shifts
Francis’s Algorithm – p. 6
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Francis’s AlgorithmSecond paper of Francis
real matrices with complex pairs of eigenvalues
complex shifts
want to stay in real arithmetic
Francis’s Algorithm – p. 6
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Francis’s AlgorithmSecond paper of Francis
real matrices with complex pairs of eigenvalues
complex shifts
want to stay in real arithmetic
two steps at once
Francis’s Algorithm – p. 6
![Page 35: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/35.jpg)
Francis’s AlgorithmSecond paper of Francis
real matrices with complex pairs of eigenvalues
complex shifts
want to stay in real arithmetic
two steps at once
double-shiftQR algorithm
Francis’s Algorithm – p. 6
![Page 36: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/36.jpg)
Francis’s AlgorithmSecond paper of Francis
real matrices with complex pairs of eigenvalues
complex shifts
want to stay in real arithmetic
two steps at once
double-shiftQR algorithm
radically different from basic QR
Francis’s Algorithm – p. 6
![Page 37: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/37.jpg)
Francis’s AlgorithmSecond paper of Francis
real matrices with complex pairs of eigenvalues
complex shifts
want to stay in real arithmetic
two steps at once
double-shiftQR algorithm
radically different from basic QR
Usual justification: Francis’s implicit-Q theorem
Francis’s Algorithm – p. 6
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Francis’s Algorithm
Francis’s Algorithm – p. 7
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Francis’s Algorithmupper Hessenberg form
Francis’s Algorithm – p. 7
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Francis’s Algorithmupper Hessenberg form
pick some shiftsρ1, . . . ,ρm (m = 1, 2, 4, 6)
Francis’s Algorithm – p. 7
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Francis’s Algorithmupper Hessenberg form
pick some shiftsρ1, . . . ,ρm (m = 1, 2, 4, 6)
p(A) = (A − ρ1I) · · · (A − ρmI)
Francis’s Algorithm – p. 7
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Francis’s Algorithmupper Hessenberg form
pick some shiftsρ1, . . . ,ρm (m = 1, 2, 4, 6)
p(A) = (A − ρ1I) · · · (A − ρmI) expensive!
Francis’s Algorithm – p. 7
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Francis’s Algorithmupper Hessenberg form
pick some shiftsρ1, . . . ,ρm (m = 1, 2, 4, 6)
p(A) = (A − ρ1I) · · · (A − ρmI) expensive!
computep(A)e1
Francis’s Algorithm – p. 7
![Page 44: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/44.jpg)
Francis’s Algorithmupper Hessenberg form
pick some shiftsρ1, . . . ,ρm (m = 1, 2, 4, 6)
p(A) = (A − ρ1I) · · · (A − ρmI) expensive!
computep(A)e1 cheap!
Francis’s Algorithm – p. 7
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Francis’s Algorithmupper Hessenberg form
pick some shiftsρ1, . . . ,ρm (m = 1, 2, 4, 6)
p(A) = (A − ρ1I) · · · (A − ρmI) expensive!
computep(A)e1 cheap!
Build unitaryQ0 with q1 = αp(A)e1.
Francis’s Algorithm – p. 7
![Page 46: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/46.jpg)
Francis’s Algorithmupper Hessenberg form
pick some shiftsρ1, . . . ,ρm (m = 1, 2, 4, 6)
p(A) = (A − ρ1I) · · · (A − ρmI) expensive!
computep(A)e1 cheap!
Build unitaryQ0 with q1 = αp(A)e1.
Perform similarity transformA → Q−1
0AQ0.
Francis’s Algorithm – p. 7
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Francis’s Algorithmupper Hessenberg form
pick some shiftsρ1, . . . ,ρm (m = 1, 2, 4, 6)
p(A) = (A − ρ1I) · · · (A − ρmI) expensive!
computep(A)e1 cheap!
Build unitaryQ0 with q1 = αp(A)e1.
Perform similarity transformA → Q−1
0AQ0.
Hessenberg form is disturbed.
Francis’s Algorithm – p. 7
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An Upper Hessenberg Matrix@
@@
@@
@@
@@
@@
@@
Francis’s Algorithm – p. 8
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After the Transformation ( Q−10 AQ0)
@@
@@
@@
@@
@@
Francis’s Algorithm – p. 9
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After the Transformation ( Q−10 AQ0)
@@
@@
@@
@@
@@
Now return the matrix to Hessenberg form.
Francis’s Algorithm – p. 9
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Chasing the Bulge@
@@@
@@
@@
@@@
Francis’s Algorithm – p. 10
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Chasing the Bulge@
@@
@@
@@
@@
@
Francis’s Algorithm – p. 11
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Done@
@@
@@
@@
@@
@@
@@
Francis’s Algorithm – p. 12
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Done@
@@
@@
@@
@@
@@
@@
The Francis iteration is complete!
Francis’s Algorithm – p. 12
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Summary of Francis Iteration
Francis’s Algorithm – p. 13
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Summary of Francis IterationPick some shifts.
Francis’s Algorithm – p. 13
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Summary of Francis IterationPick some shifts.
Computep(A)e1. (p determined by shifts)
Francis’s Algorithm – p. 13
![Page 58: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/58.jpg)
Summary of Francis IterationPick some shifts.
Computep(A)e1. (p determined by shifts)
Build Q0 with first columnq1 = αp(A)e1.
Francis’s Algorithm – p. 13
![Page 59: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/59.jpg)
Summary of Francis IterationPick some shifts.
Computep(A)e1. (p determined by shifts)
Build Q0 with first columnq1 = αp(A)e1.
Make a bulge. (A → Q−1
0AQ0)
Francis’s Algorithm – p. 13
![Page 60: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/60.jpg)
Summary of Francis IterationPick some shifts.
Computep(A)e1. (p determined by shifts)
Build Q0 with first columnq1 = αp(A)e1.
Make a bulge. (A → Q−1
0AQ0)
Chase the bulge. (return to Hessenberg form)
Francis’s Algorithm – p. 13
![Page 61: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/61.jpg)
Summary of Francis IterationPick some shifts.
Computep(A)e1. (p determined by shifts)
Build Q0 with first columnq1 = αp(A)e1.
Make a bulge. (A → Q−1
0AQ0)
Chase the bulge. (return to Hessenberg form)
A = Q−1AQ
Francis’s Algorithm – p. 13
![Page 62: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/62.jpg)
Quicker Summary
Francis’s Algorithm – p. 14
![Page 63: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/63.jpg)
Quicker SummaryMake a bulge.
Francis’s Algorithm – p. 14
![Page 64: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/64.jpg)
Quicker SummaryMake a bulge.
Chase it.
Francis’s Algorithm – p. 14
![Page 65: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/65.jpg)
Remarks
Francis’s Algorithm – p. 15
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RemarksThis is pretty simple.
Francis’s Algorithm – p. 15
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RemarksThis is pretty simple.
noQR decomposition in sight!
Francis’s Algorithm – p. 15
![Page 68: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/68.jpg)
RemarksThis is pretty simple.
noQR decomposition in sight!
Why call it theQR algorithm?
Francis’s Algorithm – p. 15
![Page 69: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/69.jpg)
RemarksThis is pretty simple.
noQR decomposition in sight!
Why call it theQR algorithm?
Confusion!
Francis’s Algorithm – p. 15
![Page 70: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/70.jpg)
RemarksThis is pretty simple.
noQR decomposition in sight!
Why call it theQR algorithm?
Confusion!
Can we think of another name?
Francis’s Algorithm – p. 15
![Page 71: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/71.jpg)
RemarksThis is pretty simple.
noQR decomposition in sight!
Why call it theQR algorithm?
Confusion!
Can we think of another name?
I’m calling it Francis’s Algorithm.
Francis’s Algorithm – p. 15
![Page 72: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/72.jpg)
RemarksThis is pretty simple.
noQR decomposition in sight!
Why call it theQR 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
![Page 73: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/73.jpg)
Question
Francis’s Algorithm – p. 16
![Page 74: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/74.jpg)
QuestionHow should we view Francis’s algorithm?
Francis’s Algorithm – p. 16
![Page 75: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/75.jpg)
QuestionHow should we view Francis’s algorithm?
Do we have to start with the basicQR algorithm?
Francis’s Algorithm – p. 16
![Page 76: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/76.jpg)
QuestionHow should we view Francis’s algorithm?
Do we have to start with the basicQR algorithm?
Couldn’t we just as well introduce Francis’salgorithm directly?
Francis’s Algorithm – p. 16
![Page 77: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/77.jpg)
QuestionHow should we view Francis’s algorithm?
Do we have to start with the basicQR algorithm?
Couldn’t we just as well introduce Francis’salgorithm directly? . . . bypassing the basicQRalgorithm entirely?
Francis’s Algorithm – p. 16
![Page 78: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/78.jpg)
QuestionHow should we view Francis’s algorithm?
Do we have to start with the basicQR algorithm?
Couldn’t we just as well introduce Francis’salgorithm directly? . . . bypassing the basicQRalgorithm entirely?
. . . and the answer is:
Francis’s Algorithm – p. 16
![Page 79: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/79.jpg)
QuestionHow should we view Francis’s algorithm?
Do we have to start with the basicQR algorithm?
Couldn’t we just as well introduce Francis’salgorithm directly? . . . bypassing the basicQRalgorithm entirely?
. . . and the answer is:Why not?
Francis’s Algorithm – p. 16
![Page 80: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/80.jpg)
QuestionHow should we view Francis’s algorithm?
Do we have to start with the basicQR algorithm?
Couldn’t we just as well introduce Francis’salgorithm directly? . . . bypassing the basicQRalgorithm entirely?
. . . and the answer is:Why not?
This simplifies the presentation.
Francis’s Algorithm – p. 16
![Page 81: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/81.jpg)
QuestionHow should we view Francis’s algorithm?
Do we have to start with the basicQR algorithm?
Couldn’t we just as well introduce Francis’salgorithm directly? . . . bypassing the basicQRalgorithm 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
![Page 82: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/82.jpg)
Francis’s Algorithm – p. 17
![Page 83: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/83.jpg)
I’m putting my money where my mouth is . . .
Francis’s Algorithm – p. 17
![Page 84: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/84.jpg)
I’m putting my money where my mouth is . . .
. . . and saving one entire section!
Francis’s Algorithm – p. 17
![Page 85: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/85.jpg)
Pedagogical Pathway
Francis’s Algorithm – p. 18
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Pedagogical Pathwayreduction to Hessenberg form
Francis’s Algorithm – p. 18
![Page 87: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/87.jpg)
Pedagogical Pathwayreduction to Hessenberg form
Francis’s algorithm
Francis’s Algorithm – p. 18
![Page 88: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/88.jpg)
Pedagogical Pathwayreduction to Hessenberg form
Francis’s algorithm
Try it out!
Francis’s Algorithm – p. 18
![Page 89: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/89.jpg)
Pedagogical Pathwayreduction to Hessenberg form
Francis’s algorithm
Try it out!
It works great!
Francis’s Algorithm – p. 18
![Page 90: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/90.jpg)
Pedagogical Pathwayreduction to Hessenberg form
Francis’s algorithm
Try it out!
It works great!
Why does it work?
Francis’s Algorithm – p. 18
![Page 91: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/91.jpg)
Ingredients of Francis’s Algorithm
Francis’s Algorithm – p. 19
![Page 92: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/92.jpg)
Ingredients of Francis’s Algorithmsubspace iteration (power method)
Francis’s Algorithm – p. 19
![Page 93: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/93.jpg)
Ingredients of Francis’s Algorithmsubspace iteration (power method)
subspace iterationwith changes of coordinate system
Francis’s Algorithm – p. 19
![Page 94: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/94.jpg)
Ingredients of Francis’s Algorithmsubspace iteration (power method)
subspace iterationwith changes of coordinate system
Krylov subspaces
Francis’s Algorithm – p. 19
![Page 95: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/95.jpg)
Ingredients of Francis’s Algorithmsubspace iteration (power method)
subspace iterationwith changes of coordinate system
Krylov subspaces(instead of the implicit-Q theorem)
Francis’s Algorithm – p. 19
![Page 96: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/96.jpg)
Ingredients of Francis’s Algorithmsubspace iteration (power method)
subspace iterationwith changes of coordinate system
Krylov subspaces(instead of the implicit-Q theorem)
Krylov subspaces and subspace iteration
Francis’s Algorithm – p. 19
![Page 97: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/97.jpg)
Ingredients of Francis’s Algorithmsubspace iteration (power method)
subspace iterationwith 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
![Page 98: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/98.jpg)
Power Method, Subspace Iteration
Francis’s Algorithm – p. 20
![Page 99: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/99.jpg)
Power Method, Subspace Iterationv, Av, A2v, A3v, . . .
Francis’s Algorithm – p. 20
![Page 100: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/100.jpg)
Power Method, Subspace Iterationv, Av, A2v, A3v, . . .
convergence rate|λ2/λ1 |
Francis’s Algorithm – p. 20
![Page 101: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/101.jpg)
Power Method, Subspace Iterationv, Av, A2v, A3v, . . .
convergence rate|λ2/λ1 |
S, AS, A2S, A3S, . . .
Francis’s Algorithm – p. 20
![Page 102: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/102.jpg)
Power Method, Subspace Iterationv, Av, A2v, A3v, . . .
convergence rate|λ2/λ1 |
S, AS, A2S, A3S, . . .
subspaces of dimensionj
Francis’s Algorithm – p. 20
![Page 103: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/103.jpg)
Power Method, Subspace Iterationv, Av, A2v, A3v, . . .
convergence rate|λ2/λ1 |
S, AS, A2S, A3S, . . .
subspaces of dimensionj (|λj+1/λj |)
Francis’s Algorithm – p. 20
![Page 104: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/104.jpg)
Power Method, Subspace Iterationv, Av, A2v, A3v, . . .
convergence rate|λ2/λ1 |
S, AS, A2S, A3S, . . .
subspaces of dimensionj (|λj+1/λj |)
Substitutep(A) for A
Francis’s Algorithm – p. 20
![Page 105: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/105.jpg)
Power Method, Subspace Iterationv, Av, A2v, A3v, . . .
convergence rate|λ2/λ1 |
S, AS, A2S, A3S, . . .
subspaces of dimensionj (|λj+1/λj |)
Substitutep(A) for A (shifts, multiple steps)
Francis’s Algorithm – p. 20
![Page 106: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/106.jpg)
Power Method, Subspace Iterationv, Av, A2v, A3v, . . .
convergence rate|λ2/λ1 |
S, AS, A2S, A3S, . . .
subspaces of dimensionj (|λj+1/λj |)
Substitutep(A) for A (shifts, multiple steps)
S, p(A)S, p(A)2S, p(A)3S, . . .
Francis’s Algorithm – p. 20
![Page 107: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/107.jpg)
Power Method, Subspace Iterationv, Av, A2v, A3v, . . .
convergence rate|λ2/λ1 |
S, AS, A2S, A3S, . . .
subspaces of dimensionj (|λj+1/λj |)
Substitutep(A) for A (shifts, multiple steps)
S, p(A)S, p(A)2S, p(A)3S, . . .
convergence rate|p(λj+1)/p(λj) |
Francis’s Algorithm – p. 20
![Page 108: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/108.jpg)
Subspace Iterationwith changes of coordinate system
Francis’s Algorithm – p. 21
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Subspace Iterationwith changes of coordinate system
takeS = span{e1, . . . , ej}
Francis’s Algorithm – p. 21
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Subspace Iterationwith changes of coordinate system
takeS = span{e1, . . . , ej}
p(A)S = span{p(A)e1, . . . , p(A)ej}
= span{q1, . . . , qj} (orthonormal)
Francis’s Algorithm – p. 21
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Subspace Iterationwith changes of coordinate system
takeS = span{e1, . . . , ej}
p(A)S = span{p(A)e1, . . . , p(A)ej}
= span{q1, . . . , qj} (orthonormal)
build unitaryQ = [q1 · · · qj · · ·]
Francis’s Algorithm – p. 21
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Subspace Iterationwith changes of coordinate system
takeS = span{e1, . . . , ej}
p(A)S = span{p(A)e1, . . . , p(A)ej}
= span{q1, . . . , qj} (orthonormal)
build unitaryQ = [q1 · · · qj · · ·]
change coordinate system:A = Q−1AQ
Francis’s Algorithm – p. 21
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Subspace Iterationwith changes of coordinate system
takeS = span{e1, . . . , ej}
p(A)S = span{p(A)e1, . . . , p(A)ej}
= span{q1, . . . , qj} (orthonormal)
build unitaryQ = [q1 · · · qj · · ·]
change coordinate system:A = Q−1AQ
qk → Q−1qk = Q∗qk = ek
Francis’s Algorithm – p. 21
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Subspace Iterationwith changes of coordinate system
takeS = span{e1, . . . , ej}
p(A)S = span{p(A)e1, . . . , p(A)ej}
= span{q1, . . . , qj} (orthonormal)
build unitaryQ = [q1 · · · qj · · ·]
change coordinate system:A = Q−1AQ
qk → Q−1qk = Q∗qk = ek
span{q1, . . . , qj} → span{e1, . . . , ej}
Francis’s Algorithm – p. 21
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Subspace Iterationwith changes of coordinate system
takeS = span{e1, . . . , ej}
p(A)S = span{p(A)e1, . . . , p(A)ej}
= span{q1, . . . , qj} (orthonormal)
build unitaryQ = [q1 · · · qj · · ·]
change coordinate system:A = Q−1AQ
qk → Q−1qk = Q∗qk = ek
span{q1, . . . , qj} → span{e1, . . . , ej}
ready for next iterationFrancis’s Algorithm – p. 21
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This version of subspace iteration . . .
Francis’s Algorithm – p. 22
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This version of subspace iteration . . .
. . . holds the subspace fixed
Francis’s Algorithm – p. 22
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This version of subspace iteration . . .
. . . holds the subspace fixed
while the matrix changes.
Francis’s Algorithm – p. 22
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This version of subspace iteration . . .
. . . holds the subspace fixed
while the matrix changes.
. . . moving toward a matrix under which
span{e1, . . . , ej}
is invariant.
Francis’s Algorithm – p. 22
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This version of subspace iteration . . .
. . . holds the subspace fixed
while the matrix changes.
. . . moving toward a matrix under which
span{e1, . . . , ej}
is invariant.
A →
[
A11 A12
0 A22
]
(A11 is j × j.)
Francis’s Algorithm – p. 22
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Application to Francis’s Iteration(first pass)
Francis’s Algorithm – p. 23
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Application to Francis’s Iteration(first pass)
A = Q−1AQ where q1 = αp(A)e1.
Francis’s Algorithm – p. 23
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Application to Francis’s Iteration(first pass)
A = Q−1AQ where q1 = αp(A)e1.
power method
Francis’s Algorithm – p. 23
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Application to Francis’s Iteration(first pass)
A = Q−1AQ where q1 = αp(A)e1.
power method+ change of coordinates
Francis’s Algorithm – p. 23
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Application to Francis’s Iteration(first pass)
A = Q−1AQ where q1 = αp(A)e1.
power method+ change of coordinates
q1 → Q−1q1 = e1
Francis’s Algorithm – p. 23
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Application to Francis’s Iteration(first pass)
A = Q−1AQ where q1 = αp(A)e1.
power method+ change of coordinates
q1 → Q−1q1 = e1
casej = 1 of subspace iteration with a change ofcoordinate system
Francis’s Algorithm – p. 23
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Application to Francis’s Iteration(first pass)
A = Q−1AQ where q1 = αp(A)e1.
power method+ change of coordinates
q1 → Q−1q1 = e1
casej = 1 of subspace iteration with a change ofcoordinate system
. . . but this is just a small part of the story.
Francis’s Algorithm – p. 23
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Krylov Subspaces . . .
Francis’s Algorithm – p. 24
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Krylov Subspaces . . .. . . and Subspace Iteration
Francis’s Algorithm – p. 24
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Krylov Subspaces . . .. . . and Subspace IterationDef: Kj(A, q) = span
{
q, Aq,A2q, . . . , Aj−1q}
Francis’s Algorithm – p. 24
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Krylov Subspaces . . .. . . and Subspace IterationDef: Kj(A, q) = span
{
q, Aq,A2q, . . . , Aj−1q}
j = 1, 2, 3, . . . (nested subspaces)
Francis’s Algorithm – p. 24
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Krylov Subspaces . . .. . . and Subspace IterationDef: Kj(A, q) = span
{
q, Aq,A2q, . . . , Aj−1q}
j = 1, 2, 3, . . . (nested subspaces)
Kj(A, q) are “determined byq”.
Francis’s Algorithm – p. 24
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Krylov Subspaces . . .. . . and Subspace IterationDef: Kj(A, q) = span
{
q, Aq,A2q, . . . , Aj−1q}
j = 1, 2, 3, . . . (nested subspaces)
Kj(A, q) are “determined byq”.
p(A)Kj(A, q) = Kj(A, p(A)q)
Francis’s Algorithm – p. 24
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Krylov Subspaces . . .. . . and Subspace IterationDef: Kj(A, q) = span
{
q, Aq,A2q, . . . , Aj−1q}
j = 1, 2, 3, . . . (nested subspaces)
Kj(A, q) are “determined byq”.
p(A)Kj(A, q) = Kj(A, p(A)q)
. . . becausep(A)A = Ap(A)
Francis’s Algorithm – p. 24
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Krylov Subspaces . . .. . . and Subspace IterationDef: Kj(A, q) = span
{
q, Aq,A2q, . . . , Aj−1q}
j = 1, 2, 3, . . . (nested subspaces)
Kj(A, q) are “determined byq”.
p(A)Kj(A, q) = Kj(A, p(A)q)
. . . becausep(A)A = Ap(A)
Conclusion: Power method induces nested subspaceiterations on Krylov subspaces.
Francis’s Algorithm – p. 24
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power method: q → p(A)kq
Francis’s Algorithm – p. 25
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power method: q → p(A)kq
nested subspace iterations:
p(A)kKj(A, q) = Kj(A, p(A)kq) j = 1, 2, 3, . . .
Francis’s Algorithm – p. 25
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power method: q → p(A)kq
nested subspace iterations:
p(A)kKj(A, q) = Kj(A, p(A)kq) j = 1, 2, 3, . . .
convergence rates:
|p(λj+1)/p(λj) |, j = 1, 2, 3, . . . , n − 1
Francis’s Algorithm – p. 25
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Krylov Subspaces . . .
Francis’s Algorithm – p. 26
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Krylov Subspaces . . .. . . and Hessenberg matrices . . .
Francis’s Algorithm – p. 26
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Krylov Subspaces . . .. . . and Hessenberg matrices . . .
. . . go hand in hand.
Francis’s Algorithm – p. 26
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Krylov Subspaces . . .. . . and Hessenberg matrices . . .
. . . go hand in hand.
A properly upper Hessenberg=⇒
Kj(A, e1) = span{e1, . . . , ej}.
Francis’s Algorithm – p. 26
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Krylov Subspaces . . .. . . and Hessenberg matrices . . .
. . . go hand in hand.
A properly upper Hessenberg=⇒
Kj(A, e1) = span{e1, . . . , ej}.
More generally . . .
Francis’s Algorithm – p. 26
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Krylov-Hessenberg Relationship
Francis’s Algorithm – p. 27
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Krylov-Hessenberg Relationship
If A = Q−1AQ,
Francis’s Algorithm – p. 27
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Krylov-Hessenberg Relationship
If A = Q−1AQ,
andA is properly upper Hessenberg,
Francis’s Algorithm – p. 27
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Krylov-Hessenberg Relationship
If A = Q−1AQ,
andA is properly upper Hessenberg,
then forj = 1, 2, 3, . . . ,
Francis’s Algorithm – p. 27
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Krylov-Hessenberg Relationship
If A = Q−1AQ,
andA is properly upper Hessenberg,
then forj = 1, 2, 3, . . . ,
span{q1, . . . , qj} = Kj(A, q1).
Francis’s Algorithm – p. 27
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Application to Francis’s Iteration
Francis’s Algorithm – p. 28
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Application to Francis’s Iteration
A = Q−1AQ where q1 = αp(A)e1.
Francis’s Algorithm – p. 28
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Application to Francis’s Iteration
A = Q−1AQ where q1 = αp(A)e1.
power method with a change of coordinate system.
Francis’s Algorithm – p. 28
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Application to Francis’s Iteration
A = Q−1AQ where q1 = αp(A)e1.
power method with a change of coordinate system.Moreover . . .
Francis’s Algorithm – p. 28
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Application to Francis’s Iteration
A = Q−1AQ where q1 = αp(A)e1.
power method with a change of coordinate system.Moreover . . .
p(A)Kj(A, e1) = Kj(A, p(A)e1)
Francis’s Algorithm – p. 28
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Application to Francis’s Iteration
A = Q−1AQ where q1 = αp(A)e1.
power method with a change of coordinate system.Moreover . . .
p(A)Kj(A, e1) = Kj(A, p(A)e1)
i.e.p(A)span{e1, . . . , ej} = span{q1, . . . , qj}
Francis’s Algorithm – p. 28
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Application to Francis’s Iteration
A = Q−1AQ where q1 = αp(A)e1.
power method with a change of coordinate system.Moreover . . .
p(A)Kj(A, e1) = Kj(A, p(A)e1)
i.e.p(A)span{e1, . . . , ej} = span{q1, . . . , qj}
subspace iteration with a change of coordinatesystem
Francis’s Algorithm – p. 28
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Application to Francis’s Iteration
A = Q−1AQ where q1 = αp(A)e1.
power method with a change of coordinate system.Moreover . . .
p(A)Kj(A, e1) = Kj(A, p(A)e1)
i.e.p(A)span{e1, . . . , ej} = span{q1, . . . , qj}
subspace iteration with a change of coordinatesystem forj = 1, 2, 3, . . . ,n − 1
Francis’s Algorithm – p. 28
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Application to Francis’s Iteration
A = Q−1AQ where q1 = αp(A)e1.
power method with a change of coordinate system.Moreover . . .
p(A)Kj(A, e1) = Kj(A, p(A)e1)
i.e.p(A)span{e1, . . . , ej} = span{q1, . . . , qj}
subspace iteration with a change of coordinatesystem forj = 1, 2, 3, . . . ,n − 1
|p(λj+1)/p(λj) | j = 1, 2, 3, . . . ,n − 1
Francis’s Algorithm – p. 28
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Details
Francis’s Algorithm – p. 29
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Detailschoice of shifts
Francis’s Algorithm – p. 29
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Detailschoice of shifts
We change the shifts at each step.
Francis’s Algorithm – p. 29
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Detailschoice of shifts
We change the shifts at each step.
⇒ quadratic or cubic convergence
Francis’s Algorithm – p. 29
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Detailschoice of shifts
We change the shifts at each step.
⇒ quadratic or cubic convergence
Watkins (2007, 2010)
Francis’s Algorithm – p. 29
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Detailschoice of shifts
We change the shifts at each step.
⇒ quadratic or cubic convergence
Watkins (2007, 2010)
Francis’s Algorithm – p. 29
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Where is John Francis?
Francis’s Algorithm – p. 30
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Where is John Francis?question asked frequently by Gene Golub
Francis’s Algorithm – p. 30
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Where is John Francis?question asked frequently by Gene Golub
inquiries by Golub and Uhlig
Francis’s Algorithm – p. 30
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Where is John Francis?question asked frequently by Gene Golub
inquiries by Golub and Uhlig
Francis is alive and well,retired in the South of England.
Francis’s Algorithm – p. 30
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Where is John Francis?question asked frequently by Gene Golub
inquiries by Golub and Uhlig
Francis is alive and well,retired in the South of England.
was unaware of the impact of his algorithm
Francis’s Algorithm – p. 30
![Page 169: Francis’s AlgorithmFrancis’s Algorithm David S. Watkins watkins@math.wsu.edu Department of Mathematics Washington State University Francis’s Algorithm – p. 1Published in: American](https://reader030.vdocuments.net/reader030/viewer/2022040504/5e37321644e8d645255ef464/html5/thumbnails/169.jpg)
Where is John Francis?question asked frequently by Gene Golub
inquiries by Golub and Uhlig
Francis is alive and well,retired in the South of England.
was unaware of the impact of his algorithm
appearance at the Biennial Numerical AnalysisConference in Glasgow in June of 2009
Francis’s Algorithm – p. 30
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John Francis speaking in Glasgow
Francis’s Algorithm – p. 31
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A Portion of the Audience
Francis’s Algorithm – p. 32
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Afterwards
Photos courtesy of Frank UhligFrancis’s Algorithm – p. 33