Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington...
Transcript of Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington...
Happy Birthday, Volker!
David S. Watkins
Department of MathematicsWashington State University
Berlin, May, 2015
David S. Watkins Happy Birthday, Volker!
Contribution from Michael Overton
Volker Mehrmann
Author or coauthor of more than 160 scientific articles
Author or coauthor of 5 monographs/textbooks
Coeditor of 5 books
Co-editor-in-chief of Linear Algebra and its Applications
Member of the German Academy of Engineering
Recent President of GAMM
Recent Director of MATHEON
An outstanding scientist with broad interests and knowledge
A superlatively nice person who always has time for everyone!
A good person to be with in a jeep . . .when an elephant is charging it!!
David S. Watkins Happy Birthday, Volker!
Contribution from Michael Overton
Volker Mehrmann
Author or coauthor of more than 160 scientific articles
Author or coauthor of 5 monographs/textbooks
Coeditor of 5 books
Co-editor-in-chief of Linear Algebra and its Applications
Member of the German Academy of Engineering
Recent President of GAMM
Recent Director of MATHEON
An outstanding scientist with broad interests and knowledge
A superlatively nice person who always has time for everyone!
A good person to be with in a jeep . . .when an elephant is charging it!!
David S. Watkins Happy Birthday, Volker!
Contribution from Michael Overton
Volker Mehrmann
Author or coauthor of more than 160 scientific articles
Author or coauthor of 5 monographs/textbooks
Coeditor of 5 books
Co-editor-in-chief of Linear Algebra and its Applications
Member of the German Academy of Engineering
Recent President of GAMM
Recent Director of MATHEON
An outstanding scientist with broad interests and knowledge
A superlatively nice person who always has time for everyone!
A good person to be with in a jeep . . .when an elephant is charging it!!
David S. Watkins Happy Birthday, Volker!
Contribution from Michael Overton
Volker Mehrmann
Author or coauthor of more than 160 scientific articles
Author or coauthor of 5 monographs/textbooks
Coeditor of 5 books
Co-editor-in-chief of Linear Algebra and its Applications
Member of the German Academy of Engineering
Recent President of GAMM
Recent Director of MATHEON
An outstanding scientist with broad interests and knowledge
A superlatively nice person who always has time for everyone!
A good person to be with in a jeep . . .when an elephant is charging it!!
David S. Watkins Happy Birthday, Volker!
Contribution from Michael Overton
Volker Mehrmann
Author or coauthor of more than 160 scientific articles
Author or coauthor of 5 monographs/textbooks
Coeditor of 5 books
Co-editor-in-chief of Linear Algebra and its Applications
Member of the German Academy of Engineering
Recent President of GAMM
Recent Director of MATHEON
An outstanding scientist with broad interests and knowledge
A superlatively nice person who always has time for everyone!
A good person to be with in a jeep . . .when an elephant is charging it!!
David S. Watkins Happy Birthday, Volker!
Contribution from Michael Overton
Volker Mehrmann
Author or coauthor of more than 160 scientific articles
Author or coauthor of 5 monographs/textbooks
Coeditor of 5 books
Co-editor-in-chief of Linear Algebra and its Applications
Member of the German Academy of Engineering
Recent President of GAMM
Recent Director of MATHEON
An outstanding scientist with broad interests and knowledge
A superlatively nice person who always has time for everyone!
A good person to be with in a jeep . . .when an elephant is charging it!!
David S. Watkins Happy Birthday, Volker!
Contribution from Michael Overton
Volker Mehrmann
Author or coauthor of more than 160 scientific articles
Author or coauthor of 5 monographs/textbooks
Coeditor of 5 books
Co-editor-in-chief of Linear Algebra and its Applications
Member of the German Academy of Engineering
Recent President of GAMM
Recent Director of MATHEON
An outstanding scientist with broad interests and knowledge
A superlatively nice person who always has time for everyone!
A good person to be with in a jeep . . .when an elephant is charging it!!
David S. Watkins Happy Birthday, Volker!
Contribution from Michael Overton
Volker Mehrmann
Author or coauthor of more than 160 scientific articles
Author or coauthor of 5 monographs/textbooks
Coeditor of 5 books
Co-editor-in-chief of Linear Algebra and its Applications
Member of the German Academy of Engineering
Recent President of GAMM
Recent Director of MATHEON
An outstanding scientist with broad interests and knowledge
A superlatively nice person who always has time for everyone!
A good person to be with in a jeep . . .when an elephant is charging it!!
David S. Watkins Happy Birthday, Volker!
Contribution from Michael Overton
Volker Mehrmann
Author or coauthor of more than 160 scientific articles
Author or coauthor of 5 monographs/textbooks
Coeditor of 5 books
Co-editor-in-chief of Linear Algebra and its Applications
Member of the German Academy of Engineering
Recent President of GAMM
Recent Director of MATHEON
An outstanding scientist with broad interests and knowledge
A superlatively nice person who always has time for everyone!
A good person to be with in a jeep . . .when an elephant is charging it!!
David S. Watkins Happy Birthday, Volker!
Contribution from Michael Overton
Volker Mehrmann
Author or coauthor of more than 160 scientific articles
Author or coauthor of 5 monographs/textbooks
Coeditor of 5 books
Co-editor-in-chief of Linear Algebra and its Applications
Member of the German Academy of Engineering
Recent President of GAMM
Recent Director of MATHEON
An outstanding scientist with broad interests and knowledge
A superlatively nice person who always has time for everyone!
A good person to be with in a jeep . . .when an elephant is charging it!!
David S. Watkins Happy Birthday, Volker!
Contribution from Michael Overton
Volker Mehrmann
Author or coauthor of more than 160 scientific articles
Author or coauthor of 5 monographs/textbooks
Coeditor of 5 books
Co-editor-in-chief of Linear Algebra and its Applications
Member of the German Academy of Engineering
Recent President of GAMM
Recent Director of MATHEON
An outstanding scientist with broad interests and knowledge
A superlatively nice person who always has time for everyone!
A good person to be with in a jeep . . .when an elephant is charging it!!
David S. Watkins Happy Birthday, Volker!
Contribution from Michael Overton
Volker Mehrmann
Author or coauthor of more than 160 scientific articles
Author or coauthor of 5 monographs/textbooks
Coeditor of 5 books
Co-editor-in-chief of Linear Algebra and its Applications
Member of the German Academy of Engineering
Recent President of GAMM
Recent Director of MATHEON
An outstanding scientist with broad interests and knowledge
A superlatively nice person who always has time for everyone!
A good person to be with in a jeep . . .
when an elephant is charging it!!
David S. Watkins Happy Birthday, Volker!
Contribution from Michael Overton
Volker Mehrmann
Author or coauthor of more than 160 scientific articles
Author or coauthor of 5 monographs/textbooks
Coeditor of 5 books
Co-editor-in-chief of Linear Algebra and its Applications
Member of the German Academy of Engineering
Recent President of GAMM
Recent Director of MATHEON
An outstanding scientist with broad interests and knowledge
A superlatively nice person who always has time for everyone!
A good person to be with in a jeep . . .when an elephant is charging it!!
David S. Watkins Happy Birthday, Volker!
Kaziranga National Park (Thanks to Shreemayee Bora)
David S. Watkins Happy Birthday, Volker!
and now moving back in time . . .
David S. Watkins Happy Birthday, Volker!
1986
David S. Watkins Happy Birthday, Volker!
1986
David S. Watkins Happy Birthday, Volker!
A question
An engineer, an algebraist, and Volker . . .
. . . walk into a bar.
Engineer: I have this interesting problem where I need to findthe roots of polynomials of high degree.
Algebraist: What a coincidence! I have a problem just likethat.
Question: What does Volker say?
David S. Watkins Happy Birthday, Volker!
A question
An engineer,
an algebraist, and Volker . . .
. . . walk into a bar.
Engineer: I have this interesting problem where I need to findthe roots of polynomials of high degree.
Algebraist: What a coincidence! I have a problem just likethat.
Question: What does Volker say?
David S. Watkins Happy Birthday, Volker!
A question
An engineer, an algebraist,
and Volker . . .
. . . walk into a bar.
Engineer: I have this interesting problem where I need to findthe roots of polynomials of high degree.
Algebraist: What a coincidence! I have a problem just likethat.
Question: What does Volker say?
David S. Watkins Happy Birthday, Volker!
A question
An engineer, an algebraist, and Volker . . .
. . . walk into a bar.
Engineer: I have this interesting problem where I need to findthe roots of polynomials of high degree.
Algebraist: What a coincidence! I have a problem just likethat.
Question: What does Volker say?
David S. Watkins Happy Birthday, Volker!
A question
An engineer, an algebraist, and Volker . . .
. . . walk into a bar.
Engineer: I have this interesting problem where I need to findthe roots of polynomials of high degree.
Algebraist: What a coincidence! I have a problem just likethat.
Question: What does Volker say?
David S. Watkins Happy Birthday, Volker!
A question
An engineer, an algebraist, and Volker . . .
. . . walk into a bar.
Engineer:
I have this interesting problem where I need to findthe roots of polynomials of high degree.
Algebraist: What a coincidence! I have a problem just likethat.
Question: What does Volker say?
David S. Watkins Happy Birthday, Volker!
A question
An engineer, an algebraist, and Volker . . .
. . . walk into a bar.
Engineer: I have this interesting problem where I need to findthe roots of polynomials of high degree.
Algebraist: What a coincidence! I have a problem just likethat.
Question: What does Volker say?
David S. Watkins Happy Birthday, Volker!
A question
An engineer, an algebraist, and Volker . . .
. . . walk into a bar.
Engineer: I have this interesting problem where I need to findthe roots of polynomials of high degree.
Algebraist:
What a coincidence! I have a problem just likethat.
Question: What does Volker say?
David S. Watkins Happy Birthday, Volker!
A question
An engineer, an algebraist, and Volker . . .
. . . walk into a bar.
Engineer: I have this interesting problem where I need to findthe roots of polynomials of high degree.
Algebraist: What a coincidence! I have a problem just likethat.
Question: What does Volker say?
David S. Watkins Happy Birthday, Volker!
A question
An engineer, an algebraist, and Volker . . .
. . . walk into a bar.
Engineer: I have this interesting problem where I need to findthe roots of polynomials of high degree.
Algebraist: What a coincidence! I have a problem just likethat.
Question:
What does Volker say?
David S. Watkins Happy Birthday, Volker!
A question
An engineer, an algebraist, and Volker . . .
. . . walk into a bar.
Engineer: I have this interesting problem where I need to findthe roots of polynomials of high degree.
Algebraist: What a coincidence! I have a problem just likethat.
Question: What does Volker say?
David S. Watkins Happy Birthday, Volker!
Show me the eigenvalue problem!
Nevertheless, there is a demand for the product.
MATLAB roots (companion matrix)
Chebfun roots (colleague matrix)
I never thought I would get caught up in this racket,
. . . but somehow I got sucked in.
... and we’ve done some good stuff.
David S. Watkins Happy Birthday, Volker!
Show me the eigenvalue problem!
Nevertheless,
there is a demand for the product.
MATLAB roots (companion matrix)
Chebfun roots (colleague matrix)
I never thought I would get caught up in this racket,
. . . but somehow I got sucked in.
... and we’ve done some good stuff.
David S. Watkins Happy Birthday, Volker!
Show me the eigenvalue problem!
Nevertheless, there is a demand for the product.
MATLAB roots (companion matrix)
Chebfun roots (colleague matrix)
I never thought I would get caught up in this racket,
. . . but somehow I got sucked in.
... and we’ve done some good stuff.
David S. Watkins Happy Birthday, Volker!
Show me the eigenvalue problem!
Nevertheless, there is a demand for the product.
MATLAB roots
(companion matrix)
Chebfun roots (colleague matrix)
I never thought I would get caught up in this racket,
. . . but somehow I got sucked in.
... and we’ve done some good stuff.
David S. Watkins Happy Birthday, Volker!
Show me the eigenvalue problem!
Nevertheless, there is a demand for the product.
MATLAB roots (companion matrix)
Chebfun roots (colleague matrix)
I never thought I would get caught up in this racket,
. . . but somehow I got sucked in.
... and we’ve done some good stuff.
David S. Watkins Happy Birthday, Volker!
Show me the eigenvalue problem!
Nevertheless, there is a demand for the product.
MATLAB roots (companion matrix)
Chebfun roots
(colleague matrix)
I never thought I would get caught up in this racket,
. . . but somehow I got sucked in.
... and we’ve done some good stuff.
David S. Watkins Happy Birthday, Volker!
Show me the eigenvalue problem!
Nevertheless, there is a demand for the product.
MATLAB roots (companion matrix)
Chebfun roots (colleague matrix)
I never thought I would get caught up in this racket,
. . . but somehow I got sucked in.
... and we’ve done some good stuff.
David S. Watkins Happy Birthday, Volker!
Show me the eigenvalue problem!
Nevertheless, there is a demand for the product.
MATLAB roots (companion matrix)
Chebfun roots (colleague matrix)
I never thought I would get caught up in this racket,
. . . but somehow I got sucked in.
... and we’ve done some good stuff.
David S. Watkins Happy Birthday, Volker!
Show me the eigenvalue problem!
Nevertheless, there is a demand for the product.
MATLAB roots (companion matrix)
Chebfun roots (colleague matrix)
I never thought I would get caught up in this racket,
. . . but somehow I got sucked in.
... and we’ve done some good stuff.
David S. Watkins Happy Birthday, Volker!
Show me the eigenvalue problem!
Nevertheless, there is a demand for the product.
MATLAB roots (companion matrix)
Chebfun roots (colleague matrix)
I never thought I would get caught up in this racket,
. . . but somehow I got sucked in.
... and we’ve done some good stuff.
David S. Watkins Happy Birthday, Volker!
Our International Research Group
This is joint work with
Jared Aurentz (Oxford)
Thomas Mach (KU Leuven)
Raf Vandebril (KU Leuven)
David S. Watkins Happy Birthday, Volker!
MATLAB
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
balance, then . . .
. . . get the zeros of p by computing the eigenvalues.
This is not always the best thing to do.
David S. Watkins Happy Birthday, Volker!
MATLAB
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
balance, then . . .
. . . get the zeros of p by computing the eigenvalues.
This is not always the best thing to do.
David S. Watkins Happy Birthday, Volker!
MATLAB
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
balance, then . . .
. . . get the zeros of p by computing the eigenvalues.
This is not always the best thing to do.
David S. Watkins Happy Birthday, Volker!
MATLAB
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
balance, then . . .
. . . get the zeros of p by computing the eigenvalues.
This is not always the best thing to do.
David S. Watkins Happy Birthday, Volker!
Chebfun
p(x) = Tn(x) + bn−1Tn−1(x) + · · · b0T0(x)
Chebyshev polynomials
colleague matrix
This is sometimes better.
David S. Watkins Happy Birthday, Volker!
Chebfun
p(x) = Tn(x) + bn−1Tn−1(x) + · · · b0T0(x)
Chebyshev polynomials
colleague matrix
This is sometimes better.
David S. Watkins Happy Birthday, Volker!
Chebfun
p(x) = Tn(x) + bn−1Tn−1(x) + · · · b0T0(x)
Chebyshev polynomials
colleague matrix
This is sometimes better.
David S. Watkins Happy Birthday, Volker!
What we’ve been doing
companion matrix or
companion pencil
p(x) = anxn + an−1x
n−1 + an−2xn−2 + · · ·+ a0
0 · · · 0 −a01 0 · · · 0 −a1
1. . .
......
0 −an−2
1 −an−1
− λ
1 · · · 0 0
1 · · · 0 0. . .
......
. . . 1 0an
. . . and variants.
Today we restrict attention to the companion matrix.
David S. Watkins Happy Birthday, Volker!
What we’ve been doing
companion matrix or
companion pencil
p(x) = anxn + an−1x
n−1 + an−2xn−2 + · · ·+ a0
0 · · · 0 −a01 0 · · · 0 −a1
1. . .
......
0 −an−2
1 −an−1
− λ
1 · · · 0 0
1 · · · 0 0. . .
......
. . . 1 0an
. . . and variants.
Today we restrict attention to the companion matrix.
David S. Watkins Happy Birthday, Volker!
What we’ve been doing
companion matrix or
companion pencil
p(x) = anxn + an−1x
n−1 + an−2xn−2 + · · ·+ a0
0 · · · 0 −a01 0 · · · 0 −a1
1. . .
......
0 −an−2
1 −an−1
− λ
1 · · · 0 0
1 · · · 0 0. . .
......
. . . 1 0an
. . . and variants.
Today we restrict attention to the companion matrix.
David S. Watkins Happy Birthday, Volker!
What we’ve been doing
companion matrix or
companion pencil
p(x) = anxn + an−1x
n−1 + an−2xn−2 + · · ·+ a0
0 · · · 0 −a01 0 · · · 0 −a1
1. . .
......
0 −an−2
1 −an−1
− λ
1 · · · 0 0
1 · · · 0 0. . .
......
. . . 1 0an
. . . and variants.
Today we restrict attention to the companion matrix.
David S. Watkins Happy Birthday, Volker!
What we’ve been doing
companion matrix or
companion pencil
p(x) = anxn + an−1x
n−1 + an−2xn−2 + · · ·+ a0
0 · · · 0 −a01 0 · · · 0 −a1
1. . .
......
0 −an−2
1 −an−1
− λ
1 · · · 0 0
1 · · · 0 0. . .
......
. . . 1 0an
. . . and variants.
Today we restrict attention to the companion matrix.
David S. Watkins Happy Birthday, Volker!
Cost of solving companion eigenvalue problem
If structure not exploited:
O(n2) storage, O(n3) flopsFrancis’s implicitly-shifted QR algorithm
If structure exploited:
O(n) storage, O(n2) flopsdata-sparse representation + Francis’s algorithmSeveral methods proposed
David S. Watkins Happy Birthday, Volker!
Cost of solving companion eigenvalue problem
If structure not exploited:
O(n2) storage, O(n3) flopsFrancis’s implicitly-shifted QR algorithm
If structure exploited:
O(n) storage, O(n2) flopsdata-sparse representation + Francis’s algorithmSeveral methods proposed
David S. Watkins Happy Birthday, Volker!
Cost of solving companion eigenvalue problem
If structure not exploited:
O(n2) storage, O(n3) flopsFrancis’s implicitly-shifted QR algorithm
If structure exploited:
O(n) storage, O(n2) flopsdata-sparse representation + Francis’s algorithm
Several methods proposed
David S. Watkins Happy Birthday, Volker!
Cost of solving companion eigenvalue problem
If structure not exploited:
O(n2) storage, O(n3) flopsFrancis’s implicitly-shifted QR algorithm
If structure exploited:
O(n) storage, O(n2) flopsdata-sparse representation + Francis’s algorithmSeveral methods proposed
David S. Watkins Happy Birthday, Volker!
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.
David S. Watkins Happy Birthday, Volker!
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.
David S. Watkins Happy Birthday, Volker!
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.
David S. Watkins Happy Birthday, Volker!
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.
David S. Watkins Happy Birthday, Volker!
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.
David S. Watkins Happy Birthday, Volker!
Our Contribution
We present
Yet another O(n) representation
Francis algorithm in O(n) flops/iteration
Fortran codes (we’re faster)
normwise backward stable (We can prove it.)
David S. Watkins Happy Birthday, Volker!
Our Contribution
We present
Yet another O(n) representation
Francis algorithm in O(n) flops/iteration
Fortran codes (we’re faster)
normwise backward stable (We can prove it.)
David S. Watkins Happy Birthday, Volker!
Our Contribution
We present
Yet another O(n) representation
Francis algorithm in O(n) flops/iteration
Fortran codes (we’re faster)
normwise backward stable (We can prove it.)
David S. Watkins Happy Birthday, Volker!
Our Contribution
We present
Yet another O(n) representation
Francis algorithm in O(n) flops/iteration
Fortran codes (we’re faster)
normwise backward stable (We can prove it.)
David S. Watkins Happy Birthday, Volker!
Structure
Companion matrix is unitary-plus-rank-one0 · · · 0 e iθ
1 0. . .
...1 0
+
0 · · · 0 −e iθ − a00 0 −a1...
......
0 · · · 0 −an−1
preserved by unitary similarities
Companion matrix is also upper Hessenberg.
preserved by Francis algorithm
We exploit this structure.
David S. Watkins Happy Birthday, Volker!
Structure
Companion matrix is unitary-plus-rank-one0 · · · 0 e iθ
1 0. . .
...1 0
+
0 · · · 0 −e iθ − a00 0 −a1...
......
0 · · · 0 −an−1
preserved by unitary similarities
Companion matrix is also upper Hessenberg.
preserved by Francis algorithm
We exploit this structure.
David S. Watkins Happy Birthday, Volker!
Structure
Companion matrix is unitary-plus-rank-one0 · · · 0 e iθ
1 0. . .
...1 0
+
0 · · · 0 −e iθ − a00 0 −a1...
......
0 · · · 0 −an−1
preserved by unitary similarities
Companion matrix is also upper Hessenberg.
preserved by Francis algorithm
We exploit this structure.
David S. Watkins Happy Birthday, Volker!
Structure
Companion matrix is unitary-plus-rank-one0 · · · 0 e iθ
1 0. . .
...1 0
+
0 · · · 0 −e iθ − a00 0 −a1...
......
0 · · · 0 −an−1
preserved by unitary similarities
Companion matrix is also upper Hessenberg.
preserved by Francis algorithm
We exploit this structure.
David S. Watkins Happy Birthday, Volker!
Structure
Companion matrix is unitary-plus-rank-one0 · · · 0 e iθ
1 0. . .
...1 0
+
0 · · · 0 −e iθ − a00 0 −a1...
......
0 · · · 0 −an−1
preserved by unitary similarities
Companion matrix is also upper Hessenberg.
preserved by Francis algorithm
We exploit this structure.
David S. Watkins Happy Birthday, Volker!
Structure
Chandrasekaran, Gu, Xia, Zhu (2007)
A = QR
Q is upper Hessenberg and unitary.
R is upper triangular and unitary-plus-rank-one.
We do this too.
David S. Watkins Happy Birthday, Volker!
Structure
Chandrasekaran, Gu, Xia, Zhu (2007)
A = QR
Q is upper Hessenberg and unitary.
R is upper triangular and unitary-plus-rank-one.
We do this too.
David S. Watkins Happy Birthday, Volker!
Structure
Chandrasekaran, Gu, Xia, Zhu (2007)
A = QR
Q is upper Hessenberg and unitary.
R is upper triangular and unitary-plus-rank-one.
We do this too.
David S. Watkins Happy Birthday, Volker!
The Unitary Part
x x x xx x x x
x x xx x
=
x xx x
11
1x xx x
1
11
x xx x
Q =��
����
O(n) storage
David S. Watkins Happy Birthday, Volker!
The Unitary Part
x x x xx x x x
x x xx x
=
x xx x
11
1x xx x
1
11
x xx x
Q =��
����
O(n) storage
David S. Watkins Happy Birthday, Volker!
The Unitary Part
x x x xx x x x
x x xx x
=
x xx x
11
1x xx x
1
11
x xx x
Q =��
����
O(n) storage
David S. Watkins Happy Birthday, Volker!
The Upper Triangular Part
R = U + xyT unitary-plus-rank-one, so
R has quasiseparable rank 2.
R =
x · · · x x · · · x. . .
......
...x x · · · x
x · · · x. . .
...x
quasiseparable generator representation (O(n) storage)
Chandrasekaran et. al. exploit this structure.
We do it differently.
David S. Watkins Happy Birthday, Volker!
The Upper Triangular Part
R = U + xyT unitary-plus-rank-one, so
R has quasiseparable rank 2.
R =
x · · · x x · · · x. . .
......
...x x · · · x
x · · · x. . .
...x
quasiseparable generator representation (O(n) storage)
Chandrasekaran et. al. exploit this structure.
We do it differently.
David S. Watkins Happy Birthday, Volker!
The Upper Triangular Part
R = U + xyT unitary-plus-rank-one, so
R has quasiseparable rank 2.
R =
x · · · x x · · · x. . .
......
...x x · · · x
x · · · x. . .
...x
quasiseparable generator representation (O(n) storage)
Chandrasekaran et. al. exploit this structure.
We do it differently.
David S. Watkins Happy Birthday, Volker!
The Upper Triangular Part
R = U + xyT unitary-plus-rank-one, so
R has quasiseparable rank 2.
R =
x · · · x x · · · x. . .
......
...x x · · · x
x · · · x. . .
...x
quasiseparable generator representation (O(n) storage)
Chandrasekaran et. al. exploit this structure.
We do it differently.
David S. Watkins Happy Birthday, Volker!
The Upper Triangular Part
R = U + xyT unitary-plus-rank-one, so
R has quasiseparable rank 2.
R =
x · · · x x · · · x. . .
......
...x x · · · x
x · · · x. . .
...x
quasiseparable generator representation (O(n) storage)
Chandrasekaran et. al. exploit this structure.
We do it differently.
David S. Watkins Happy Birthday, Volker!
Our Representation
Add a row/column for extra wiggle room
A =
0 −a0 11 −a1 0
. . ....
...1 −an−1 0
0 0
Extra zero root can be deflated immediately.
A = QR, where
Q =
0 ±1 01 0 0
. . ....
...1 0 0
0 1
R =
1 −a1 0
. . ....
...1 −an−1 0±a0 ∓1
0 0
David S. Watkins Happy Birthday, Volker!
Our Representation
Add a row/column for extra wiggle room
A =
0 −a0 11 −a1 0
. . ....
...1 −an−1 0
0 0
Extra zero root can be deflated immediately.
A = QR, where
Q =
0 ±1 01 0 0
. . ....
...1 0 0
0 1
R =
1 −a1 0
. . ....
...1 −an−1 0±a0 ∓1
0 0
David S. Watkins Happy Birthday, Volker!
Our Representation
Q =
0 ±1 01 0 0
. . ....
...1 0 0
0 1
Q is stored in factored form
Q =��
����
Q = Q1Q2 · · ·Qn−1
David S. Watkins Happy Birthday, Volker!
Our Representation
Q =
0 ±1 01 0 0
. . ....
...1 0 0
0 1
Q is stored in factored form
Q =��
����
Q = Q1Q2 · · ·Qn−1
David S. Watkins Happy Birthday, Volker!
Our Representation
R =
1 −a1 0
. . ....
...1 −an−1 0±a0 ∓1
0 0
R is unitary-plus-rank-one:
1 0 0. . .
......
1 0 00 ∓1
±1 0
+
0 −a1 0
. . ....
...0 −an−1 0±a0 0
∓1 0
David S. Watkins Happy Birthday, Volker!
Representation of R
R = U + xyT , where
xyT =
−a1
...−an−1
±a0∓1
[
0 · · · 0 1 0]
Next step: Roll up x .
David S. Watkins Happy Birthday, Volker!
Representation of R
R = U + xyT , where
xyT =
−a1
...−an−1
±a0∓1
[
0 · · · 0 1 0]
Next step: Roll up x .
David S. Watkins Happy Birthday, Volker!
Representation of R
R = U + xyT , where
xyT =
−a1
...−an−1
±a0∓1
[
0 · · · 0 1 0]
Next step: Roll up x .
David S. Watkins Happy Birthday, Volker!
Representation of R
xxxx
=
xxxx
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Happy Birthday, Volker!
Representation of R
��
xxxx
=
xxx0
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Happy Birthday, Volker!
Representation of R
����
xxxx
=
xx00
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Happy Birthday, Volker!
Representation of R
����
��
xxxx
=
x000
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Happy Birthday, Volker!
Representation of R
����
��
xxxx
=
x000
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Happy Birthday, Volker!
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = C ∗(B + e1yT ) = C ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Happy Birthday, Volker!
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = C ∗(B + e1yT ) = C ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Happy Birthday, Volker!
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = C ∗(B + e1yT ) = C ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Happy Birthday, Volker!
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = C ∗(B + e1yT ) = C ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Happy Birthday, Volker!
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = C ∗(B + e1yT ) = C ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Happy Birthday, Volker!
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R = C ∗(B + e1yT )
B is upper Hessenberg (and unitary)
so B = B1 · · ·Bn.
R = C ∗(B + e1yT ) = C ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Happy Birthday, Volker!
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = C ∗(B + e1yT ) = C ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Happy Birthday, Volker!
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = C ∗(B + e1yT ) = C ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Happy Birthday, Volker!
Representation of R
C1 · · ·Cn−1Cnx = e1
Cx = e1
C ∗e1 = x
R = U + xyT = U + C ∗e1yT = C ∗(CU + e1y
T )
R = C ∗(B + e1yT )
B is upper Hessenberg (and unitary) so B = B1 · · ·Bn.
R = C ∗(B + e1yT ) = C ∗
n · · ·C ∗1 (B1 · · ·Bn + e1y
T )
O(n) storage
Bonus: Redundancy! No need to keep track of y .
David S. Watkins Happy Birthday, Volker!
Representation of A
Altogether we have
A = QR = Q C ∗ (B + e1yT )
A = Q1 · · ·Qn−1 C∗n · · ·C ∗
1 (B1 · · ·Bn + e1yT )
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+ · · ·
David S. Watkins Happy Birthday, Volker!
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 Happy Birthday, Volker!
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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�
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����
David S. Watkins Happy Birthday, Volker!
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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�
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����
David S. Watkins Happy Birthday, Volker!
Two Basic Operations
Two basic operations:
Fusion� �� � ⇒ ��
Turnover (aka shift through, Givens swap, . . . )
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��� �
David S. Watkins Happy Birthday, Volker!
Two Basic Operations
Two basic operations:
Fusion� �� � ⇒ ��
Turnover (aka shift through, Givens swap, . . . )
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��� �
David S. Watkins Happy Birthday, Volker!
Two Basic Operations
Two basic operations:
Fusion� �� � ⇒ ��
Turnover (aka shift through, Givens swap, . . . )
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David S. Watkins Happy Birthday, Volker!
The Bulge Chase
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David S. Watkins Happy Birthday, Volker!
The Bulge Chase
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The Bulge Chase
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The Bulge Chase
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The Bulge Chase
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David S. Watkins Happy Birthday, Volker!
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 Happy Birthday, Volker!
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 Happy Birthday, Volker!
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 Happy Birthday, Volker!
See our papers for . . .
Paper to appear in SIMAX has
. . . timings,
. . . accuracy comparisons,
. . . backward error analysis.
Paper on companion pencils is in progress.
David S. Watkins Happy Birthday, Volker!
See our papers for . . .
Paper to appear in SIMAX has
. . . timings,
. . . accuracy comparisons,
. . . backward error analysis.
Paper on companion pencils is in progress.
David S. Watkins Happy Birthday, Volker!
See our papers for . . .
Paper to appear in SIMAX has
. . . timings,
. . . accuracy comparisons,
. . . backward error analysis.
Paper on companion pencils is in progress.
David S. Watkins Happy Birthday, Volker!
See our papers for . . .
Paper to appear in SIMAX has
. . . timings,
. . . accuracy comparisons,
. . . backward error analysis.
Paper on companion pencils is in progress.
David S. Watkins Happy Birthday, Volker!
See our papers for . . .
Paper to appear in SIMAX has
. . . timings,
. . . accuracy comparisons,
. . . backward error analysis.
Paper on companion pencils is in progress.
David S. Watkins Happy Birthday, Volker!
Summary
We have a new fast method for companion eigenvalueproblems
and unitary-plus-rank-one matrices (or pencils) in general.
Method is normwise backward stable, accurate,
and faster than other fast methods.
Thank you for your attention.
David S. Watkins Happy Birthday, Volker!
Summary
We have a new fast method for companion eigenvalueproblems
and unitary-plus-rank-one matrices (or pencils) in general.
Method is normwise backward stable, accurate,
and faster than other fast methods.
Thank you for your attention.
David S. Watkins Happy Birthday, Volker!
Summary
We have a new fast method for companion eigenvalueproblems
and unitary-plus-rank-one matrices (or pencils) in general.
Method is normwise backward stable, accurate,
and faster than other fast methods.
Thank you for your attention.
David S. Watkins Happy Birthday, Volker!
Summary
We have a new fast method for companion eigenvalueproblems
and unitary-plus-rank-one matrices (or pencils) in general.
Method is normwise backward stable, accurate,
and faster than other fast methods.
Thank you for your attention.
David S. Watkins Happy Birthday, Volker!
Summary
We have a new fast method for companion eigenvalueproblems
and unitary-plus-rank-one matrices (or pencils) in general.
Method is normwise backward stable, accurate,
and faster than other fast methods.
Thank you for your attention.
David S. Watkins Happy Birthday, Volker!
Summary
We have a new fast method for companion eigenvalueproblems
and unitary-plus-rank-one matrices (or pencils) in general.
Method is normwise backward stable, accurate,
and faster than other fast methods.
Thank you for your attention.
David S. Watkins Happy Birthday, Volker!