happy birthday, volker!happy birthday, volker! david s. watkins department of mathematics washington...
TRANSCRIPT
![Page 1: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/1.jpg)
Happy Birthday, Volker!
David S. Watkins
Department of MathematicsWashington State University
Berlin, May, 2015
David S. Watkins Happy Birthday, Volker!
![Page 2: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/2.jpg)
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!
![Page 3: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/3.jpg)
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!
![Page 4: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/4.jpg)
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!
![Page 5: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/5.jpg)
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!
![Page 6: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/6.jpg)
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!
![Page 7: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/7.jpg)
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!
![Page 8: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/8.jpg)
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!
![Page 9: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/9.jpg)
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!
![Page 10: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/10.jpg)
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!
![Page 11: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/11.jpg)
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!
![Page 12: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/12.jpg)
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!
![Page 13: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/13.jpg)
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!
![Page 14: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/14.jpg)
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!
![Page 15: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/15.jpg)
Kaziranga National Park (Thanks to Shreemayee Bora)
David S. Watkins Happy Birthday, Volker!
![Page 16: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/16.jpg)
and now moving back in time . . .
David S. Watkins Happy Birthday, Volker!
![Page 17: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/17.jpg)
1986
David S. Watkins Happy Birthday, Volker!
![Page 18: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/18.jpg)
1986
David S. Watkins Happy Birthday, Volker!
![Page 19: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/19.jpg)
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!
![Page 20: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/20.jpg)
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!
![Page 21: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/21.jpg)
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!
![Page 22: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/22.jpg)
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!
![Page 23: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/23.jpg)
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!
![Page 24: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/24.jpg)
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!
![Page 25: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/25.jpg)
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!
![Page 26: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/26.jpg)
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!
![Page 27: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/27.jpg)
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!
![Page 28: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/28.jpg)
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!
![Page 29: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/29.jpg)
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!
![Page 30: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/30.jpg)
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!
![Page 31: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/31.jpg)
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!
![Page 32: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/32.jpg)
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!
![Page 33: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/33.jpg)
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!
![Page 34: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/34.jpg)
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!
![Page 35: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/35.jpg)
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!
![Page 36: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/36.jpg)
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!
![Page 37: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/37.jpg)
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!
![Page 38: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/38.jpg)
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!
![Page 39: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/39.jpg)
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!
![Page 40: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/40.jpg)
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!
![Page 41: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/41.jpg)
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!
![Page 42: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/42.jpg)
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!
![Page 43: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/43.jpg)
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!
![Page 44: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/44.jpg)
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!
![Page 45: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/45.jpg)
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!
![Page 46: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/46.jpg)
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!
![Page 47: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/47.jpg)
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!
![Page 48: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/48.jpg)
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!
![Page 49: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/49.jpg)
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!
![Page 50: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/50.jpg)
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!
![Page 51: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/51.jpg)
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!
![Page 52: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/52.jpg)
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!
![Page 53: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/53.jpg)
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!
![Page 54: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/54.jpg)
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!
![Page 55: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/55.jpg)
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!
![Page 56: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/56.jpg)
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!
![Page 57: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/57.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.
David S. Watkins Happy Birthday, Volker!
![Page 58: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/58.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.
David S. Watkins Happy Birthday, Volker!
![Page 59: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/59.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.
David S. Watkins Happy Birthday, Volker!
![Page 60: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/60.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.
David S. Watkins Happy Birthday, Volker!
![Page 61: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/61.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.
David S. Watkins Happy Birthday, Volker!
![Page 62: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/62.jpg)
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!
![Page 63: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/63.jpg)
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!
![Page 64: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/64.jpg)
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!
![Page 65: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/65.jpg)
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!
![Page 66: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/66.jpg)
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!
![Page 67: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/67.jpg)
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!
![Page 68: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/68.jpg)
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!
![Page 69: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/69.jpg)
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!
![Page 70: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/70.jpg)
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!
![Page 71: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/71.jpg)
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!
![Page 72: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/72.jpg)
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!
![Page 73: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/73.jpg)
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!
![Page 74: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/74.jpg)
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!
![Page 75: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/75.jpg)
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!
![Page 76: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/76.jpg)
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!
![Page 77: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/77.jpg)
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!
![Page 78: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/78.jpg)
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!
![Page 79: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/79.jpg)
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!
![Page 80: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/80.jpg)
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!
![Page 81: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/81.jpg)
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!
![Page 82: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/82.jpg)
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!
![Page 83: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/83.jpg)
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!
![Page 84: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/84.jpg)
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!
![Page 85: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/85.jpg)
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!
![Page 86: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/86.jpg)
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!
![Page 87: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/87.jpg)
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!
![Page 88: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/88.jpg)
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!
![Page 89: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/89.jpg)
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!
![Page 90: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/90.jpg)
Representation of R
xxxx
=
xxxx
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Happy Birthday, Volker!
![Page 91: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/91.jpg)
Representation of R
��
xxxx
=
xxx0
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Happy Birthday, Volker!
![Page 92: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/92.jpg)
Representation of R
����
xxxx
=
xx00
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Happy Birthday, Volker!
![Page 93: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/93.jpg)
Representation of R
����
��
xxxx
=
x000
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Happy Birthday, Volker!
![Page 94: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/94.jpg)
Representation of R
����
��
xxxx
=
x000
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
David S. Watkins Happy Birthday, Volker!
![Page 95: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/95.jpg)
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!
![Page 96: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/96.jpg)
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!
![Page 97: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/97.jpg)
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!
![Page 98: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/98.jpg)
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!
![Page 99: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/99.jpg)
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!
![Page 100: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/100.jpg)
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!
![Page 101: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/101.jpg)
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!
![Page 102: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/102.jpg)
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!
![Page 103: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/103.jpg)
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!
![Page 104: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/104.jpg)
Representation of A
Altogether we have
A = QR = Q C ∗ (B + e1yT )
A = Q1 · · ·Qn−1 C∗n · · ·C ∗
1 (B1 · · ·Bn + e1yT )
����
��
��
�
��
��
�
��
����
��
+ · · ·
David S. Watkins Happy Birthday, Volker!
![Page 105: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/105.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 =
����
��
��
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 106: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/106.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 =
����
��
��
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 107: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/107.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 =
����
��
��
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 108: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/108.jpg)
Two Basic Operations
Two basic operations:
Fusion� �� � ⇒ ��
Turnover (aka shift through, Givens swap, . . . )
� ���
�� ⇔
��
��� �
David S. Watkins Happy Birthday, Volker!
![Page 109: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/109.jpg)
Two Basic Operations
Two basic operations:
Fusion� �� � ⇒ ��
Turnover (aka shift through, Givens swap, . . . )
� ���
�� ⇔
��
��� �
David S. Watkins Happy Birthday, Volker!
![Page 110: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/110.jpg)
Two Basic Operations
Two basic operations:
Fusion� �� � ⇒ ��
Turnover (aka shift through, Givens swap, . . . )
� ���
�� ⇔
��
��� �
David S. Watkins Happy Birthday, Volker!
![Page 111: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/111.jpg)
The Bulge Chase
����
��
��
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 112: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/112.jpg)
The Bulge Chase
� �� ���
��
��
�
��
��
�
� ���
��
����
David S. Watkins Happy Birthday, Volker!
![Page 113: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/113.jpg)
The Bulge Chase
� �� ���
��
��
�
��
��
�
� ���
��
����
David S. Watkins Happy Birthday, Volker!
![Page 114: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/114.jpg)
The Bulge Chase
����
��
��
�
��
��
�
� ���
��
����
David S. Watkins Happy Birthday, Volker!
![Page 115: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/115.jpg)
The Bulge Chase
����
��
��
�
��
��
�
� ���
��
����
David S. Watkins Happy Birthday, Volker!
![Page 116: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/116.jpg)
The Bulge Chase
����
��
��
�
��
��
�
��
��� �
����
David S. Watkins Happy Birthday, Volker!
![Page 117: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/117.jpg)
The Bulge Chase
����
��
��
�
��
��
�
��
��� �
����
David S. Watkins Happy Birthday, Volker!
![Page 118: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/118.jpg)
The Bulge Chase
����
��
��
��
�� �
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 119: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/119.jpg)
The Bulge Chase
����
��
��
��
�� �
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 120: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/120.jpg)
The Bulge Chase
����
��
� ���
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 121: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/121.jpg)
The Bulge Chase
����
��
� ���
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 122: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/122.jpg)
The Bulge Chase
� ���
��
��
��
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 123: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/123.jpg)
The Bulge Chase
� ���
��
��
��
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 124: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/124.jpg)
The Bulge Chase
��
��� �
��
��
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 125: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/125.jpg)
The Bulge Chase
��
��� �
��
��
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 126: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/126.jpg)
The Bulge Chase
����
��
��
�
��
��
�
��� ��
��
���
David S. Watkins Happy Birthday, Volker!
![Page 127: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/127.jpg)
The Bulge Chase
����
��
��
�
��
��
�
���
��
�� ���
David S. Watkins Happy Birthday, Volker!
![Page 128: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/128.jpg)
The Bulge Chase
����
��
��
�
��
��
� ��
����
����
David S. Watkins Happy Birthday, Volker!
![Page 129: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/129.jpg)
The Bulge Chase
����
��
�� �
��
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 130: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/130.jpg)
The Bulge Chase
��� ��
��
�
��
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 131: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/131.jpg)
The Bulge Chase
���
��
�� �
��
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 132: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/132.jpg)
The Bulge Chase
����
��
��
�
��
��
�
����
� ���
��
David S. Watkins Happy Birthday, Volker!
![Page 133: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/133.jpg)
The Bulge Chase
����
��
��
�
��
��
�
����
��
��� �
David S. Watkins Happy Birthday, Volker!
![Page 134: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/134.jpg)
The Bulge Chase
����
��
��
�
��
��
�� �
����
����
David S. Watkins Happy Birthday, Volker!
![Page 135: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/135.jpg)
The Bulge Chase
����
��
��
�
� ��
��
��
����
����
David S. Watkins Happy Birthday, Volker!
![Page 136: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/136.jpg)
The Bulge Chase
����
� �� �
��
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 137: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/137.jpg)
The Bulge Chase
����
� �� �
��
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 138: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/138.jpg)
The Bulge Chase
����
��
��
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 139: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/139.jpg)
The Bulge Chase
����
��
��
�
��
��
�
����
����
David S. Watkins Happy Birthday, Volker!
![Page 140: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/140.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 Happy Birthday, Volker!
![Page 141: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/141.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 Happy Birthday, Volker!
![Page 142: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/142.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 Happy Birthday, Volker!
![Page 143: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/143.jpg)
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!
![Page 144: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/144.jpg)
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!
![Page 145: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/145.jpg)
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!
![Page 146: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/146.jpg)
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!
![Page 147: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/147.jpg)
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!
![Page 148: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/148.jpg)
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!
![Page 149: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/149.jpg)
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!
![Page 150: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/150.jpg)
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!
![Page 151: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/151.jpg)
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!
![Page 152: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/152.jpg)
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!
![Page 153: Happy Birthday, Volker!Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution](https://reader034.vdocuments.mx/reader034/viewer/2022042910/5f408f104e9d8a03d36cf5ee/html5/thumbnails/153.jpg)
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!