Happy Birthday, Volker! David S. Watkins Department of Mathematics - - PowerPoint PPT Presentation

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Happy Birthday, Volker! David S. Watkins Department of Mathematics - - PowerPoint PPT Presentation

Happy Birthday, Volker! David S. Watkins Department of Mathematics Washington State University Berlin, May, 2015 David S. Watkins Happy Birthday, Volker! Contribution from Michael Overton David S. Watkins Happy Birthday, Volker!


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Happy Birthday, Volker!

David S. Watkins

Department of Mathematics Washington State University

Berlin, May, 2015

David S. Watkins Happy Birthday, Volker!

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Contribution from Michael Overton

David S. Watkins Happy Birthday, Volker!

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Contribution from Michael Overton

Volker Mehrmann

David S. Watkins Happy Birthday, Volker!

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Contribution from Michael Overton

Volker Mehrmann Author or coauthor of more than 160 scientific articles

David S. Watkins Happy Birthday, Volker!

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Contribution from Michael Overton

Volker Mehrmann Author or coauthor of more than 160 scientific articles Author or coauthor of 5 monographs/textbooks

David S. Watkins Happy Birthday, Volker!

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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

David S. Watkins Happy Birthday, Volker!

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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

David S. Watkins Happy Birthday, Volker!

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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

David S. Watkins Happy Birthday, Volker!

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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

David S. Watkins Happy Birthday, Volker!

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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

David S. Watkins Happy Birthday, Volker!

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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

David S. Watkins Happy Birthday, Volker!

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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!

David S. Watkins Happy Birthday, Volker!

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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 . . .

David S. Watkins Happy Birthday, Volker!

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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!

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Kaziranga National Park (Thanks to Shreemayee Bora)

David S. Watkins Happy Birthday, Volker!

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and now moving back in time . . .

David S. Watkins Happy Birthday, Volker!

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1986

David S. Watkins Happy Birthday, Volker!

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1986

David S. Watkins Happy Birthday, Volker!

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A question

David S. Watkins Happy Birthday, Volker!

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A question

An engineer,

David S. Watkins Happy Birthday, Volker!

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A question

An engineer, an algebraist,

David S. Watkins Happy Birthday, Volker!

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A question

An engineer, an algebraist, and Volker . . .

David S. Watkins Happy Birthday, Volker!

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A question

An engineer, an algebraist, and Volker . . . . . . walk into a bar.

David S. Watkins Happy Birthday, Volker!

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A question

An engineer, an algebraist, and Volker . . . . . . walk into a bar. Engineer:

David S. Watkins Happy Birthday, Volker!

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A question

An engineer, an algebraist, and Volker . . . . . . walk into a bar. Engineer: I have this interesting problem where I need to find the roots of polynomials of high degree.

David S. Watkins Happy Birthday, Volker!

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A question

An engineer, an algebraist, and Volker . . . . . . walk into a bar. Engineer: I have this interesting problem where I need to find the roots of polynomials of high degree. Algebraist:

David S. Watkins Happy Birthday, Volker!

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SLIDE 27

A question

An engineer, an algebraist, and Volker . . . . . . walk into a bar. Engineer: I have this interesting problem where I need to find the roots of polynomials of high degree. Algebraist: What a coincidence! I have a problem just like that.

David S. Watkins Happy Birthday, Volker!

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A question

An engineer, an algebraist, and Volker . . . . . . walk into a bar. Engineer: I have this interesting problem where I need to find the roots of polynomials of high degree. Algebraist: What a coincidence! I have a problem just like that. Question:

David S. Watkins Happy Birthday, Volker!

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A question

An engineer, an algebraist, and Volker . . . . . . walk into a bar. Engineer: I have this interesting problem where I need to find the roots of polynomials of high degree. Algebraist: What a coincidence! I have a problem just like that. Question: What does Volker say?

David S. Watkins Happy Birthday, Volker!

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Show me the eigenvalue problem!

David S. Watkins Happy Birthday, Volker!

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Show me the eigenvalue problem!

Nevertheless,

David S. Watkins Happy Birthday, Volker!

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Show me the eigenvalue problem!

Nevertheless, there is a demand for the product.

David S. Watkins Happy Birthday, Volker!

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Show me the eigenvalue problem!

Nevertheless, there is a demand for the product. MATLAB roots

David S. Watkins Happy Birthday, Volker!

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Show me the eigenvalue problem!

Nevertheless, there is a demand for the product. MATLAB roots (companion matrix)

David S. Watkins Happy Birthday, Volker!

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SLIDE 35

Show me the eigenvalue problem!

Nevertheless, there is a demand for the product. MATLAB roots (companion matrix) Chebfun roots

David S. Watkins Happy Birthday, Volker!

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SLIDE 36

Show me the eigenvalue problem!

Nevertheless, there is a demand for the product. MATLAB roots (companion matrix) Chebfun roots (colleague matrix)

David S. Watkins Happy Birthday, Volker!

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SLIDE 37

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,

David S. Watkins Happy Birthday, Volker!

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SLIDE 38

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.

David S. Watkins Happy Birthday, Volker!

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SLIDE 39

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!

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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!

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MATLAB

p(x) = xn + an−1xn−1 + an−2xn−2 + · · · + a0 = 0 monic polynomial

David S. Watkins Happy Birthday, Volker!

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MATLAB

p(x) = xn + an−1xn−1 + an−2xn−2 + · · · + a0 = 0 monic polynomial companion matrix A =         · · · −a0 1 · · · −a1 1 ... . . . . . . ... −an−2 1 −an−1         balance, then . . .

David S. Watkins Happy Birthday, Volker!

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MATLAB

p(x) = xn + an−1xn−1 + an−2xn−2 + · · · + a0 = 0 monic polynomial companion matrix A =         · · · −a0 1 · · · −a1 1 ... . . . . . . ... −an−2 1 −an−1         balance, then . . . . . . get the zeros of p by computing the eigenvalues.

David S. Watkins Happy Birthday, Volker!

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MATLAB

p(x) = xn + an−1xn−1 + an−2xn−2 + · · · + a0 = 0 monic polynomial companion matrix A =         · · · −a0 1 · · · −a1 1 ... . . . . . . ... −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!

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Chebfun

p(x) = Tn(x) + bn−1Tn−1(x) + · · · b0T0(x) Chebyshev polynomials

David S. Watkins Happy Birthday, Volker!

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Chebfun

p(x) = Tn(x) + bn−1Tn−1(x) + · · · b0T0(x) Chebyshev polynomials colleague matrix

David S. Watkins Happy Birthday, Volker!

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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!

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What we’ve been doing

David S. Watkins Happy Birthday, Volker!

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What we’ve been doing

companion matrix or

David S. Watkins Happy Birthday, Volker!

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What we’ve been doing

companion matrix or companion pencil p(x) = anxn + an−1xn−1 + an−2xn−2 + · · · + a0        · · · −a0 1 · · · −a1 1 ... . . . . . . −an−2 1 −an−1        − λ         1 · · · 1 · · · ... . . . . . . ... 1 an        

David S. Watkins Happy Birthday, Volker!

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What we’ve been doing

companion matrix or companion pencil p(x) = anxn + an−1xn−1 + an−2xn−2 + · · · + a0        · · · −a0 1 · · · −a1 1 ... . . . . . . −an−2 1 −an−1        − λ         1 · · · 1 · · · ... . . . . . . ... 1 an         . . . and variants.

David S. Watkins Happy Birthday, Volker!

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What we’ve been doing

companion matrix or companion pencil p(x) = anxn + an−1xn−1 + an−2xn−2 + · · · + a0        · · · −a0 1 · · · −a1 1 ... . . . . . . −an−2 1 −an−1        − λ         1 · · · 1 · · · ... . . . . . . ... 1 an         . . . and variants. Today we restrict attention to the companion matrix.

David S. Watkins Happy Birthday, Volker!

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Cost of solving companion eigenvalue problem

David S. Watkins Happy Birthday, Volker!

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Cost of solving companion eigenvalue problem

If structure not exploited:

O(n2) storage, O(n3) flops Francis’s implicitly-shifted QR algorithm

David S. Watkins Happy Birthday, Volker!

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Cost of solving companion eigenvalue problem

If structure not exploited:

O(n2) storage, O(n3) flops Francis’s implicitly-shifted QR algorithm

If structure exploited:

O(n) storage, O(n2) flops data-sparse representation + Francis’s algorithm

David S. Watkins Happy Birthday, Volker!

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Cost of solving companion eigenvalue problem

If structure not exploited:

O(n2) storage, O(n3) flops Francis’s implicitly-shifted QR algorithm

If structure exploited:

O(n) storage, O(n2) flops data-sparse representation + Francis’s algorithm Several methods proposed

David S. Watkins Happy Birthday, Volker!

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Some of the Competitors

Chandrasekaran, Gu, Xia, Zhu (2007) Bini, Boito, Eidelman, Gemignani, Gohberg (2010) Boito, Eidelman, Gemignani, Gohberg (2012)

David S. Watkins Happy Birthday, Volker!

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SLIDE 58

Some of the Competitors

Chandrasekaran, Gu, Xia, Zhu (2007) Bini, Boito, Eidelman, Gemignani, Gohberg (2010) Boito, Eidelman, Gemignani, Gohberg (2012) Fortran codes available

David S. Watkins Happy Birthday, Volker!

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SLIDE 59

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

David S. Watkins Happy Birthday, Volker!

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SLIDE 60

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

David S. Watkins Happy Birthday, Volker!

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SLIDE 61

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!

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SLIDE 62

Our Contribution

We present Yet another O(n) representation

David S. Watkins Happy Birthday, Volker!

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SLIDE 63

Our Contribution

We present Yet another O(n) representation Francis algorithm in O(n) flops/iteration

David S. Watkins Happy Birthday, Volker!

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SLIDE 64

Our Contribution

We present Yet another O(n) representation Francis algorithm in O(n) flops/iteration Fortran codes (we’re faster)

David S. Watkins Happy Birthday, Volker!

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SLIDE 65

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!

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SLIDE 66

Structure

Companion matrix is unitary-plus-rank-one      · · · eiθ 1 ... . . . 1      +      · · · −eiθ − a0 −a1 . . . . . . . . . · · · −an−1     

David S. Watkins Happy Birthday, Volker!

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SLIDE 67

Structure

Companion matrix is unitary-plus-rank-one      · · · eiθ 1 ... . . . 1      +      · · · −eiθ − a0 −a1 . . . . . . . . . · · · −an−1      preserved by unitary similarities

David S. Watkins Happy Birthday, Volker!

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SLIDE 68

Structure

Companion matrix is unitary-plus-rank-one      · · · eiθ 1 ... . . . 1      +      · · · −eiθ − a0 −a1 . . . . . . . . . · · · −an−1      preserved by unitary similarities Companion matrix is also upper Hessenberg.

David S. Watkins Happy Birthday, Volker!

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SLIDE 69

Structure

Companion matrix is unitary-plus-rank-one      · · · eiθ 1 ... . . . 1      +      · · · −eiθ − a0 −a1 . . . . . . . . . · · · −an−1      preserved by unitary similarities Companion matrix is also upper Hessenberg. preserved by Francis algorithm

David S. Watkins Happy Birthday, Volker!

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SLIDE 70

Structure

Companion matrix is unitary-plus-rank-one      · · · eiθ 1 ... . . . 1      +      · · · −eiθ − a0 −a1 . . . . . . . . . · · · −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!

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SLIDE 71

Structure

Chandrasekaran, Gu, Xia, Zhu (2007) A = QR

David S. Watkins Happy Birthday, Volker!

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SLIDE 72

Structure

Chandrasekaran, Gu, Xia, Zhu (2007) A = QR Q is upper Hessenberg and unitary. R is upper triangular and unitary-plus-rank-one.

David S. Watkins Happy Birthday, Volker!

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SLIDE 73

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!

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The Unitary Part

    x x x x x x x x x x x x x     =     x x x x 1 1         1 x x x x 1         1 1 x x x x    

David S. Watkins Happy Birthday, Volker!

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SLIDE 75

The Unitary Part

    x x x x x x x x x x x x x     =     x x x x 1 1         1 x x x x 1         1 1 x x x x     Q =

  • David S. Watkins

Happy Birthday, Volker!

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SLIDE 76

The Unitary Part

    x x x x x x x x x x x x x     =     x x x x 1 1         1 x x x x 1         1 1 x x x x     Q =

  • O(n) storage

David S. Watkins Happy Birthday, Volker!

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SLIDE 77

The Upper Triangular Part

R = U + xyT unitary-plus-rank-one, so

David S. Watkins Happy Birthday, Volker!

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SLIDE 78

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          

David S. Watkins Happy Birthday, Volker!

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SLIDE 79

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)

David S. Watkins Happy Birthday, Volker!

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SLIDE 80

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.

David S. Watkins Happy Birthday, Volker!

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SLIDE 81

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!

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SLIDE 82

Our Representation

Add a row/column for extra wiggle room A =        −a0 1 1 −a1 ... . . . . . . 1 −an−1        Extra zero root can be deflated immediately.

David S. Watkins Happy Birthday, Volker!

slide-83
SLIDE 83

Our Representation

Add a row/column for extra wiggle room A =        −a0 1 1 −a1 ... . . . . . . 1 −an−1        Extra zero root can be deflated immediately. A = QR, where Q =        ±1 1 ... . . . . . . 1 1        R =        1 −a1 ... . . . . . . 1 −an−1 ±a0 ∓1       

David S. Watkins Happy Birthday, Volker!

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SLIDE 84

Our Representation

Q =        ±1 1 ... . . . . . . 1 1        Q is stored in factored form Q =

  • David S. Watkins

Happy Birthday, Volker!

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SLIDE 85

Our Representation

Q =        ±1 1 ... . . . . . . 1 1        Q is stored in factored form Q =

  • Q = Q1Q2 · · · Qn−1

David S. Watkins Happy Birthday, Volker!

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SLIDE 86

Our Representation

R =        1 −a1 ... . . . . . . 1 −an−1 ±a0 ∓1        R is unitary-plus-rank-one:        1 ... . . . . . . 1 ∓1 ±1        +        −a1 ... . . . . . . −an−1 ±a0 ∓1       

David S. Watkins Happy Birthday, Volker!

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SLIDE 87

Representation of R

R = U + xyT, where

David S. Watkins Happy Birthday, Volker!

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SLIDE 88

Representation of R

R = U + xyT, where xyT =        −a1 . . . −an−1 ±a0 ∓1       

  • · · ·

1

  • David S. Watkins

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SLIDE 89

Representation of R

R = U + xyT, where xyT =        −a1 . . . −an−1 ±a0 ∓1       

  • · · ·

1

  • Next step: Roll up x.

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SLIDE 90

Representation of R

   x x x x    =    x x x x   

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SLIDE 91

Representation of R

  x x x x    =    x x x   

David S. Watkins Happy Birthday, Volker!

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SLIDE 92

Representation of R

  x x x x    =    x x   

David S. Watkins Happy Birthday, Volker!

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SLIDE 93

Representation of R

  x x x x    =    x   

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SLIDE 94

Representation of R

  x x x x    =    x    C1 · · · Cn−1Cnx = αe1 (w.l.g. α = 1)

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SLIDE 95

Representation of R

C1 · · · Cn−1Cnx = e1

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SLIDE 96

Representation of R

C1 · · · Cn−1Cnx = e1 Cx = e1

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SLIDE 97

Representation of R

C1 · · · Cn−1Cnx = e1 Cx = e1 C ∗e1 = x

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SLIDE 98

Representation of R

C1 · · · Cn−1Cnx = e1 Cx = e1 C ∗e1 = x R = U + xyT = U + C ∗e1yT = C ∗(CU + e1yT)

David S. Watkins Happy Birthday, Volker!

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SLIDE 99

Representation of R

C1 · · · Cn−1Cnx = e1 Cx = e1 C ∗e1 = x R = U + xyT = U + C ∗e1yT = C ∗(CU + e1yT) R = C ∗(B + e1yT)

David S. Watkins Happy Birthday, Volker!

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SLIDE 100

Representation of R

C1 · · · Cn−1Cnx = e1 Cx = e1 C ∗e1 = x R = U + xyT = U + C ∗e1yT = C ∗(CU + e1yT) R = C ∗(B + e1yT) B is upper Hessenberg (and unitary)

David S. Watkins Happy Birthday, Volker!

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SLIDE 101

Representation of R

C1 · · · Cn−1Cnx = e1 Cx = e1 C ∗e1 = x R = U + xyT = U + C ∗e1yT = C ∗(CU + e1yT) R = C ∗(B + e1yT) B is upper Hessenberg (and unitary) so B = B1 · · · Bn.

David S. Watkins Happy Birthday, Volker!

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SLIDE 102

Representation of R

C1 · · · Cn−1Cnx = e1 Cx = e1 C ∗e1 = x R = U + xyT = U + C ∗e1yT = C ∗(CU + e1yT) R = C ∗(B + e1yT) B is upper Hessenberg (and unitary) so B = B1 · · · Bn. R = C ∗(B + e1yT) = C ∗

n · · · C ∗ 1 (B1 · · · Bn + e1yT)

David S. Watkins Happy Birthday, Volker!

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SLIDE 103

Representation of R

C1 · · · Cn−1Cnx = e1 Cx = e1 C ∗e1 = x R = U + xyT = U + C ∗e1yT = C ∗(CU + e1yT) R = C ∗(B + e1yT) B is upper Hessenberg (and unitary) so B = B1 · · · Bn. R = C ∗(B + e1yT) = C ∗

n · · · C ∗ 1 (B1 · · · Bn + e1yT)

O(n) storage Bonus: Redundancy! No need to keep track of y.

David S. Watkins Happy Birthday, Volker!

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SLIDE 104

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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SLIDE 105

Francis Iterations

We have complex single-shift code . . . real double-shift code.

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SLIDE 106

Francis Iterations

We have complex single-shift code . . . real double-shift code. We describe single-shift case for simplicity.

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SLIDE 107

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

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SLIDE 108

Two Basic Operations

Two basic operations:

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SLIDE 109

Two Basic Operations

Two basic operations: Fusion

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SLIDE 110

Two Basic Operations

Two basic operations: Fusion

  • Turnover

(aka shift through, Givens swap, . . . )

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SLIDE 111

The Bulge Chase

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SLIDE 112

The Bulge Chase

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SLIDE 113

The Bulge Chase

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SLIDE 114

The Bulge Chase

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SLIDE 115

The Bulge Chase

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SLIDE 116

The Bulge Chase

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SLIDE 117

The Bulge Chase

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SLIDE 118

The Bulge Chase

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slide-119
SLIDE 119

The Bulge Chase

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slide-120
SLIDE 120

The Bulge Chase

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SLIDE 121

The Bulge Chase

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SLIDE 122

The Bulge Chase

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SLIDE 123

The Bulge Chase

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slide-124
SLIDE 124

The Bulge Chase

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SLIDE 125

The Bulge Chase

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SLIDE 126

The Bulge Chase

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SLIDE 127

The Bulge Chase

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SLIDE 128

The Bulge Chase

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slide-129
SLIDE 129

The Bulge Chase

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slide-130
SLIDE 130

The Bulge Chase

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slide-131
SLIDE 131

The Bulge Chase

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slide-132
SLIDE 132

The Bulge Chase

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slide-133
SLIDE 133

The Bulge Chase

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slide-134
SLIDE 134

The Bulge Chase

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slide-135
SLIDE 135

The Bulge Chase

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slide-136
SLIDE 136

The Bulge Chase

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slide-137
SLIDE 137

The Bulge Chase

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slide-138
SLIDE 138

The Bulge Chase

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slide-139
SLIDE 139

The Bulge Chase

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SLIDE 140

Done!

iteration complete!

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SLIDE 141

Done!

iteration complete! Cost: 3n turnovers/iteration, so O(n) flops/iteration

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SLIDE 142

Done!

iteration complete! Cost: 3n turnovers/iteration, so O(n) flops/iteration Double-shift iteration is similar. (Chase two core transformations instead of one.)

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SLIDE 143

See our papers for . . .

Paper to appear in SIMAX has

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SLIDE 144

See our papers for . . .

Paper to appear in SIMAX has . . . timings,

David S. Watkins Happy Birthday, Volker!

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SLIDE 145

See our papers for . . .

Paper to appear in SIMAX has . . . timings, . . . accuracy comparisons,

David S. Watkins Happy Birthday, Volker!

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SLIDE 146

See our papers for . . .

Paper to appear in SIMAX has . . . timings, . . . accuracy comparisons, . . . backward error analysis.

David S. Watkins Happy Birthday, Volker!

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SLIDE 147

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!

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SLIDE 148

Summary

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SLIDE 149

Summary

We have a new fast method for companion eigenvalue problems

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SLIDE 150

Summary

We have a new fast method for companion eigenvalue problems and unitary-plus-rank-one matrices (or pencils) in general.

David S. Watkins Happy Birthday, Volker!

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SLIDE 151

Summary

We have a new fast method for companion eigenvalue problems and unitary-plus-rank-one matrices (or pencils) in general. Method is normwise backward stable, accurate,

David S. Watkins Happy Birthday, Volker!

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SLIDE 152

Summary

We have a new fast method for companion eigenvalue problems and unitary-plus-rank-one matrices (or pencils) in general. Method is normwise backward stable, accurate, and faster than other fast methods.

David S. Watkins Happy Birthday, Volker!

slide-153
SLIDE 153

Summary

We have a new fast method for companion eigenvalue problems 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!