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!

Contribution from Michael Overton

Volker Mehrmann

David S. Watkins Happy Birthday, Volker!

Contribution from Michael Overton

Volker Mehrmann Author or coauthor of more than 160 scientific articles

David S. Watkins Happy Birthday, Volker!

Contribution from Michael Overton

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

David S. Watkins Happy Birthday, Volker!

Contribution from Michael Overton

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

David S. Watkins Happy Birthday, Volker!

Contribution from Michael Overton

Volker Mehrmann Author or coauthor of more than 160 scientific articles Author or coauthor of 5 monographs/textbooks Coeditor of 5 books Co-editor-in-chief of Linear Algebra and its Applications

David S. Watkins Happy Birthday, Volker!

Contribution from Michael Overton

Volker Mehrmann Author or coauthor of more than 160 scientific articles Author or coauthor of 5 monographs/textbooks Coeditor of 5 books Co-editor-in-chief of Linear Algebra and its Applications Member of the German Academy of Engineering

David S. Watkins Happy Birthday, Volker!

Contribution from Michael Overton

Volker Mehrmann Author or coauthor of more than 160 scientific articles Author or coauthor of 5 monographs/textbooks Coeditor of 5 books Co-editor-in-chief of Linear Algebra and its Applications Member of the German Academy of Engineering Recent President of GAMM

David S. Watkins Happy Birthday, Volker!

Contribution from Michael Overton

Volker Mehrmann Author or coauthor of more than 160 scientific articles Author or coauthor of 5 monographs/textbooks Coeditor of 5 books Co-editor-in-chief of Linear Algebra and its Applications Member of the German Academy of Engineering Recent President of GAMM Recent Director of MATHEON

David S. Watkins Happy Birthday, Volker!

Contribution from Michael Overton

Volker Mehrmann Author or coauthor of more than 160 scientific articles Author or coauthor of 5 monographs/textbooks Coeditor of 5 books Co-editor-in-chief of Linear Algebra and its Applications Member of the German Academy of Engineering Recent President of GAMM Recent Director of MATHEON An outstanding scientist with broad interests and knowledge

David S. Watkins Happy Birthday, Volker!

Contribution from Michael Overton

Volker Mehrmann Author or coauthor of more than 160 scientific articles Author or coauthor of 5 monographs/textbooks Coeditor of 5 books Co-editor-in-chief of Linear Algebra and its Applications Member of the German Academy of Engineering Recent President of GAMM Recent Director of MATHEON An outstanding scientist with broad interests and knowledge A superlatively nice person who always has time for everyone!

David S. Watkins Happy Birthday, Volker!

Contribution from Michael Overton

Volker Mehrmann Author or coauthor of more than 160 scientific articles Author or coauthor of 5 monographs/textbooks Coeditor of 5 books Co-editor-in-chief of Linear Algebra and its Applications Member of the German Academy of Engineering Recent President of GAMM Recent Director of MATHEON An outstanding scientist with broad interests and knowledge A superlatively nice person who always has time for everyone! A good person to be with in a jeep . . .

David S. Watkins Happy Birthday, Volker!

Contribution from Michael Overton

Volker Mehrmann Author or coauthor of more than 160 scientific articles Author or coauthor of 5 monographs/textbooks Coeditor of 5 books Co-editor-in-chief of Linear Algebra and its Applications Member of the German Academy of Engineering Recent President of GAMM Recent Director of MATHEON An outstanding scientist with broad interests and knowledge A superlatively nice person who always has time for everyone! A good person to be with in a jeep . . . when an elephant is charging it!!

David S. Watkins Happy Birthday, Volker!

Kaziranga National Park (Thanks to Shreemayee Bora)

David S. Watkins Happy Birthday, Volker!

and now moving back in time . . .

David S. Watkins Happy Birthday, Volker!

1986

David S. Watkins Happy Birthday, Volker!

1986

David S. Watkins Happy Birthday, Volker!

A question

David S. Watkins Happy Birthday, Volker!

A question

An engineer,

David S. Watkins Happy Birthday, Volker!

A question

An engineer, an algebraist,

David S. Watkins Happy Birthday, Volker!

A question

An engineer, an algebraist, and Volker . . .

David S. Watkins Happy Birthday, Volker!

A question

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

David S. Watkins Happy Birthday, Volker!

A question

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

David S. Watkins Happy Birthday, Volker!

A question

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

David S. Watkins Happy Birthday, Volker!

A question

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

David S. Watkins Happy Birthday, Volker!

A question

An engineer, an algebraist, and Volker . . . . . . walk into a bar. Engineer: I have this interesting problem where I need to 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!

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!

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!

Show me the eigenvalue problem!

David S. Watkins Happy Birthday, Volker!

Show me the eigenvalue problem!

Nevertheless,

David S. Watkins Happy Birthday, Volker!

Show me the eigenvalue problem!

Nevertheless, there is a demand for the product.

David S. Watkins Happy Birthday, Volker!

Show me the eigenvalue problem!

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

David S. Watkins Happy Birthday, Volker!

Show me the eigenvalue problem!

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

David S. Watkins Happy Birthday, Volker!

Show me the eigenvalue problem!

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

David S. Watkins Happy Birthday, Volker!

Show me the eigenvalue problem!

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

David S. Watkins Happy Birthday, Volker!

Show me the eigenvalue problem!

Nevertheless, there is a demand for the product. MATLAB roots (companion matrix) Chebfun roots (colleague matrix) I never thought I would get caught up in this racket,

David S. Watkins Happy Birthday, Volker!

Show me the eigenvalue problem!

Nevertheless, there is a demand for the product. MATLAB roots (companion matrix) Chebfun roots (colleague matrix) I never thought I would get caught up in this racket, . . . but somehow I got sucked in.

David S. Watkins Happy Birthday, Volker!

Show me the eigenvalue problem!

Nevertheless, there is a demand for the product. MATLAB roots (companion matrix) Chebfun roots (colleague matrix) I never thought I would get caught up in this racket, . . . but somehow I got sucked in. ... and we’ve done some good stuff.

David S. Watkins Happy Birthday, Volker!

Our International Research Group

This is joint work with Jared Aurentz (Oxford) Thomas Mach (KU Leuven) Raf Vandebril (KU Leuven)

David S. Watkins Happy Birthday, Volker!

MATLAB

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

David S. Watkins Happy Birthday, Volker!

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!

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!

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!

Chebfun

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

David S. Watkins Happy Birthday, Volker!

Chebfun

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

David S. Watkins Happy Birthday, Volker!

Chebfun

p(x) = Tn(x) + bn−1Tn−1(x) + · · · b0T0(x) Chebyshev polynomials colleague matrix This is sometimes better.

David S. Watkins Happy Birthday, Volker!

What we’ve been doing

David S. Watkins Happy Birthday, Volker!

What we’ve been doing

companion matrix or

David S. Watkins Happy Birthday, Volker!

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!

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!

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!

Cost of solving companion eigenvalue problem

David S. Watkins Happy Birthday, Volker!

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!

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!

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!

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!

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!

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!

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!

Some of the Competitors

Chandrasekaran, Gu, Xia, Zhu (2007) Bini, Boito, Eidelman, Gemignani, Gohberg (2010) Boito, Eidelman, Gemignani, Gohberg (2012) Fortran codes available evidence of backward stability quasiseparable generator representation We will do something else.

David S. Watkins Happy Birthday, Volker!

Our Contribution

We present Yet another O(n) representation

David S. Watkins Happy Birthday, Volker!

Our Contribution

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

David S. Watkins Happy Birthday, Volker!

Our Contribution

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

David S. Watkins Happy Birthday, Volker!

Our Contribution

We present Yet another O(n) representation Francis algorithm in O(n) flops/iteration Fortran codes (we’re faster) normwise backward stable (We can prove it.)

David S. Watkins Happy Birthday, Volker!

Structure

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

David S. Watkins Happy Birthday, Volker!

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!

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!

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!

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!

Structure

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

David S. Watkins Happy Birthday, Volker!

Structure

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

David S. Watkins Happy Birthday, Volker!

Structure

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

David S. Watkins Happy Birthday, Volker!

The Unitary Part

    x x x 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!

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!

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!

The Upper Triangular Part

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

David S. Watkins Happy Birthday, Volker!

The Upper Triangular Part

R = U + xyT unitary-plus-rank-one, so R has quasiseparable rank 2. R =           x · · · x x · · · x ... . . . . . . . . . x x · · · x x · · · x ... . . . x          

David S. Watkins Happy Birthday, Volker!

The Upper Triangular Part

R = U + xyT unitary-plus-rank-one, so R has quasiseparable rank 2. R =           x · · · x x · · · x ... . . . . . . . . . x x · · · x x · · · x ... . . . x           quasiseparable generator representation (O(n) storage)

David S. Watkins Happy Birthday, Volker!

The Upper Triangular Part

R = U + xyT unitary-plus-rank-one, so R has quasiseparable rank 2. R =           x · · · x x · · · x ... . . . . . . . . . x x · · · x x · · · x ... . . . x           quasiseparable generator representation (O(n) storage) Chandrasekaran et. al. exploit this structure.

David S. Watkins Happy Birthday, Volker!

The Upper Triangular Part

R = U + xyT unitary-plus-rank-one, so R has quasiseparable rank 2. R =           x · · · x x · · · x ... . . . . . . . . . x x · · · x x · · · x ... . . . x           quasiseparable generator representation (O(n) storage) Chandrasekaran et. al. exploit this structure. We do it differently.

David S. Watkins Happy Birthday, Volker!

Our Representation

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

David S. Watkins Happy Birthday, Volker!

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!

Our Representation

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

• David S. Watkins

Happy Birthday, Volker!

Our Representation

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

• Q = Q1Q2 · · · Qn−1

David S. Watkins Happy Birthday, Volker!

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!

Representation of R

R = U + xyT, where

David S. Watkins Happy Birthday, Volker!

Representation of R

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

• · · ·

1

• David S. Watkins

Happy Birthday, Volker!

Representation of R

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

• · · ·

1

• Next step: Roll up x.

David S. Watkins Happy Birthday, Volker!

Representation of R

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

David S. Watkins Happy Birthday, Volker!

Representation of R

• 

  x x x x    =    x x x   

David S. Watkins Happy Birthday, Volker!

Representation of R

• 

  x x x x    =    x x   

David S. Watkins Happy Birthday, Volker!

Representation of R

• 

  x x x x    =    x   

David S. Watkins Happy Birthday, Volker!

Representation of R

• 

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

David S. Watkins Happy Birthday, Volker!

Representation of R

C1 · · · Cn−1Cnx = e1

David S. Watkins Happy Birthday, Volker!

Representation of R

C1 · · · Cn−1Cnx = e1 Cx = e1

David S. Watkins Happy Birthday, Volker!

Representation of R

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

David S. Watkins Happy Birthday, Volker!

Representation of R

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

David S. Watkins Happy Birthday, Volker!

Representation of R

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

David S. Watkins Happy Birthday, Volker!

Representation of R

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

David S. Watkins Happy Birthday, Volker!

Representation of R

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

David S. Watkins Happy Birthday, Volker!

Representation of R

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

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!

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!

Francis Iterations

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

David S. Watkins Happy Birthday, Volker!

Francis Iterations

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

David S. Watkins Happy Birthday, Volker!

Francis Iterations

We have complex single-shift code . . . real double-shift code. We describe single-shift case for simplicity. ignoring rank-one part . . . A =

• David S. Watkins

Happy Birthday, Volker!

Two Basic Operations

Two basic operations:

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Two Basic Operations

Two basic operations: Fusion

• ⇒
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Two Basic Operations

Two basic operations: Fusion

• ⇒
• Turnover

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

• ⇔
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The Bulge Chase

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The Bulge Chase

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The Bulge Chase

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The Bulge Chase

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The Bulge Chase

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

• David S. Watkins

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The Bulge Chase

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The Bulge Chase

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The Bulge Chase

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Done!

iteration complete!

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Done!

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

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Done!

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

David S. Watkins Happy Birthday, Volker!

See our papers for . . .

Paper to appear in SIMAX has

David S. Watkins Happy Birthday, Volker!

See our papers for . . .

Paper to appear in SIMAX has . . . timings,

David S. Watkins Happy Birthday, Volker!

See our papers for . . .

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

David S. Watkins Happy Birthday, Volker!

See our papers for . . .

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

David S. Watkins Happy Birthday, Volker!

See our papers for . . .

Paper to appear in SIMAX has . . . timings, . . . accuracy comparisons, . . . backward error analysis. Paper on companion pencils is in progress.

David S. Watkins Happy Birthday, Volker!

Summary

David S. Watkins Happy Birthday, Volker!

Summary

We have a new fast method for companion eigenvalue problems

David S. Watkins Happy Birthday, Volker!

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!

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!

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!

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!