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Fast Stable Computation of the Eigenvalues of Companion Matrices - - PowerPoint PPT Presentation

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Fast Stable Computation of the Eigenvalues of Companion Matrices David S. Watkins Department of Mathematics Washington State University Spa, Belgium, 2014 David S. Watkins Eigenvalues of Companion Matrices Collaborators This is joint work

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SLIDE 1

Fast Stable Computation of the Eigenvalues

  • f Companion Matrices

David S. Watkins

Department of Mathematics Washington State University

Spa, Belgium, 2014

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 2

Collaborators

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 3

Collaborators

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 4

The Problem

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 5

The Problem

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 6

Cost

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 including some by us (lightning fast but not stable)

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 7

Cost

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 including some by us (lightning fast but not stable)

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 8

Cost

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 including some by us (lightning fast but not stable)

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 9

Cost

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 including some by us (lightning fast but not stable)

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 10

Cost

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 including some by us (lightning fast but not stable)

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 11

Some of the Competitors

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 12

Some of the Competitors

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 13

Some of the Competitors

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 14

Some of the Competitors

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 15

Some of the Competitors

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 16

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 Eigenvalues of Companion Matrices

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SLIDE 17

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 Eigenvalues of Companion Matrices

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SLIDE 18

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 Eigenvalues of Companion Matrices

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SLIDE 19

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 Eigenvalues of Companion Matrices

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SLIDE 20

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 Eigenvalues of Companion Matrices

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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 Eigenvalues of Companion Matrices

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SLIDE 22

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 Eigenvalues of Companion Matrices

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SLIDE 23

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 Eigenvalues of Companion Matrices

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SLIDE 24

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 Eigenvalues of Companion Matrices

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SLIDE 25

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 Eigenvalues of Companion Matrices

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SLIDE 26

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 Eigenvalues of Companion Matrices

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SLIDE 27

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 Eigenvalues of Companion Matrices

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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     Q =

  • O(n) storage

David S. Watkins Eigenvalues of Companion Matrices

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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     Q =

  • O(n) storage

David S. Watkins Eigenvalues of Companion Matrices

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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     Q =

  • O(n) storage

David S. Watkins Eigenvalues of Companion Matrices

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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 Eigenvalues of Companion Matrices

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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 Eigenvalues of Companion Matrices

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SLIDE 33

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 Eigenvalues of Companion Matrices

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SLIDE 34

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 Eigenvalues of Companion Matrices

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SLIDE 35

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 Eigenvalues of Companion Matrices

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SLIDE 36

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 Eigenvalues of Companion Matrices

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SLIDE 37

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 Eigenvalues of Companion Matrices

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Our Representation

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

  • Q = Q1Q2 · · · Qn−1

David S. Watkins Eigenvalues of Companion Matrices

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Our Representation

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

  • Q = Q1Q2 · · · Qn−1

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 40

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 Eigenvalues of Companion Matrices

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Representation of R

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

  • · · ·

1

  • Next step: Roll up x.

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 42

Representation of R

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

  • · · ·

1

  • Next step: Roll up x.

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 43

Representation of R

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

  • · · ·

1

  • Next step: Roll up x.

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 44

Representation of R

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 45

Representation of R

  x x x x    =    x x x   

David S. Watkins Eigenvalues of Companion Matrices

slide-46
SLIDE 46

Representation of R

  x x x x    =    x x   

David S. Watkins Eigenvalues of Companion Matrices

slide-47
SLIDE 47

Representation of R

  x x x x    =    x   

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 48

Representation of R

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 49

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 Eigenvalues of Companion Matrices

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SLIDE 50

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 Eigenvalues of Companion Matrices

slide-51
SLIDE 51

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 Eigenvalues of Companion Matrices

slide-52
SLIDE 52

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 Eigenvalues of Companion Matrices

slide-53
SLIDE 53

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 Eigenvalues of Companion Matrices

slide-54
SLIDE 54

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 Eigenvalues of Companion Matrices

slide-55
SLIDE 55

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 Eigenvalues of Companion Matrices

slide-56
SLIDE 56

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 Eigenvalues of Companion Matrices

slide-57
SLIDE 57

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 Eigenvalues of Companion Matrices

slide-58
SLIDE 58

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 Eigenvalues of Companion Matrices

slide-59
SLIDE 59

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

Eigenvalues of Companion Matrices

slide-60
SLIDE 60

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

Eigenvalues of Companion Matrices

slide-61
SLIDE 61

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

Eigenvalues of Companion Matrices

slide-62
SLIDE 62

Two Basic Operations

Two basic operations: Fusion

  • ⇒
  • Turnover

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

  • ⇔
  • David S. Watkins

Eigenvalues of Companion Matrices

slide-63
SLIDE 63

Two Basic Operations

Two basic operations: Fusion

  • ⇒
  • Turnover

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

  • ⇔
  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 64

Two Basic Operations

Two basic operations: Fusion

  • ⇒
  • Turnover

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

  • ⇔
  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 65

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 66

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 67

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 68

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 69

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 70

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 71

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 72

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 73

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 74

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 75

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 76

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 77

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 78

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 79

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 80

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 81

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 82

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 83

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 84

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 85

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 86

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 87

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 88

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 89

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 90

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 91

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 92

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 93

The Bulge Chase

  • David S. Watkins

Eigenvalues of Companion Matrices

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SLIDE 94

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 Eigenvalues of Companion Matrices

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SLIDE 95

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 Eigenvalues of Companion Matrices

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SLIDE 96

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 Eigenvalues of Companion Matrices

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SLIDE 97

Speed Comparison, Complex Case

Contestants

LAPACK code ZHSEQR (O(n3)) BEGG (Boito et. al. 2012) AMVW (our single-shift code)

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 98

Speed Comparison, Complex Case

10 10

1

10

2

10

3

10

4

10

5

10

−5

10

−4

10

−3

10

−2

10

−1

10 10

1

10

2

10

3

degree time (seconds) LAPACK BEGG AMVW David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 99

Speed Comparison, Complex Case

At degree 1024: method time LAPACK 7.14 BEGG 1.02 AMVW 0.18

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 100

Speed Comparison, Real Case

Results similar for double-shift code . . . . . . but we’re only about twice as fast as the competition. (compared with Chandrasekaran et. al.)

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 101

Accuracy

On these big (and well conditioned) problems . . . we are about as accurate as LAPACK, . . . and more accurate than the other fast methods. Our codes act backward stable . . . because they are backward stable. much more in paper (almost done) also tried harder problems (we do OK)

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 102

Accuracy

On these big (and well conditioned) problems . . . we are about as accurate as LAPACK, . . . and more accurate than the other fast methods. Our codes act backward stable . . . because they are backward stable. much more in paper (almost done) also tried harder problems (we do OK)

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 103

Accuracy

On these big (and well conditioned) problems . . . we are about as accurate as LAPACK, . . . and more accurate than the other fast methods. Our codes act backward stable . . . because they are backward stable. much more in paper (almost done) also tried harder problems (we do OK)

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 104

Accuracy

On these big (and well conditioned) problems . . . we are about as accurate as LAPACK, . . . and more accurate than the other fast methods. Our codes act backward stable . . . because they are backward stable. much more in paper (almost done) also tried harder problems (we do OK)

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 105

Accuracy

On these big (and well conditioned) problems . . . we are about as accurate as LAPACK, . . . and more accurate than the other fast methods. Our codes act backward stable . . . because they are backward stable. much more in paper (almost done) also tried harder problems (we do OK)

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 106

Summary

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 107

Summary

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 108

Summary

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 109

Summary

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 110

Summary

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

David S. Watkins Eigenvalues of Companion Matrices

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SLIDE 111

Summary

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

Thank you for your attention.

David S. Watkins Eigenvalues of Companion Matrices