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1. . .
1
−a0−a1...
−an−1
Fast and stable computation of theroots of polynomials
Jared L. Aurentz1, Thomas Mach2, Raf Vandebril2, andDavid S. Watkins1
1Dept. of Mathematics, Washington State University2Dept. Computer Science, KU Leuven
Structured Numerical Linear and Multilinear Algebra:Analysis, Algorithms and Applications11 September 2014
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 1/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Collaborators
This is joint work with
David S. Watkins (WSU)
Jared L. Aurentz (WSU)
Raf Vandebril (KU Leuven)
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 2/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Problem
p(x) = xn + an−1xn−1 + an−2x
n−2 + · · ·+ a0 = 0
Companion matrix
A =
−a01 −a1
1 −a2. . .
...1 −an−2
1 −an−1
Get the zeros of p by computing the eigenvalues of A.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 3/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Problem
p(x) = xn + an−1xn−1 + an−2x
n−2 + · · ·+ a0 = 0
Companion matrix
A =
−a01 −a1
1 −a2. . .
...1 −an−2
1 −an−1
Get the zeros of p by computing the eigenvalues of A.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 3/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Cost
If structure not exploited:
O(n2) storageO(n3) flopsFrancis’s implicitly-shifted QR algorithmMatlab’s roots command.
If structure exploited:
O(n) storageO(n2) flopsdata-sparse representation + Francis’s algorithmseveral methods proposedincluding some by my coauthors (fast but potentially unstable)
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 4/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Cost
If structure not exploited:
O(n2) storageO(n3) flopsFrancis’s implicitly-shifted QR algorithmMatlab’s roots command.
If structure exploited:
O(n) storageO(n2) flopsdata-sparse representation + Francis’s algorithmseveral methods proposedincluding some by my coauthors (fast but potentially unstable)
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 4/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
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 use essentially 2× 2 unitary matrices
results on backward stability
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 5/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Our Contribution
We present
yet another O(n) representation
Francis’s algorithm in O(n) flops/iteration
Fortran codes (we’re faster)
normwise “almost” backward stable (We can prove it.)
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 6/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Structure
companion matrix is unitary-plus-rank-one
A =
0 · · · 0 e iθ
1 0. . .
...1 0
+
0 · · · 0 −e iθ − a00 0 −a1...
......
0 · · · 0 −an−1
preserved by unitary similarities
companion matrix is also upper Hessenberg.
preserved by Francis’s algorithm
We exploit this structure.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 7/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Structure
companion matrix is unitary-plus-rank-one
A =
0 · · · 0 e iθ
1 0. . .
...1 0
+
0 · · · 0 −e iθ − a00 0 −a1...
......
0 · · · 0 −an−1
preserved by unitary similarities
companion matrix is also upper Hessenberg.
preserved by Francis’s algorithm
We exploit this structure.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 7/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Structure
companion matrix is unitary-plus-rank-one
A =
0 · · · 0 e iθ
1 0. . .
...1 0
+
0 · · · 0 −e iθ − a00 0 −a1...
......
0 · · · 0 −an−1
preserved by unitary similarities
companion matrix is also upper Hessenberg.
preserved by Francis’s algorithm
We exploit this structure.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 7/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Structure
Chandrasekaran, Gu, Xia, Zhu (2007)
Compute the companion’s QR factorization
A = QR
Q is upper Hessenberg and unitary.
R is upper triangular and unitary-plus-rank-one.
We do this too.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 8/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Structure
Chandrasekaran, Gu, Xia, Zhu (2007)
Compute the companion’s QR factorization
A = QR
Q is upper Hessenberg and unitary.
R is upper triangular and unitary-plus-rank-one.
We do this too.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 8/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Unitary Part (Matrix Q)
Q factored in core transformations× × × ×× × × ×
× × ×× ×
=
× ×× ×
11
1
× ×× ×
1
1
1× ×× ×
Q =��
����
O(n) storage
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 9/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Upper Triangular Part (Matrix R)
R = U + xyT unitary-plus-rank-one, so
R has quasiseparable rank 2.
R =
× · · · × × · · · ×. . .
......
...× × · · · ×
× · · · ×. . .
...×
quasiseparable generator representation (O(n) storage)
Chandrasekaran, Gu, Xia, and Zhu exploit this structure.
We do it differently.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 10/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Our Representation (Matrix A)
Add a row/column for extra wiggle room
A =
0 −a0 11 −a1 0
. . ....
...1 −an−1 0
0 0
Extra zero root can be deflated immediately.
A = QR, where
Q =
0 ±1 01 0 0
. . ....
...1 0 0
0 1
R =
1 −a1 0
. . ....
...1 −an−1 0
±a0 ∓1
0 0
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 11/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Our Representation (Matrix A)
Add a row/column for extra wiggle room
A =
0 −a0 11 −a1 0
. . ....
...1 −an−1 0
0 0
Extra zero root can be deflated immediately.
A = QR, where
Q =
0 ±1 01 0 0
. . ....
...1 0 0
0 1
R =
1 −a1 0
. . ....
...1 −an−1 0
±a0 ∓1
0 0
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 11/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Our Representation (Matrix Q)
Q =
0 ±1 01 0 0
. . ....
...1 0 0
0 1
Q is stored in factored form
Q =��
����
Q = Q1Q2 · · ·Qn−1 (mark the n − 1 instead of n)
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 12/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Our Representation (Matrix R)
R =
1 −a1 0
. . ....
...1 −an−1 0
±a0 ∓1
0 0
R is unitary-plus-rank-one:
1 0 0. . .
......
1 0 00 ∓1
±1 0
+
0 −a1 0
. . ....
...0 −an−1 0
±a0 0
∓1 0
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 13/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Representation of R
R = U + xyT , where
xyT =
−a1...
−an−1
±a0∓1
[0 · · · 0 1 0
]
Next step: Roll up x .
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 14/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Representation of R
����
��
××××
=
××××
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Representation of R
����
��
××××
=
×××0
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Representation of R
��
����
××××
=
××00
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Representation of R
����
��
××××
=
×000
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Representation of R
����
��
××××
=
×000
C1 · · ·Cn−1Cnx = αe1 (w.l.g. α = 1)
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 15/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Representation of A
Altogether we have
A = QR = Q C ∗ (B + e1yT )
A = Q1 · · ·Qn−1 C∗n · · ·C ∗
1 (B1 · · ·Bn + e1yT )
����
��
Q1 · · ·Qn−1
����
����
C ∗n · · ·C ∗
1
����
����
B = B1 · · ·Bn
+
1
0
0
0
0
e1
× × × × ×
yT
R
.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 16/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Francis’s Iterations
We have complex single-shift code,
real double-shift code.
We describe single-shift case for simplicity
ignoring rank-one part.
Two basic operations:
Fusion� �� � ⇒ ��
Turnover� ��
��
� ⇔ ��
��� �
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 17/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Francis’s Iterations
We have complex single-shift code,
real double-shift code.
We describe single-shift case for simplicity
ignoring rank-one part.
Two basic operations:
Fusion� �� � ⇒ ��
Turnover� ��
��
� ⇔ ��
��� �
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 17/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Francis’s Iterations
We have complex single-shift code,
real double-shift code.
We describe single-shift case for simplicity
ignoring rank-one part.
Two basic operations:
Fusion� �� � ⇒ ��
Turnover� ��
��
� ⇔ ��
��� �
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 17/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ��
����
������ ����
������
�� ������
��
����
��
Q1 · · ·Qn−1
����
����
C ∗n · · ·C ∗
1
����
����
B = B1 · · ·Bn
+
1
0
0
0
0
e1
× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
��
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��
Q1 · · ·Qn−1
����
����
C ∗n · · ·C ∗
1
����
����
B = B1 · · ·Bn
+
1
0
0
0
0
e1
× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ��
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��
Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
����
����
B = B1 · · ·Bn
+
1
0
0
0
0
e1
× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ��
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��
Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
0
0
0
0
e1
× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ����
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
0
0
0
0
e1
× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ����
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
0
0
0
0
e1
× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ������
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
0
0
0
0
e1
× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ������
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
0
0
0
0
e1
× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ������
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
0
0
0
0
e1
× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ������
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
0
0
0
0
e1
× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ������
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
0
0
0
0
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× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ������
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
0
0
0
0
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× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ������
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
0
0
0
0
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× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ������
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
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B = B1 · · ·Bn
+
1
0
0
0
0
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× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ������
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
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0
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× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ������
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
0
0
0
0
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× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ������
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
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B = B1 · · ·Bn
+
1
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0
0
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× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ������
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
0
0
0
0
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× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ������
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Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
0
0
0
0
e1
× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
The Bulge Chase
�� ������
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�� ������
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��
Q1 · · ·Qn−1
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C ∗n · · ·C ∗
1
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B = B1 · · ·Bn
+
1
0
0
0
0
e1
× × × × ×
yT
R
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 18/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Done!
Iteration complete!
Cost: 3n turnovers/iteration, so O(n) flops/iteration
Double-shift iteration is similar:
chase two core transformations instead of one, 7n turnovers/iteration.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 19/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Speed Comparison, Fortran implementation
ContestantsLAPACK code xHSEQR (O(n3))
AMVW (our single-shift code)
BBEGG (Bini, Boito, et al. 2010)
only single-shift code
BEGG (Boito, Eidelman, et al. 2012)
only double-shift code
CGXZ (Chandrasekaran, Gu, et al. 2007)
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 20/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
complex single-shift—random coefficients
AMVW: 4.269
BBEGG: 23.297BEGG: 15.442
101 102 103 104
10−5
10−2
101
Tim
ein
seconds
AMVWZHSEQRBBEGGBEGG
101 102 103 10410−16
10−12
10−8
Accuracy
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 21/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
real double-shift—random coefficients
AMVW: 3.817
BBEGG: 7.899CGXZ: 9.023
101 102 103 104
10−5
10−2
101
Tim
ein
seconds
AMVWDHSEQRBBEGGCGXZ
101 102 103 10410−16
10−12
10−8
Accuracy
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 22/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
single-shift—random aeb, b ∈ [−R ,R], R = 5, . . . , 25
101 102 103 104
10−5
10−2
101
Tim
ein
seconds
AMVWZHSEQRBBEGGBEGG
101 102 103 10410−16
10−12
10−8
Accuracy
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 23/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
double-shift—random aeb, b ∈ [−R ,R], R = 5, . . . , 25
101 102 103 104
10−5
10−2
101
Tim
ein
seconds
AMVWDHSEQRBBEGGCGXZ
101 102 103 10410−16
10−12
10−8
Accuracy
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 24/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
single-shift—zn − 1
AMVW: 3.198
BBEGG: 27.847BEGG: 16.933
101 102 103 104
10−5
10−2
101
Tim
ein
seconds
AMVWZHSEQRBBEGGBEGG
101 102 103 10410−16
10−12
10−8
Accuracy
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 25/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
double-shift—zn − 1
AMVW: 3.554
BBEGG: 6.595CGXZ: 8.883
101 102 103 104
10−5
10−2
101
Tim
ein
seconds
AMVWDHSEQRBBEGGCGXZ
101 102 103 10410−16
10−12
10−8
Accuracy
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 26/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Special Polynomials
Roots known!
No. Description Deg.
1 Wilkinson polynomial 102 Wilkinson polynomial 153 Wilkinson polynomial 204 scaled and shifted Wilkinson poly. 205 reverse Wilkinson polynomial 106 reverse Wilkinson polynomial 157 reverse Wilkinson polynomial 208 prescribed roots of varying scale 209 prescribed roots of varying scale −3 2010 Chebyshev polynomial 2011 z20 + z19 + · · ·+ z + 1 20
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 27/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Special Polynomials (2)
Roots known!
No. Description Deg.
MPSolve12 trv m, C. Traverso 2413 mand31 Mandelbrot example (k = 5) 3114 mand63 Mandelbrot example (k = 6) 63
Vanni Noferini15 polynomial from V. Noferini 1216 polynomial from V. Noferini 35
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 28/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Special Polynomials (3)
Roots known!
No. Description Deg.
Jenkins and Traub17 p1(z) with a =1 e−8 318 p1(z) with a =1 e−15 319 p1(z) with a =1 e+8 320 p1(z) with a =1 e+15 321 p3(z) underflow test 1022 p3(z) underflow test 2023 p10(z) deflation test a = 10 e+3 324 p10(z) deflation test a = 10 e+6 325 p10(z) deflation test a = 10 e+9 326 p11(z) deflation test m = 15 60
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 29/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Special Polynomials (4)
Roots unknown!
No. Description Deg.
27 Bernoulli polynomial (k = 20) 2028 truncated exponential (k = 20) 20
used in Bevilacqua, Del Corso, and Gemignani29–33 p1(z) with m = 10, 20, 30, 256, 512 2m34–38 p2(z) with m = 10, 20, 30, 256, 512 l 2m39–43 p3(z) with m + 1 = 20, . . . , 1024, λ = 0.9 m + 144–48 p3(z) with m + 1 = 20, . . . , 1024, λ = 0.999 m + 1
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 30/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
single-shift, unbalanced, relative backward error
No AMVW ZHSEQR ZGEEV BBEGG BEGG
1 −15 −16 −16 −01 −152 −15 −15 −15 −01 −153 −14 −15 −14 −01 −014 −15 −15 −15 −01 −145 −16 −15 −15 −06 −156 −15 −15 −15 −06 −147 −15 −15 −15 −04 −148 −15 −16 −15 −01 −149 −14 −15 −15 −01 −0110 −15 −15 −15 −01 −1411 −15 −15 −15 −01 −1412 −15 −13 −15 −01 −06
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 31/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
single-shift, unbalanced, relative backward error (2)
No AMVW ZHSEQR ZGEEV BBEGG BEGG
13 −15 −15 −15 −01 −1414 −15 −15 −15 −01 −1415 −15 −15 −12 −01 −0116 −15 −15 −10 −01 −0117 −17 −16 −22 −01 −1718 −17 −16 −30 −01 −1619 −16 −01 −24 +07 −0120 −17 +13 −17 +14 −0121 −16 −16 −18 −03 −1522 −16 −15 −17 −02 −1323 −16 −16 −15 +03 −1624 −16 −16 −16 +06 −15
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 32/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
single-shift, unbalanced, relative backward error (3)
No AMVW ZHSEQR ZGEEV BBEGG BEGG
25 −26 −16 −16 +09 −1126 −15 −15 −14 −01 −1327 −15 −15 −15 −01 −1428 −12 −14 −15 −01 −0329 −15 −15 −15 −01 −1430 −15 −15 −14 −01 −1331 −15 −14 −14 −02 −1332 −13 −13 −13 −01 −1133 −13 −13 −13 −01 −1034 −15 −15 −15 −01 −1435 −15 −15 −15 −02 −1436 −15 −14 −15 −01 −14
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 33/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
single-shift, unbalanced, relative backward error (4)
No AMVW ZHSEQR ZGEEV BBEGG BEGG
37 −14 −14 −14 −02 −1238 −14 −13 −14 −02 −1139 −15 −15 −15 −01 −1440 −15 −14 −14 −01 −1341 −15 −14 −14 −01 −1342 −13 −13 −13 +00 −1143 −13 −13 −13 −01 −1044 −15 −14 −14 +00 −1345 −15 −14 −14 +00 −1346 −15 −14 −14 −01 −1347 −13 −13 −13 +00 −1148 −12 −12 −13 +00 −10
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 34/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
double-shift, unbalanced, relative backward error
No AMVW DHSEQR DGEEV BBEGG CGXZ
1 −15 −15 −15 −09 −112 −15 −15 −15 −01 −063 −14 −15 −15 −01 −014 −15 −15 −15 −13 −145 −16 −16 −16 −15 −156 −15 −15 −16 −02 −147 −15 −15 −15 −02 −158 −15 −15 −15 −01 −149 −15 −15 −15 −01 −0410 −15 −15 −15 −14 −1411 −15 −15 −15 −15 −1512 −06 −12 −15 −01 −01
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 35/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
double-shift, unbalanced, relative backward error (2)
No AMVW DHSEQR DGEEV BBEGG CGXZ
13 −15 −15 −16 −02 −1414 −15 −15 −15 −04 −1315 −15 −15 −10 −01 −1016 −15 −15 −10 −01 −0517 −17 −16 −24 −16 −1618 −18 −17 +00 −16 −1619 −02 +00 −16 −01 −0120 −01 −17 −16 −01 −0121 −17 −16 −17 −15 −0522 −16 −16 −17 −02 −0123 −16 −16 −16 −16 −1624 −14 −16 −15 −15 −15
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 36/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
double-shift, unbalanced, relative backward error (3)
No AMVW DHSEQR DGEEV BBEGG CGXZ
25 −16 −16 −24 −16 −1126 −15 −14 −15 −14 −1427 −15 −15 −15 −11 −1428 −14 −14 −15 −01 −0129 −15 −15 −15 −15 −1530 −15 −14 −14 −15 −1431 −15 −14 −14 −15 −1432 −14 −14 −13 −14 −1133 −13 −13 −13 −13 −0934 −15 −15 −15 −15 −1535 −15 −15 −15 −14 −1536 −15 −14 −15 −15 −15
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 37/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
double-shift, unbalanced, relative backward error (4)
No AMVW DHSEQR DGEEV BBEGG CGXZ
37 −14 −14 −14 −12 −1238 −14 −13 −14 −11 +0739 −15 −14 −15 −15 −1540 −15 −14 −14 −13 −1441 −15 −14 −14 −13 −1442 −14 −13 −13 −11 −1343 −13 −12 −12 −10 −1244 −15 −14 −14 −13 −1445 −15 −14 −14 −13 −1346 −14 −14 −14 −13 −1347 −14 −13 −13 −12 −1248 −13 −12 −12 −10 −12
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 38/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Backward stability
We have a proof forU∗(A+∆A)U = A,
where‖∆A‖2 ≤ ‖coefficients of p(x)‖22O(εm).
The square is annoying, and
in the analysis also a dependency of |a0|.But in the numerical experiments
Tests with small |a0| no problem: as accurate as LAPACK.Tests with growing ‖x‖2 no problem: as accurate as LAPACK.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 39/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Backward stability
We have a proof forU∗(A+∆A)U = A,
where‖∆A‖2 ≤ ‖coefficients of p(x)‖22O(εm).
The square is annoying, and
in the analysis also a dependency of |a0|.
But in the numerical experiments
Tests with small |a0| no problem: as accurate as LAPACK.Tests with growing ‖x‖2 no problem: as accurate as LAPACK.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 39/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Backward stability
We have a proof forU∗(A+∆A)U = A,
where‖∆A‖2 ≤ ‖coefficients of p(x)‖22O(εm).
The square is annoying, and
in the analysis also a dependency of |a0|.But in the numerical experiments
Tests with small |a0| no problem: as accurate as LAPACK.Tests with growing ‖x‖2 no problem: as accurate as LAPACK.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 39/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Conclusion
We have a new fast companion eigenvalue method:
about as accurate as LAPACK,
and more accurate than the other fast methods,
does OK on harder problems
faster than LAPACK from size 16 on,
faster than all other fast methods,
”almost” backward stable (see preprint).
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 40/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Conclusion
We have a new fast companion eigenvalue method:
about as accurate as LAPACK,
and more accurate than the other fast methods,
does OK on harder problems
faster than LAPACK from size 16 on,
faster than all other fast methods,
”almost” backward stable (see preprint).
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 40/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
Conclusion
We have a new fast companion eigenvalue method:
about as accurate as LAPACK,
and more accurate than the other fast methods,
does OK on harder problems
faster than LAPACK from size 16 on,
faster than all other fast methods,
”almost” backward stable (see preprint).
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 40/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
More ...
Preprint: http://www.cs.kuleuven.be/publicaties/rapporten/tw/TW654.abs.html.
Fortran code: http://people.cs.kuleuven.be/~raf.vandebril/homepage/software/companion_qr.php.
Package by Andreas Noack in julia (http://julialang.org/):Pkg.clone("https://github.com/andreasnoackjensen/AMVW.jl")
Pkg.build("AMVW").
Thank you for your attention.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 41/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
More ...
Preprint: http://www.cs.kuleuven.be/publicaties/rapporten/tw/TW654.abs.html.
Fortran code: http://people.cs.kuleuven.be/~raf.vandebril/homepage/software/companion_qr.php.
Package by Andreas Noack in julia (http://julialang.org/):Pkg.clone("https://github.com/andreasnoackjensen/AMVW.jl")
Pkg.build("AMVW").
Thank you for your attention.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 41/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
More ...
Preprint: http://www.cs.kuleuven.be/publicaties/rapporten/tw/TW654.abs.html.
Fortran code: http://people.cs.kuleuven.be/~raf.vandebril/homepage/software/companion_qr.php.
Package by Andreas Noack in julia (http://julialang.org/):Pkg.clone("https://github.com/andreasnoackjensen/AMVW.jl")
Pkg.build("AMVW").
Thank you for your attention.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 41/41
Introduction Factorization & Facts Francis’s Algorithm Numerical experiments
More ...
Preprint: http://www.cs.kuleuven.be/publicaties/rapporten/tw/TW654.abs.html.
Fortran code: http://people.cs.kuleuven.be/~raf.vandebril/homepage/software/companion_qr.php.
Package by Andreas Noack in julia (http://julialang.org/):Pkg.clone("https://github.com/andreasnoackjensen/AMVW.jl")
Pkg.build("AMVW").
Thank you for your attention.
J. Aurentz, T. Mach, R. Vandebril, D. Watkins, Fast and stable computation of the roots of polynomials 41/41
![COMPUTATION OF THE HIGHEST COEFFICIENTS OF … · arxiv:1011.1602v1 [math.co] 7 nov 2010 computation of the highest coefficients of weighted ehrhart quasi-polynomials of rational](https://img.dokumen.tips/doc/110x75/5c66744709d3f230488c20a9/computation-of-the-highest-coefficients-of-arxiv10111602v1-mathco-7-nov.jpg)
