What are roots of the Wilkinson polynomial

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What are roots of the Wilkinson polynomial?
Zhonggang Zeng
Northeastern Illinois University
May 12, 2001
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Can you solve (x-1.0 )100 0
Can you solve x100-100 x99 4950 x98 - 161700
x973921225x96 - ... - 100 x 1 0
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The Wilkinson polynomial p(x)
(x-1)(x-2)...(x-20) x20 - 210 x19
20615 x18 ...
Wilkinson wrote in 1984 Speaking for
myself I regard it as the most traumatic
experience in my career as a numerical analyst.
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Classical textook methods for multiple roots
Newtons iteration xj1 xj -
f(xj)/f(xj), j0,1,2,... converges locally
to a multiple root of f(x) with a linear rate.
The modified Newtons iteration xj1 xj
- mf(xj)/f(xj), j0,1,2,... converges
locally to a m-fold root of f(x) with a quadratic
rate.
Newtons iteration applied to g(x)
f(x)/f(x) converges locally and quadratically
to a root of f(x) regardliss of its multiplicity.
None of them work!
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Example f(x) (x-2)7(x-3)(x-4) in
expanded form.
Modified Newtons iteration with m 7 intended
for root x 2 x1 1.9981 x2 1.7481 x3
1.9892 x4 0.4726 x5 1.8029 x6 1.9931 x7
4.2681 x8 3.3476 ... ...
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How do we justify the answer?
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The forward error 5
The backward error 5 x 10-10
Conclusion the problem is bad
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If the answer is highly sensitive to
perturbations, you have probably asked the wrong
question. Maxims about numerical mathematics,
computers, science and life, L. N. Trefethen.
SIAM News
A Customer B Numerical analyst
Who is asking a wrong question?
A The polynomial B The computing subject
What is the wrong question?
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The question we used to ask
Given a polynomial p(x) xn a1
xn-1...an-1 x an find ( z1, ..., zn )
such that p(x) ( x - z1 )( x - z2 ) ... ( x -
zn )
Right - or - Wrong ?
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Kahans pejorative manifolds
All n-polynomials having certain multiplicity
structure form a pejorative manifold
xn a1 xn-1...an-1 x an ltgt (a1 , ...,
an-1 , an )
Example ( x-t )2 x2 (-2t)x t2
Pejorative manifold a1 -2t a2 t2
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Pejorative manifolds of 3-polynomials
The edge a1 -3s a2 3s2 a3 -s3
The wings a1 -s-2t a22stt2 a3 -st2
General form of pejorative manifolds u G(z)
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W. Kahan, Conserving confluence curbs
ill-condition, 1972
1. Ill-condition occurs when a polynomial is near
a pejorative manifold.
2. A small drift by a polynomial on that
pejorative manifold does not cause large forward
error to the multiple roots, except
3. If a multiple root is sensitive to small
perturbation on the pejorative manifold, then the
polynomial is near a pejorative submanifold of
higher multiplicity.
Ill-condition is caused by solving
polynomial equations on a wrong manifold
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Given a polynomial p(x) xn a1
xn-1...an-1 x an
Find ( z1, ..., zn ) such that p(x) ( x - z1
)( x - z2 ) ... ( x - zn )
/ / / / / / / / / / / / / / / / /
/ / / / / / / / / / / / / / / / /
/ /
The wrong question
because you are asking for simple roots!
Find ( z1, ..., zm ) such that p(x) ( x - z1 )
s1( x - z2 )s2 ... ( x - zm )sm s1...
sm n, m lt n
The right question
do it on the pejorative manifold!
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For ill-conditioned polynomial p(x) xn
a1 xn-1...an-1 x an a (a1 ,
..., an-1 , an )
The objective find uG(z) that is nearest to
p(x)a
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Let ( x - z1 ) s1( x - z2 )s2 ... ( x - zm
)sm xn g1 ( z1, ..., zm ) xn-1...gn-1 (
z1, ..., zm ) x gn ( z1, ..., zm )
Then, p(x) ( x - z1 ) s1( x - z2 )s2 ...
( x - zm )sm ltgt
g1 ( z1, ..., zm ) a1 g2( z1, ..., zm ) a2 ...
... ... gn ( z1, ..., zm ) an
I.e. An over determined polynomial system G(z)
a
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The Gauss-Newton iteration
zi1zi - J(zi ) G(zi )-a , i0,1,2 ...
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zi1zi - J(zi ) G(zi )-a , i0,1,2 ...
Theorem If z(z1, ..., zm) with z1, ..., zm
distinct, then the Jacobian J(z) of G(z) is of
full rank.
Theorem Let uG(z) be nearest to p(x)a,
if 1. z(z1, ..., zm) with z1, ..., zm
distinct 2. z0 is sufficiently close to
z 3. a is sufficiently close to u then the
Gauss-Newton iteration converges with a linear
rate. Further assume that a u , then
the convergence is quadratic.
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The edge u1 -3s u2 3s2 u3 -s3
The wings u1 -s-2t u22stt2 u3 -st2
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Example p(x) ( x- 0.5)18( x-1.0 )10( x-1.5
)16
The Gauss-Newton iteration x1 x2 x3 ---- ----
---- 0.45 1.05 1.55 0.51 0.86 1.57 0.5002 0.
9983 1.5007 0.4999996 0.999997 1.500002 0.4999999
97 1.00000001 1.499999993
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What are the roots of the Wilkinson polynomial?
(x-1)(x-2)...(x-19)(x-20) (x-z1 )(x-z2 )(x-z3
)2(x-z4 )3(x-z5 )4(x-z6 )4(x-z7 )3(x-z8 )2
Where roots multiplicity 1.00031227
1.98468140 3.36763082
5.99316993 9.29701289
13.85522338 16.66437572
19.84916622
These roots are not sensitive to perturbation
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Conclusion
Ill-condition is caused by a wrong identity.
Multiple roots are stable and can be computed
with high accuracy, if they are calculated on a
proper pejorative manifold.
As a related work, isolated multiple
roots/eigenvalues can be computed as simple,
stable zeros of an extended polynomial system
with high accuracy.