1. Northeastern Illinois University
What are roots of the Wilkinson polynomial?
Zhonggang Zeng
Northeastern Illinois University
May 12, 2001

2. Can you solve (x-1.0 )100 = 0
Can you solve
x x x x x x +1 = 0

3. The Wilkinson polynomial
p(x) = (x-1)(x-2)...(x-20)
= x x x
Wilkinson wrote in 1984:
Speaking for myself I regard it as the most traumatic
experience in my career as a numerical analyst.

4. Classical textook methods for multiple roots
Newton’s iteration
xj+1 = xj - f(xj)/f’(xj), j=0,1,2,...
converges locally to a multiple root of f(x) with a linear rate.
The modified Newton’s iteration
xj+1 = xj - mf(xj)/f’(xj), j=0,1,2,...
converges locally to a m-fold root of f(x) with a quadratic rate.
Newton’s 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!

5. Example: f(x) = (x-2)7(x-3)(x-4) in expanded form.
Modified Newton’s iteration with m = 7
intended for root x = 2:
x1 =
x2 =
x3 =
x4 =
x5 =
x6 =
x7 =
x8 =

6. How do we justify the answer?

7. Conclusion: the problem is “bad”
The forward error:
The backward error: x 10-10
Conclusion: the problem is “bad”

8. Who is asking a wrong question? What is the wrong question?
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?

9. The question we used to ask:
Given a polynomial
p(x) = xn + a1 xn an-1 x + an
find ( z1, ..., zn ) such that
p(x) = ( x - z1 )( x - z2 ) ... ( x - zn )
Right or Wrong ?

10. Kahan’s pejorative manifolds
All n-polynomials having certain multiplicity
structure form a pejorative manifold
xn + a1 xn an-1 x + an <=> (a1 , ..., an-1 , an )
Example: ( x-t )2 = x2 + (-2t)x + t2
Pejorative manifold:
a1= -2t
a2= t2

11. Pejorative manifolds of 3-polynomials
The edge:
a1 = -3s
a2 = 3s2
a3 = -s3
The wings:
a1= -s-2t
a2=2st+t2
a3= -st2
General form of
pejorative manifolds
u = G(z)

12. Ill-condition is caused by solving polynomial
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

13. 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
s sm = n, m < n
The right question:
do it on the pejorative manifold!

14. For ill-conditioned polynomial
p(x)= xn + a1 xn an-1 x + an ~ a = (a1 , ..., an-1 , an )
The objective: find u*=G(z*) that is nearest to p(x)~a

15. I.e. An over determined polynomial system
Let ( x - z1 ) s1( x - z2 )s2 ... ( x - zm )sm =
xn + g1 ( z1, ..., zm ) xn gn-1 ( z1, ..., zm ) x + gn ( z1, ..., zm )
Then, p(x) = ( x - z1 ) s1( x - z2 )s2 ... ( x - zm )sm <==>
g1 ( z1, ..., zm ) =a1
g2( z1, ..., zm ) =a2
gn ( z1, ..., zm ) =an
I.e. An over determined polynomial system
G(z) = a

16. zi+1=zi - J(zi )+[ G(zi )-a ], i=0,1,2 ...
The Gauss-Newton iteration
zi+1=zi - J(zi )+[ G(zi )-a ], i=0,1,2 ...

17. zi+1=zi - J(zi )+[ G(zi )-a ], i=0,1,2 ...
Theorem: If z=(z1, ..., zm) with z1, ..., zm distinct, then the Jacobian
J(z) of G(z) is of full rank.
Theorem: Let u*=G(z*) be nearest to p(x)~a, if
1. z*=(z*1, ..., z*m) with z*1, ..., z*m 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.

18. The edge:
u1 = -3s
u2 = 3s2
u3 = -s3
The wings:
u1= -s-2t
u2=2st+t2
u3= -st2

19. Example: p(x) = ( x- 0.5)18( x-1.0 )10( x-1.5 )16
The Gauss-Newton iteration:
x1 x2 x3

20. 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
These roots are not sensitive to perturbation

21. 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.