1. Computing ill-Conditioned Eigenvalues
and Polynomial Roots
Zhonggang Zeng
Northeastern Illinois University
International Conference on Matrix Theory and its Applications -- Shanghai

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

3. Eigenvalues of
1 1
A = X
X-1

4. 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.

5. Myths on multiple eigenvalues/roots:
- multiple e’values/roots are ill-conditioned, or even intractable
- extension of machine precision is necessary to
calculate multiple roots
- there is an “attainable precision” for multiple
eigenvalues/roots:
machine precision
attainable precision =
multiplicity
Example: for a 100-fold eigenvalue, to get 5 digits right
500 digits in machine precision
5 digits precision =
100 in multiplicity

6. Conclusion: the problem is “bad”
The forward error:
-- Ouch! Who’s responsible?
The backward error: x 10-10
-- method is good!
Conclusion: the problem is “bad”

7. 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 or matrix
B: The computing objective
What is the wrong question?

8. 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

9. Pejorative manifolds of 3-polynomials
( x - s )( x - t )2 = x3 + (-s-2t) x2 + (2st+t2) x + (-st2)
a1= -s-2t
a2= 2st+t2
a3= -st2
Pejorative manifold of
multiplicity structure [1,2]
( x - s )3 = x3 + (-3s) x2 + (3s2) x + (-s3)
a1 = -3s
a2 = 3s2
a3 = -s3
Pejorative manifold of
multiplicity structure [ 3 ]

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

11. Ill-condition is caused by solving polynomial
W. Kahan, Conserving confluence curbs ill-condition, 1972
1. Ill-condition occurs when a polynomial/matrix is near a
pejorative manifold.
2. A small “drift” of the problem on that pejorative manifold does not cause large forward error to the multiple roots, except
3. If a multiple root/eigenvalue is sensitive to small perturbation
on the pejorative manifold, then the polynomial/matrix is near a
pejorative submanifold of higher multiplicity.
Ill-condition is caused by solving polynomial
equations on a wrong manifold

12. Pejorative manifolds of 3-polynomials
The wings:
a1= -s-2t
a2= 2st+t2
a3= -st2
The edge:
a1 = -3s
a2 = 3s2
a3 = -s3

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 distinct 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 n
(m<n) m
I.e. An over determined polynomial system
G(z) = a

16. Project to tangent plane
The polynomial a
Project to tangent plane
u1 = G(z0)+J(z0)(z1- z0) ~
tangent plane P0 :
u = G(z0)+J(z0)(z- z0)
pejorative root
u*=G(z* )
initial iterate
u0=G(z0)
new iterate
u1=G(z1)
Pejorative manifold
u = G( z )
Solve
G(z0)+J(z0)( z - z0 ) = a
for linear least squares solution z = z1
Solve
G( z ) = a
for nonlinear least squares solution z=z*
G(z0)+J(z0)( z - z0 ) = a
J(z0)( z - z0 ) = - [G(z0) - a ]
z1 = z0 - [J(z0)+] [G(z0) - a]

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 iteration converges with a linear rate.
Further assume that a = u* , then the convergence is quadratic.

18. The “pejorative” condition number
v = G(z)
u = G(y)
|| u - v ||2 = backward error
|| y - z ||2 = forward error
u - v = G(y) - G(z) = J(z) (y - z) + h.o.t.
|| u - v ||2 = || J(z) (y - z) ||2 > s || y - z ||2
|| y - z ||2 < (1/s) || u - v ||2
1/s is the pejorative condition number
where s is the smallest singular value of J(z) .

19. Example (x-0.9)18(x-1.0)10(x-1.1)16 = 0
Step z1 z2 z3
forward error: 6 x 10-15
backward error: 8 x 10-16
Pejorative condition: 58
Even clustered multiple roots are pejoratively well conditioned

20. Roots are correct up to 7 digits!
Example (x-.3-.6i)100 (x-.1-.7i) 200 (x i) 300 (x-.3-.4i) 400 =0
Scary enough? Round coefficients to 6 digits.
Z z z3 z4
i i i i
i i i i
i i i i
i i i i
i i i i
Roots are correct up to 7 digits!
Pejorative condition: 0.58

21. What are the roots of the Wilkinson polynomial?
Example: The Wilkinson polynomial
p(x) = (x-1)(x-2)...(x-20) = x x x
There are 605 manifolds in total.
It is near some manifolds, but which ones?
Multiplicity backward error condition Estimated
structure number error
[1,1,1,1,1,1,1,1,1...,1]
[1,1,1,1,2,2,2,4,2,2,2]
[1,1,1,2,3,4,5,3]
[1,1,2,3,4,6,3]
[1,1,2,5,7,4]
[1,2,5,7,5]
[1,3,8,8]
[2,8,10]
[5,15]
[20]
What are the roots of the Wilkinson polynomial?
Choose your poison!

22. The “right” question for ill-conditioned eigenproblem
Given a matrix A
Find a structured Schur form S and a matrix U such that
AU - US = 0
U*U - I = 0
Over-determined!!!
3 1 3
l + l
A ~ m 2
S =
m + m
2 1 2
Minimize || AU - US ||F2 + || U*U - I ||F2
---- nonlinear least squares problem

23. + O(10 -7) The pejorative condition number: 22.8
Example: A + E, where ||E|| A 1.0e-7
Step l m
3 + 3
UT(A+E)U =
+ O(10 -7) 2
2 + 2
The pejorative condition number:

24. Conclusion 1. Ill-condition is cause by a wrong “identity”
2. Multiple eigenvalues/roots are pejoratively
well conditioned, thereby tractable.
3. Extension of machine precision is NOT needed,
a change in computing concept is.
4. To calculate ill-conditioned eigenvalues/roots, one
has to figure out the pejorative structure (how?)