On Computing

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On Computing Multiple Eigenvalues, the
Jordan Structure and their
Sensitivity
Zhonggang Zeng Northeastern Illinois University
Sixth International Workshop on Accurate Solution
of Eigenvalue Problems (IWASEP VI)
May 23, 2006, Penn State Univ
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Objectives
- Multiple eigenvalues
- Staircase decomposition
Jordan decomposition A XJX-1 follows,
if necessary
- Sensitivity
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JCF computing
1966 Kublanovskaya 1970 Varah 1970
Ruhe 1976 Golub Wilkinson 1979 Van
Dooren 1980 Kagstrom Ruhe 1983 Demmel
1987 Demmel Kagstrom 1993 Demmel
Kagstrom 1997 Edelman, Elmroth Kagstrom
Main difficulty accuracy of multiple eigenvalues
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Ill-posed Problem Jordan Canonical Form (JCF)
0 10 3 0 -1 -1 -4 0 0 -5 -5 0 1 0 0
-1 0 -5 -1 0 -3 -1 0 5 9 -1 3 -2 -1 1
1 -2 -2 1 -1 1 1 2 -1 -1 1 0 -1 1 3 1
7 2 -2 -11 1 0 6 -4 -3 6 0 5 -1 0 -3 -2
-1 0 0 0 -1 1 5 2 3 1 -1 0 0 0 0
0 -1 0 1 2 0 0 1 0 -1 1 -4 -2 -9 -2 6
19 -2 0 -8 8 6 -8 1 -7 1 -2 4 4 2 0 0
-1 0 -1 1 -1 1 2 1 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 1 9 -2 4 -3 3 3
1 -2 -2 1 0 1 2 1 -1 -1 1 0 -1 0 1 0
1 0 0 -2 0 3 4 0 0 3 0 2 0 0 -1 0
0 0 0 0 1 -4 -2 0 1 4 1 0 3 5 4 0
-2 0 0 1 0 3 1 0 1 1 -1 1 -2 1 -1
3 -1 -1 -3 3 0 -3 0 -2 -1 0 1 0 0 0 0
0 5 2 6 2 -3 -16 1 0 12 -5 -1 12 0 9 -1
0 -5 -3 -2 0 0 0 -1 4 0 1 -2 -4 -1 0
0 -5 -4 3 4 0 -1 -2 0 -3 -1 0 -1 -2 1 0
1 0 0 -2 0 0 2 0 0 2 3 2 0 0 -1 0
0 0 0 0 0 -1 4 -3 3 -1 1 1 0 0 0 0
-2 3 3 1 0 0 0 0 0 1 0 2 12 -1 2
-7 0 0 2 -4 -3 2 -3 2 4 6 -1 -2 0 0 -1
3 -4 -1 -5 -2 2 12 -1 0 -7 4 3 -7 0 -6 1
3 4 2 1 0 0 0 0 11 8 1 -2 -12 -3 0
6 -9 -8 6 1 5 0 -1 0 -7 -2 0 -3 -1 -2 0
7 -2 5 1 -1 1 -2 0 0 -2 0 -1 1 1 0 4
3 -1 -1 0 3 2 6 2 -2 -7 1 0 2 -5 -4 2
-2 2 0 3 -1 -3 1 2 0 2 5 -12 -10 2 -3
1 5 -1 0 6 6 0 0 0 -2 -1 0 6 0 3 5
0 4 -9 0 1 0 1 4 -1 0 4 4 0 -4 0 0
4 0 4 0 1 6 4 2 0 2 0 0 -3 0 0
3 0 0 3 0 3 0 0 -2 0 0 0 0 3
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Ill-posed problem root-finding
with coefficients in hardware precision
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Are multiple roots/eigenvalues
really sensitive to perturbations?
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pejorative manifold
A two staged algorithm
Determine the multiplicity structure (or manifold)
Reformulate / solve least squares
problem well-posed, and even well conditioned
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For polynomial
with (inexact ) coefficients in machine precision
Stage I results The backward error 6.05
x 10-10 Computed roots
multiplicities 1.000000000000353 20 2.0000000000
30904 15 3.000000000176196 10 4.000000000109542
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ACM TOMS, 2004 Z. Zeng, Algorithm 835 --
MultRoot Math. Comp. 2005 Z. Zeng, Computing
multiple roots of inexact polynomials
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JCF computing For a given matrix A
Jordan form with Segre characteristics
3,2,2,1
find X, J AX XJ
find U, S, l AU U(lIS), UHUI
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The problem is well-posed, may even
well-conditioned
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Gauss-Newton iteration on system
Example A 50x50 matrix with eigenvalues 1, 2, 3
with multiplicity 20, 15, 5
respectively
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A two-stage strategy for computing JCF
Stage I determine the Jordan structure and
an initial approximation
Stage II Solve the reformulated least
squares problem at each eigenvalue, subject to
the structural constraint, using the
Gauss-Newtion iteration.
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eigenvalue Segre characteristics l1 5 3 2 2 l2
3 3 1 l3 2

------------------------------------------
--------------------------------------------------
--------------------------------------------------
------ 10 6
3 2
Let v be a coefficient vector of p1. Then for a
random x
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By a Hessenberg reduction,
with A1 A, and a random x being the first
column of Q,
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Computing multiple roots of inexact polynomials,
Z. Zeng, Math Comp 2005
Rank-revealing
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Example 100x100 matrix A with
multiple eigenvalues 1, -1, 2, -2
50 simple eigenvalues
random
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When matrix A is possibly approximate
Stage I Determine the manifold
(matrix bundle) Stage II Solve the
well-posed least squares problem
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Sensitivity of an eigenvalue (simple or multiple)
Condition number
At a multiple eigenvalue, k(l) infinity by
definition, not by computation
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Example
gtgt eig(A) ans 2.00078250207872
0.00056834208954i 2.00078250207872 -
0.00056834208954i 1.99970128002318
0.00092011990651i 1.99970128002318 -
0.00092011990651i 1.99903243579620
2.00000000000000
gtgt eig(AE) ans 1.99785628939037
1.99891738496176 0.00185657043184i
1.99891738496176 - 0.00185657043184i
2.00107196168888 0.00187502026186i
2.00107196168888 - 0.00187502026186i
2.00216501730835
gtgt norm((xy')/(x'y)) ans 14.04290360613186
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Example
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What is the minimum perturbation E such
that an eigenvalue of A increases its
multiplicity?
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Wanted a measurement of sensitivity

• reliable in identifying a multiple eigenvalue

• can be estimated/computed efficiently

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Example
gtgt eig(A) ans 2.00078250207872
0.00056834208954i 2.00078250207872 -
0.00056834208954i 1.99970128002318
0.00092011990651i 1.99970128002318 -
0.00092011990651i 1.99903243579620
2.00000000000000
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Sensitivity of an m-fold eigenvalue
After obtaining an m-fold eigenvalue l in
staircase decomposition
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Example 12x12 Frank matrix
Eigenvalue Condition number
0.03102744447368 18281388.1
0.04950888583700 38773448.2
0.08122648015585 26647311.8
0.14364690493016 6701300.4
0.28474967198534 560311.0
0.64350532103739 14466.8
1.55398870911557 216.1
3.51185594858010 6.9
6.96153308556711 1.7
12.31107740086854 3.1
20.19898864587709 5.0
32.22889150157213 3.3
gtgt F frank(12) F 12 11 10 9 8
7 6 5 4 3 2 1 11 11
10 9 8 7 6 5 4 3 2 1
0 10 10 9 8 7 6 5 4 3
2 1 0 0 9 9 8 7 6
5 4 3 2 1 0 0 0 8 8
7 6 5 4 3 2 1 0 0
0 0 7 7 6 5 4 3 2 1
0 0 0 0 0 6 6 5 4 3
2 1 0 0 0 0 0 0 5
5 4 3 2 1 0 0 0 0 0
0 0 4 4 3 2 1 0 0
0 0 0 0 0 0 3 3 2 1
0 0 0 0 0 0 0 0 0 2
2 1 0 0 0 0 0 0 0
0 0 0 1 1
R
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Frank matrix is near matrices of an eigenvalue
with multiplicity
m 2, l 0.03864934375529, distance
3.9e-12, t2(l) 1.34e07
m 3, l 0.05043386836727, distance
4.7e-10, t3(l) 2.73e05
m 4, l 0.07030194271661, distance
3.9e-08, t4(l) 9.04e03
m 5, l 0.10767505748315, distance
2.1e-06, t5(l) 5.40e02
m 6, l 0.18704749378810, distance
7.1e-05, t5(l) 6.48e01
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Condition estimator
- Approximate
(or power iteration)
- QR decomposition
Q R
- Inverse iterations
- Condition number
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