Eigenvalue coalescence for parameter dependent matrices

Transcript

Eigenvalue coalescence for
parameter dependent matrices
Alessandro Pugliese
Università degli Studi di Bari
Joint work with Luca Dieci (Georgia Inst.
of Tech.)
Seminar @ BCAM, Bilbao – February 14, 2011
Alessandro Pugliese (Università degli Studi di Bari)
Eigenvalue coalescence
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Based on:
L. Dieci, A. P., “Two-parameter SVD: Coalescing singular values and
periodicity”, SIAM J. Matrix Analysis (2009).
L. Dieci, A. P., “Singular values of two-parameter matrices: An algorithm to
accurately find their intersections”, Mathematics and Computers in
Simulation (2008).
L. Dieci, M.G. Gasparo, A. Papini, A. P., “Locating coalescing singular values
of large two-parameter matrices”, Mathematics and Computers in Simulation
(2011).
L. Dieci, A. P. “Hermitian matrices depending on three parameters:
Coalescing eigenvalues” (submitted).
L. Dieci, M.G. Gasparo, A. Papini, A. P., “Continuation of smooth
decompositions for Hermitian functions, with application to localization of
conical intersections for 3-parameter functions” (in preparation).
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Motivation
Matrices that depend on parameters appear all the time in applied sciences
and engineering.
For instance, they are at the heart of dynamical systems studies, and spectral
properties of these matrices are key in determining stability and bifurcations
of dynamical invariants (say, equilibria and periodic orbits).
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Motivation
Of particular interest to us is the phenomenon of coalescing of eigenvalues of real
symmetric or complex Hermitian matrices (as well as coalescing singular values for
real/complex general matrices). This also has received a good deal of attention in
several, seemingly disparate, fields. For example:
Chemical Physics, conical intersections (e.g., Yarcony [2001])
Quantum Physics, diabolical points (Berry [1984])
Continuation techniques for 1D paths of decompositions (O’Neil,
Bunse-Gerstner et alia, ... [from ’90s on])
Best approximation, model reduction, data compression, ... (e.g.,
Kock-Lubich [2007], Simon-Zha [1999]). (Coalescing leads to singularity!)
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Motivation
Structural Dynamics:
Free vibration study using finite element models of beams, plates, ... (e.g.,
Srikantha Phani et al. [2006]) leads to an Hermitian eigenproblem, where:
Eigenvalues are frequencies, eigenvectors are modes of vibration.
Veering (a.k.a. “avoided crossing”) of eigenvalue curves is ubiquitous. It has been
associated to the onset of (spatial) mode localization and hypersensitive behavior.
Mode shape exchange: two distinct modes of vibration interchange upon small
modifications of a parameter. The structure responds with a different mode when
in internal/external parameter is slightly perturbed. (e.g., Pierre [1988],
Ouisse-Guyader [2003]).
Originally, it has been debated whether
veering was induced by approximation or physically
observable (Leissa [1974], Perkins-Mote [1986]).
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Mathematical Framework
We consider the following problems:
Problem 1: Let Ω be a bounded region of R2 . Given a smooth symmetric
matrix function A : Ω → Rn×n , find values x ∈ Ω where the eigenvalues of A
coalesce.
Problem 2: Let Ω be a bounded region of R3 . Given a smooth Hermitian
matrix function A : Ω → Cn×n , find values x ∈ Ω where the eigenvalues of A
coalesce.
The singular values decomposition (SVD) of A is intimately related to several
symmetric/Hermitian eigenproblems, such as those for AT A and AAT . So, we
also consider the analogous problems for the singular values.
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Background (1 parameter)
Differentiability of factors for 1 parameter functions is well established:
A analytic =⇒ Schur factors U, Λ are analytic
Kato [analytic Schur for Hermitian case, 1976]
Bunse-Gerstner, Byers, Mehrmann & Nichols [analytic SVD, 1991]
A of class C
k
=⇒
eigenvalues C k , possible loss
of differentiability of orthogonal factor
if there are degenerate eigenvalues
Dieci & Eirola [Schur and SVD for C k case, 1999]
Chern & Dieci [Schur and SVD for periodic C k case, 2000. A of period 1
with distinct eigenvalues: Factors of period 1 or 2?]
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Background (several parameters)
Example (Kato)
Complete loss of smoothness even for analytic A:
q
x1 x2
; λ± (x1 , x2 ) = ± x21 + x22
A(x1 , x2 ) =
x2 −x1
The coalescing causes loss of differentiability
in the eigenvalues and loss of continuity in
the eigenvectors. Contrast with 1-parameter case!
Little hope to detect a coalescing
point by simply continuing curves of eigenvalues
(or singular values) by freezing one parameter
at the time. A different approach is needed.
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Background (several parameters)
Important: If the eigenvalues stay distinct, it is possible to maintain
differentiability of the factors also when A depends on several parameters.
In this case, U (as well as V for the SVD) is unique except for changes of sign
of its columns (real symmetric case) or multiplication of its columns by phase
factors eiθ (complex Hermitian case).
Alternatively, one can smoothly obtain block reduction in disjoint groups of
eigenvalues/singular values. [Hsieh-Sibuya, Gingold]
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Genericity and Codimension
Coalescing of eigenvalues of real symmetric matrices is a codimension 2
phenomenon. This fact had first been observed by von Neumann and Wigner
more than 80 years ago (noncrossing rule).
Informally:
2 × 2 real symmetric matrices:
a
b
b
d
2 × 2 real symmetric matrices with coalescing eigenvalues:
a 0
0 a
(Triple coalescing has codimension 5, “two-pairs” coalescing has codimension 4,
and so on)
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Unlike the real case, coalescing of eigenvalues of complex Hermitian matrices is a
codimension 3 phenomenon.
Informally:
2 × 2 complex Hermitian matrices:
a
b + ic
b − ic
d
2 × 2 complex Hermitian matrices with coalescing eigenvalues:
a 0
0 a
formal justification of the n × n case goes through reduction to
block-diagonal form
Key fact: The codimension tells us how many parameters we need in order to
expect the phenomenon to occur.
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Consequences for real symmetric matrix functions. Generically:
one parameter
we expect a smooth matrix valued function to have distinct eigenvalues (and
consequently a smooth Schur decomposition)
when it occurs, coalescing of eigenvalues does not persist under perturbation
two parameters
coalescing of eigenvalues has to be expected
it occurs at isolated points
it persists under small perturbations
three parameters
coalescing of eigenvalues is expected to occur along curves
Similar considerations apply to the complex Hermitian case.
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1 parameter: unperturbed matrix
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1 parameter: perturbed matrix
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2 parameters: unperturbed matrix
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2 parameters: perturbed matrix
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2 parameters: zoom in on a point where eigenvalues coalesce
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Reminder
“Avoided crossing” of eigenvalue curves, veering, mode shape exchange ...
avoided crossing: a slice of the “double-cone” near a coalescing point?
This picture is ubiquitous whenever only 1 parameter is varied. It suggests that a
conical intersection could be “nearby”, i.e. that it could be obtained by “freeing
one more parameter”.
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Reminder
2 parameters: Slicing near a conical intersections
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Theoretical Results, 2 parameters
Theorem (real symmetric (2 × 2) case)
Consider P ∈ C k (Ω, R2×2 ), k ≥ 1, symmetric. For all x ∈ Ω, write
a(x) b(x)
P (x) =
,
b(x) d(x)
and let λ1 (x) ≥ λ2 (x) be its two continuous eigenvalues, for all x in Ω. Assume
that there exists a unique point ξ0 ∈ Ω where the eigenvalues coalesce:
λ1 (ξ0 ) = λ2 (ξ0 ). Consider the C k function F : Ω → R2 given by:
a(x) − d(x)
F (x) =
,
b(x)
and assume that 0 is a regular value for both functions a − d and b. Then,
consider the two C k curves Γ1 and Γ2 through ξ0 , given by:
Γ1 = {x ∈ Ω : a(x) − d(x) = 0}, Γ2 = {x ∈ Ω : b(x) = 0}. Assume that Γ1 and
Γ2 intersect transversally at ξ0 .
..
.
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Theorem (real symmetric (2 × 2) case cont’d)
..
.
Let Γ be a simple closed curve enclosing the point ξ0 , parametrized as a C p
(p ≥ 0) 1-periodic function γ in the variable t: γ : t ∈ R → Ω. Let
m = min(k, p), and let Pγ be the C m function P (γ(t)), t ∈ R. Then, for all
t ∈ R, Pγ (t) has the eigendecomposition
Pγ (t) = Uγ (t)Λγ (t)UγT (t)
such that:
(i) Λγ ∈ C1m (R, R2×2 ) and diagonal: Λγ (t) =
λ1 (γ(t))
0
0
for all
λ2 (γ(t))
t ∈ R;
(ii) Uγ ∈ C2m (R, R2×2 ) real orthogonal, and Uγ (t + 1) = −Uγ (t) for all t ∈ R.
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a(x) − d(x) = 0
b(x) = 0
Γ
C
Ω
ξ0
“Generic coalescing point”: Transversal Intersection at ξ0
Key idea: the first (respectively, second) component of the eigenvector associated
to λ1 changes sign exactly when C crosses the curve b(x) = 0 and
a(x) − d(x) < 0 (respectively, a(x) − d(x) > 0).
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In other words, the eigenvectors have changed sign, they flipped over (cfr.
Herzberg & Longuet-Higgins [1963]).
Bulding on the previous result, repeatedly using block reduction techniques,
and more ...
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Theorem (real symmetric n × n case)
Let A : Ω → Rn×n be C k and symmetric on Ω. Let γ : [0, 1] → Ω be a (smooth)
simple closed curve, s.t. A(γ(t)) has distinct eigenvalues for all t.
Let A(γ(t)) =

T

 λ (t)
1
|
|
|
|


..
= u1 (t) · · · un (t) 
 u1 (t) · · · un (t)
.
|
|
|
λn (t) | |
|
{z
}|
{z
}
{z
}
U(t)
Λ(t)
U(t)T
be the (smooth) Schur decomposition of A along γ.
The following holds: If γ encloses an odd number of generic coalescing points for
λi , then ui (t) changes sign along the circuit:
ui (0)T ui (1) = −1 .
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Theorem (Converse result)
Let A : Ω → Rn×n be C k and symmetric on Ω. Let γ : [0, 1] → Ω be a (smooth)
simple closed curve s.t. A(γ(t)) has distinct eigenvalues for all t. Let
A(γ(t)) = U (t)Λ(t)U (t)T
be the (smooth) Schur decomposition of A along γ.
Consider D = U (0)T U (1). Let i1 < i2 < · · · < i2q be the indices for which
Dii = −1. Pair them together (i1 , i2 ), . . . , (i2q−1 , i2q ).
If i2h−1 ≤ i < i2h for some 1 ≤ h ≤ q, then
λi = λi+1
at least once inside γ, and generically at an odd number of points inside γ.
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Remark: We expect all coalescing points to have multiplicity 1 (this is the generic
property).
To illustrate the previous Theorem, suppose we have n ≥ 4, and


−1


1




1
D=



−1


..
.
Then, we expect that inside the region encircled by Γ, the pairs (λ1 , λ2 ), (λ2 , λ3 ),
and (λ3 , λ4 ), have coalesced.
Notice: If λi coalesces at an even number of (generic) points inside γ, the
corresponding eigenvector does NOT change sign.
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Extension, 2 parameters
For large dimensional problems one may want to monitor only a small number of
eigenvalues, say just q n dominant eigenvalues/eigenvectors of A. We work
with the reduced Schur decomposition along γ:
T
U (q) (t) A(γ(t)) U (q) (t) = Λ(q) (t) ,
and we may have an odd number of −1 on the diagonal of
T
D(q) = U (q) (0) U (q) (1).
Example (q=4)
If we have
D(q)

1

=


1
−1
1



then (λ3 , λ4 ) and (λ4 , λ5 ) coincide at an odd number of points inside γ.
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Algorithm, 2 parameters
The previous results lend naturally to find regions of Ω inside which some
eigenvalues of A coalesce:
change of sign in the matrix of eigenvectors along a loop Γ
⇓
coalescing pair of eigenvalues inside Γ.
Observations.
Period 2 implies coalescing: Detection phase
Γ only needs to be continuous and does not have to be near the coalescing
point (not a local result)
Topological result, analogous of the I.V.T. of Calculus: “Change of sign of a
continuous function implies a root”
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1) given
A(xi,j ) = U0 Λ0 U0T
xi+1,j
compute continuous
Schur of A along red
and blue paths
xi+1,j+1
2) save orthogonal factors
at xi+1,j+1 :
x
i,j
U1 , U2
xi,j+1
3) compare U1 and U2 :
D= (U1 )T U2
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A 1D solver for computing smooth Schur decomposition paths is required (1D
continuation is not trivial)
Along the 1D paths, step-size is chosen adaptively
Robust: handle the case when eigenvalues coalesce and distinguish it from the
(computationally harder) case of near-coalescing
This is the workhorse of the algorithm
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Grid resolution is very important
Detection phase is highly parallelizable
Once we have a region where the i-th and (i + 1)-st eigenvalues coalesce, we
may proceed in two ways to accurately locate the parameter value where
coalescing occurs:
1. progressive refinement of the region
2. Newton (modified) to find a root of the gradient of
f (x) = λi (x) − λi+1 (x)
2
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Examples, 2 parameters
1
0.8
0.6
- Ω = [−1, 1] × [−1, 1]
- 2 × 2 grid
0.4
0.2
- λ1 = λ2 , λ2 = λ3
0
box
B1 1
B2 1
−0.2
−0.4
pair
2-3
2-3
iterates
7
6
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
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- 10 × 10 grid
1
- λ1 = λ2 , λ2 = λ3
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
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1
box
B4 1
B5 1
B7 2
B9 2
B4 3
B8 3
B3 8
B4 8
B2 9
B4 10
pair
1-2
1-2
1-2
1-2
2-3
2-3
2-3
2-3
1-2
1-2
iterates
8
7
–
7
6
6
7
–
7
8
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- 100 × 100 boxes
1
- λ1 = λ2 , λ2 = λ3
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
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0.8
1
box
B34 9
B48 9
B70 13
B82 19
B34 26
B72 29
B29 72
B34 75
B19 82
B34 92
pair
1-2
1-2
1-2
1-2
2-3
2-3
2-3
2-3
1-2
1-2
iterates
5
5
5
5
5
5
5
6
6
6
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−0.6
−0.7
−0.8
0.2
0.3
0.4
Local refinement: box B7 2
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300
250
200
150
100
50
0
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Figure: distribution of the workload over the region Ω
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Behavior on a large dimensional problems: A(x1 , x2 ) ∈ Rn×n ,
Ω = [−1, 1] × [−1, 1], grid 10 × 10, follow q = 6 dominant eigenvalues, pairs of
eigenvalues coalesce at ∼ 20 points.
Table: Experiments, m = 3200
n
200
400
800
1600
3200
average # of
steps
132
129
135
137
151
min-max #
steps
75-679
72-503
71-642
77-702
71-549
rejected
steps
1635
1558
1808
1682
1819
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What if A is complex Hermitian?
real symmetric
complex Hermitian
1D smooth continuation
uniquely defined
(eigenvectors defined
up to sign change)
NOT uniquely defined
(eigenvectors defined up
to phase factor eiθ )
codimension of coalescing
2
3
It is natural to work with 3 parameters, eigenvalue coalescence is expected to
occur at isolated points in the 3-dimensional space. New ideas are needed.
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How to resolve the non-uniqueness of smooth continuation along 1D paths?
Several different approaches have been proposed:
Stone [1976]:
Im u∗j (t + dt)uj (t) = O(dt2 ) for all t and j
Bunse-Gerstner et alia [1991]:
minimize
(Minimum Variation Decomposition, MVD)
Dieci & Eirola [1999]:
Rb
U̇
dt
a
F
u∗j (t)u̇j (t) = 0 for all t and j
They all lead to the same unitary Schur factor!
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Let us now consider loops in parameter space:
γ : t ∈ [0, 1] 7→ γ(t) ∈ R3 ,
smooth, with γ(0) = γ(1). Suppose, all eigenvalues of A stay distinct for all t,
and let
A(γ(t)) = U (t)Λ(t)U ∗ (t)
be a Minimum Variation Decomposition. How are U (0) and U (1) related?
We have:
 iα
e 1

U (1) = U (0) 
..

.
eiαn


For each j = 1, . . . , n, αj ∈ (−π, π] is know as the Berry phase (or geometric
phase) associated to λj .
Note: A real symmetric =⇒ αj = 0, π ⇐⇒ eiαj = ±1
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Stone insight: variation of Berry phase as a loop covers a sphere-like surface is
related to presence (or lack thereof) of coalescing points inside the surface.
To fix ideas, let us think of the family of parallels (lines of constant latitude)
{γs }s∈[0,1] that covers S2 , where γ0 , γ1 represent (respectively) north pole and
south pole.
We can define n smooth maps:
αj : s ∈ [0, 1] 7→ αj (s) ,
where αj (s) is the Berry phase associated to
λj along γs .
αj (s) must be allowed to go beyond (−π, π] (to retain smoothness)
γ0 and γ1 are constant; therefore αj (0) can (and will) be chosen to be 0, and
αj (1) has to be a multiple of 2π (A constant along γ0,1 ⇒ U(1)=U(0)).
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The same considerations apply to more general surfaces and loops’ coverings. We
call S a 2-sphere if it is the embedded images of S2 in R3 under a smooth map f .
f
−
→
Definition (Stone)
We say that a 2-sphere S is phase rotating if αj (1) 6= 0 for at least one j.
Otherwise, we say that S is phase preserving.
Remark: Definition doesn’t depend on the choice of loops’ covering.
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Theorem (complex Hermitian 2 × 2 case)
Let A ∈ C 1 (Ω, C2×2 ) be Hermitian. Let λ1 (ξ) ≤ λ2 (ξ) be its continuous
eigenvalues, for all ξ ∈ Ω. Let ξ0 ∈ Ω be a “generic” coalescing point for A in Ω,
and suppose that ξ0 is the only coalescing point in Ω. Let S ⊂ Ω be a 2-sphere. If
the interior part of S contains ξ0 , then S is phase-rotating, with α1,2 (1) = ±2π.
proof makes use of powerful topological results: show the result is true for S2
and then extend it to S via homotopy arguments.
repeatedly using block reduction techniques, and more ...
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Theorem (complex Hermitian n × n case)
Let A ∈ C 1 (Ω, Cn×n ) be Hermitian. Let λ1 (ξ), . . . , λn (ξ) be its continuous
eigenvalues, labeled in ascending order. Suppose that, for any j = 1, . . . , n − 1,
we have:
λj = λj+1
solely at dj distinct generic coalescing points in Ω, with a total numer of
Pn−1
N = j=1 dj distinct coalescing points. Let S ⊂ Ω be a 2-sphere whose interior
part contains all those coalescing points. Then we have that:
(i)
α1 (1) = 0 (resp. 2π)
(mod 4π)
(ii)
αj (1) = 0 (resp. 2π)
(mod 4π)
(iii) αn (1) = 0 (resp. 2π)
if d1 is even (resp. odd)
if dj−1 + dj is even (resp. odd)
and j = 2, . . . , n − 1
(mod 4π) if dn is even (resp. odd)
with |αj (1)| ≤ 2dj π for j = 1, n and |αj (1)| ≤ 2(dj−1 + dj )π for j = 2, . . . , n − 1.
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Observations:
Stone went in the opposite direction, and only considered the case of one
coalescing point
“phase rotations” add up nicely
a truly topological result, that lends nicely to find regions of R3 where
eigenvalues coalesce
again, an even number of coalescing points involving the same eigenvalue
may produce no phase rotation
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subdivide region in cubes
cover each cube with
square-shaped loops,
compute Berry phase along
each loop
you have obtained n
functions αj (s), where s
parametrizes the red
C-shaped meridian
if a cube is phase rotating,
zoom-in to locate
coalescing point accurately
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Algorithmic details:
1D continuation of the MVD Schur is performed via a 2 steps process.
At step k, with U (k) ≈ U (tk ):
construct “an” orthogonal Schur factor at tk+1 , call it Q(k+1)
rotate Q(k+1) to enforce minimum variation w.r.t. U (k) :
U (k+1) = Q(k+1) diag([eiθk , k = 1, . . . , n])
trivial predictor is O(h3 ) in enforcing minimum variation, surprising!
step-size is chosen adaptively along parallels (square shaped loops) and
meridian (C-shaped curve)
when a cube is declared phase rotating, multiple seeds are corrected via a
Newton-like procedure (zoom-in) to locate the coalescing point accurately
in case of zoom-in failure (or if multiple coalescing points are detected) local
refinement is performed (cube is divided in 8 sub-cubes)
the code is highly parallelizable by nature
it has proved to be very robust
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15
10
λ
5
Berry phase
1
λ
2
λ
3
0
−5
−10
0
0.5
1
1.5
2
2.5
s
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15
10
5
λ
Berry phase
1
λ
2
λ
3
0
λ
4
λ
5
λ
6
−5
−10
−15
0
0.5
1
1.5
2
2.5
s
February 14, 2011
48 / 49
Work To Do
complete the study of an alternative approach for complex Hermitian problem
(based on a perturbative idea)
analyze problems arising in Structural Engineering
generalized eigenvalue problem
non-generic coalescence of eigenvalues
study eigenvalue coalescence for non symmetric/Hermitian matrix functions
February 14, 2011
49 / 49

Eigenvalue coalescence for parameter dependent matrices

Transcript

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22 Frequency Response for Linear Systems