Nonlinear Eigenvalue Problems Solved via Contour Integrals

Nonlinear Eigenvalue Problems
and Contour Integrals
Marc Van Barel and Peter Kravanja

Nonlinear eigenvalue problems
Given
▶ an integer m > 1 (‘problem size’)
▶ a domain Ω ⊂ C
▶ a matrix-valued function T : Ω → Cm×m analytic in Ω
we consider nonlinear eigenvalue problems of the form
T(�)v = 0
where � ∈ Ω and v ∈ Cm, v ̸= 0.
1

More specifically, given
▶ a closed contour Γ ⊂ Ω that has its interior in Ω
we approximate all eigenvalues (and corresponding eigenvectors)
inside Γ.
No initial approximations of eigenvalues or eigenvectors are
needed.
2

Contour integrals
The method described in this talk uses (numerical approximations
of) contour integrals of the resolvent operator applied to a
random rectangular matrix:
1
2�i
︁
Γ
f (z)T(z)−1
V̂ dz ∈ Cm×M̃
where
▶ f : Ω → C is analytic in Ω
▶ V̂ ∈ Cm×M̃ is a random matrix
given an upper estimate M̃ for the number of eigenvalues inside
Γ.
3

Related problems and algorithms
Related problems solved via contour integrals:
▶ All the zeros of an analytic function located inside a contour
▶ Delvess and Lyness (1967) — based on Newton polynomial
▶ Kravanja and Van Barel (2000)
4

▶ All the zeros of an analytic function located inside a contour
▶ Delvess and Lyness (1967) — based on Newton polynomial
▶ Kravanja and Van Barel (2000)
▶ All the eigenvalues of the pencil A − �B, where A, B ∈ Cm×m,
located inside a circle (the unit circle)
▶ Sakurai and Sugiura (2003) — SS-method, SS-H(ankel)
▶ Ikegami and Sakurai (2010) — (block-)CIRR, SS-RR
▶ Ikegami, Sakurai and Nagashima (2010)
— filter function, block-SS
4

▶ All the eigenvalues of the pencil A − �B, where A, B ∈ Rm×m
are symmetric and B is positive definite, located inside a circle
▶ Sakurai and Tadano (2007) — (block-)CIRR
5

▶ All the eigenvalues of the pencil A − �B, where A, B ∈ Rm×m
are symmetric and B is positive definite, located inside a circle
▶ Sakurai and Tadano (2007) — (block-)CIRR
▶ All the eigenvalues of the pencil A − �I, where A ∈ Cm×m is
Hermitian, located inside a circle (the unit circle)
▶ Ohno, Kuramashi, Sakurai and Tadano (2010)
— shifted CG method
5

▶ All the eigenvalues of the pencil A − �B, where A, B ∈ Cm×m
are Hermitian and B is positive definite, located inside a
compact interval on the real axis
▶ Polizzi (2009) — the FEAST algorithm
▶ Krämer, Di Napoli, Galgon, Lang and Bientinesi (2013)
— computational analysis of FEAST
▶ Tang and Polizzi (2014) — theoretical analysis of FEAST
6

▶ All the eigenvalues of the pencil A − �B, where A, B ∈ Cm×m
are Hermitian and B is positive definite, located inside a
compact interval on the real axis
▶ Polizzi (2009) — the FEAST algorithm
▶ Krämer, Di Napoli, Galgon, Lang and Bientinesi (2013)
— computational analysis of FEAST
▶ Tang and Polizzi (2014) — theoretical analysis of FEAST
▶ All the eigenvalues of a polynomial eigenvalue problem located
inside a circle (the unit circle)
▶ Asakura, Sakurai, Tdano, Ikegami, Kimura (2010)
— Smith form (only simple eigenvalues)
6

▶ All the eigenvalues of an analytic nonlinear eigenvalue problem
located inside a circle (the unit circle)
▶ Asakura, Sakurai, Tadano, Ikegami and Kimura (2009)
— Smith form (only simple eigenvalues), also block-SS
▶ Beyn (2012) — Keldysh’ theorem (also multiple eigenvalues)
▶ Yokota and Sakurai (2013)
— SS-RR, SS-H compared to Beyn’s method
(also multiple eigenvalues)
7

Zeros of analytic functions
Before discussing the general case of nonlinear eigenvalue
problems, it is helpful to recall a quadrature method for computing
all the zeros of an analytic function that are located inside a
contour.
8

Kravanja and Van Barel (2000)
Suppose m = 1 (i.e., scalar problem T(�) = 0) and define
▶ N(Γ) as the total number of zeros (counting multiplicities) of
the analytic function T : Ω → C inside Γ
▶ n(Γ) as the number of mutually distinct zeros of T inside Γ
9

Kravanja and Van Barel (2000)
Suppose m = 1 (i.e., scalar problem T(�) = 0) and define
▶ N(Γ) as the total number of zeros (counting multiplicities) of
the analytic function T : Ω → C inside Γ
▶ n(Γ) as the number of mutually distinct zeros of T inside Γ
Consider the ordinary moments
sp =
1
2�i
︁
Γ
zp T′(z)
T(z)
dz, p = 0, 1, 2, . . .
Denoting the mutually distinct zeros as �k with multipl. �k ,
sp =
n(Γ)
︁
k=1
�k �p
k , p = 0, 1, 2, . . . .
Note that N(Γ) = s0 since T′/T has a simple pole at each zero of
T with residue equal to its multiplicity. 9

For all k ∈ N0, the k × k Hankel matrices Hk and H<
k are defined
as follows:
Hk =
︀
sr+l
︀k−1
r,l=0
and H<
k =
︀
sr+l+1
︀k−1
r,l=0
Then
▶ Hk = V T
k DVk with Vk the Vandermonde matrix based on the
zeros �k , i.e., Vk =
︁
�j
i
︁j=0,1,2,...,k−1
i=1,2,...,n(Γ)
and D = diag(�k )
▶ n(Γ) = rank Hk for all k ≥ n(Γ)
▶ The mutually distinct zeros of T inside Γ are given by the
eigenvalues of the pencil
H<
n(Γ) − �Hn(Γ) = V T
n(Γ)(�I − Λ)DVn(Γ)
The ordinary moments are approximated via a quadrature formula,
e.g., the trapezoidal rule in case Γ is the unit circle.
10

Unfortunately, there are problems:
▶ Higher-order moments are calculated less accurately than
lower-order moments
▶ Ill-conditioned Hankel matrices
▶ Clusters of zeros
11

To improve the algorithm, consider the formal inner product
⟨ �, � ⟩ =
1
2�i
︁
Γ
�(z)�(z)
T′(z)
T(z)
dz
for any polynomial functions � and �.
Proceed as follows:
▶ Construct a sequence of formal orthogonal polynomials
represented by their zeros,
▶ which are the eigenvalues of a generalized eigenvalue problem
involving (shifted) Gram matrices of formal inner products.
▶ A look-ahead criterion jumps over nearly singular sections.
▶ The final regular FOP is of degree n(Γ) and its zeros are given
by the mutually distinct zeros of T inside Γ.
12

If the derivative T′ is not readily available, then consider
⟨ �, � ⟩⋆ =
1
2�i
︁
Γ
�(z)�(z)
1
T(z)
dz
In particular,
s⋆
p =
1
2�i
︁
Γ
zp
T(z)−1
dz, p = 0, 1, 2, . . .
⇒ derivative-free method for computing all the zeros of T inside Γ,
where each zero is approximated as often as its multiplicity.
For simple eigenvalues,
s⋆
p =
︁
k
ck �p
k
with ck = T′(�k )−1.
13

Eigenvalue problems
Let us recall the nonlinear eigenvalue problem.
Given
▶ an integer m > 1 (‘problem size’)
▶ a domain Ω ⊂ C
▶ a matrix-valued function T : Ω → Cm×m analytic in Ω
we consider nonlinear eigenvalue problems of the form
T(�)v = 0
where � ∈ Ω and v ∈ Cm, v ̸= 0.
14

Contour integrals
Generalized eigenvalue problems
Consider the pencil A − zB where A, B ∈ Cm×m.
To compute all the eigenvalues located inside the contour Γ
Tetsuya Sakurai (2003, 2010) and his co-authors use
1
2�i
︁
Γ
(z − �)p
ûH
(zB − A)−1
v̂ dz, p = 0, 1, 2, . . .
where � ∈ C belongs to the interior of Γ and the vectors û, v̂ ∈ Cm
have been chosen at random.
15

Contour integrals
Given an upper estimate M̃ for the number of eigenvalues of
A − zB located inside Γ,
▶ these contour integrals are approximated via a quadrature
formula (e.g., the trapezoidal rule if Γ is the unit circle) for
p = 0, 1, . . . , 2M̃ − 1 ;
▶ a generalized eigenvalue problem (of size M̃ × M̃) involving a
Hankel matrix and a shifted Hankel matrix leads to
approximations of the eigenvalues of A − zB located inside Γ.
16

Contour integrals
To approximate the eigenvectors, specific linear combinations are
taken from the columns of the rectangular matrix given by
1
2�i
︁
Γ
(z − �)p
(zB − A)−1
v̂ dz ∈ Cm
for p = 0, 1, . . . , M̃ − 1.
17

Contour integrals
Consider the pencil A − zB where A, B ∈ Cm×m are Hermitian and
B is positive definite.
To compute all the eigenvalues located inside a compact interval
on the real axis (enclosed by the contour Γ), Eric Polizzi (2009)
considers
S =
1
2�i
︁
Γ
(zB − A)−1
BV̂ dz
given
▶ an upper estimate M̃ for the number of eigenvalues,
▶ a rectangular matrix V̂ ∈ Cm×M̃ chosen at random.
18

Contour integrals
Skeleton of the FEAST algorithm:
1. Choose V̂ ∈ Cm×M̃ of rank M̃.
2. Compute S by contour integration.
3. Orthogonalize S resulting in the matrix Q having orthonormal
columns.
4. Form the Rayleigh quotients
AQ = QH
AQ and BQ = QH
BQ
5. Solve the size-M̃ generalized eigenvalue problem
AQỸ = BQỸ Λ̃
6. Compute the approximate Ritz pairs (Λ̃ , X̃ = QỸ )
7. If convergence is not reached, then go to Step 1, with V̂ = X̃
19

Contour integrals
To compute all the eigenvalues inside the contour Γ, Tetsuya
Sakurai (2009, 2013) and his co-authors use the scalars
1
2�i
︁
Γ
zp
ûH
T(z)−1
v̂ dz, p = 0, 1, 2, . . .
where the vectors û, v̂ ∈ Cm have been chosen at random.
To approximate the eigenvectors, they consider the vectors
1
2�i
︁
Γ
zp
T(z)−1
v̂ dz, p = 0, 1, 2, . . .
The eigenvectors are specific linear combinations of these
vectors.
20

Keldysh’ theorem
Wolf-Jürgen Beyn’s method (2012) as well as our variant for
nonlinear eigenvalue problems are based on Keldysh’
theorem.
We consider only simple eigenvalues.
21

Keldysh’ theorem
Keldysh’ theorem
Let � ⊂ Ω be a compact subset and let n(�) denote the number of
eigenvalues of T in �.
Let �k for k = 1, . . . , n(�) denote these eigenvalues and suppose
that all of them are simple. Let vk and wk for k = 1, . . . , n(�)
denote their left and right eigenvectors, such that
T(�k )vk = 0 wH
k T(�k ) = 0 wH
k T′
(�k )vk = 1
Then there is a neighbourhood � of � in Ω and an analytic
function R : � → Cm×m such that
T(z)−1
=
n(�)
︁
k=1
vk wH
k (z − �k )−1
+ R(z)
for all z ∈ � ∖ {�1, . . . , �n(�)}.
22

Beyn’s method
Corollary (Beyn, 2012)
Suppose that T has no eigenvalues on the contour Γ ⊂ Ω and let
n(Γ) denote the number of eigenvalues of T inside Γ.
Let �k for k = 1, . . . , n(Γ) denote these eigenvalues and suppose
that all of them are simple. Let vk and wk for k = 1, . . . , n(Γ)
denote the corresponding left and right (normalized)
eigenvectors.
Then
1
2�i
︁
Γ
f (z)T(z)−1
dz =
n(Γ)
︁
k=1
f (�k )vk wH
k
for any function f : Ω → C that is analytic in Ω.
23

Beyn’s method
Define the matrices V , W ∈ Cm×n(Γ) as follows:
V =
︀
v1 · · · vn(Γ)
︀
W =
︀
w1 · · · wn(Γ)
︀
Assume that n(Γ) is not larger than the system dimension m.
In large-scale problems we actually expect to have
n(Γ) ≪ m.
Assume that rank(V ) = rank(W ) = n(Γ), which is the case in
typical applications.
Choose q ∈ N such that n(Γ) ≤ q ≤ m and choose the matrix
V̂ ∈ Cm×q such that W HV̂ ∈ Cn(Γ)×q has rank n(Γ).
24

Beyn’s method
Define the matrices S0, S1 ∈ Cm×q as follows:
S0 =
1
2�i
︁
Γ
T(z)−1
V̂ dz
S1 =
1
2�i
︁
Γ
zT(z)−1
V̂ dz
Then
S0 = VW H
V̂
S1 = V ΛW H
V̂
where the matrix Λ ∈ Cn(Γ)×n(Γ) is defined as
Λ = diag(�1, . . . , �n(Γ))
25

Beyn’s method
Wolf-Jürgen Beyn’s method (2012) is based on the singular value
decomposition of S0. Let
S0 = V0Σ0W H
0
where
V0 ∈ Cm×n(Γ)
V H
0 V0 = I
W0 ∈ Cq×n(Γ)
W H
0 W0 = I
Σ0 = diag(�1, . . . , �n(Γ))
Beyn has shown that
V H
0 S1W0Σ−1
0 = QΛQ−1
where Q = V H
0 V . It follows that V H
0 S1W0Σ−1
0 is diagonalizable.
Its eigenvalues are the eigenvalues of T inside the contour and its
eigenvectors lead to the corresponding eigenvectors of T.
26

Our variant is based on the canonical polyadic decomposition of a
tensor based on S0 and S1 . . .
27

Our variant is based on the canonical polyadic decomposition of a
tensor based on S0 and S1 . . .
. . . but let us first investigate the consequences of approximating
the contour integrals by a quadrature formula.
27

Numerical integration and filter functions
28

Define the matrices Sp ∈ Cm×q as follows:
Sp =
1
2�i
︁
Γ
zp
T(z)−1
V̂ dz, p = 0, 1, 2, . . .
We approximate Sp by a N-point quadrature formula with nodes zj
and corresponding weights �j for j = 0, 1, 2, . . . , N − 1.
Sp ≈ S̃p =
N−1
︁
j=0
�j zp
j T(zj )−1
V̂ , p = 0, 1, 2, . . .
Then Keldysh’ theorem implies that
S̃p =
n(�)
︁
k=1
vk wH
k V̂
N−1
︁
j=0
�j zp
j
zj − �k
+
N−1
︁
j=0
�j zp
j R(zj )V̂
29

The function bp : C → C defined as
bp(z) =
N−1
︁
j=0
�j zp
j
zj − z
, p = 0, 1, 2, . . .
is called the filter function (of order p) corresponding to the
quadrature formula.
It follows that
S̃p =
n(�)
︁
k=1
vk wH
k V̂ bp(�k ) +
N−1
︁
j=0
�j zp
j R(zj )V̂
30

If Γ is the unit circle, then the trapezoidal rule is used as
quadrature formula.
▶ Nodes:
zj = ei 2�j/N
▶ Weights:
�j =
zj
N
for j = 0, 1, 2, . . . , N − 1.
31

In this case
b0(z) =
N−1
︁
j=0
�j
zj − z
=
1
N
N−1
︁
j=0
zj
zj − z
=
1
1 − zN
and
bp(z) =
N−1
︁
j=0
�j zp
j
zj − z
=
1
N
N−1
︁
j=0
zp+1
j
zj − z
=
zp
1 − zN
for p = 1, 2, . . .
Conclusion:
bp(z) = zp
b0(z), p = 0, 1, 2, . . .
in case of the unit circle and the trapezoidal rule.
32

Properties of the filter function
Suppose that Γ is the unit circle. Then
1
2�i
︁
Γ
1
z − �
dz =
︂
1 |�| < 1
0 |�| > 1
is approximated by
b0(�) =
1
1 − �N
33

Let � > 0 be small, � ≪ 1. Then the �-level curve of b0(�),
i.e.,
{ � ∈ C : |b0(�)| = � }
is approximately a circle with its center at the origin and radius
��,N = �−1/N.
⃒
⃒
⃒
⃒
1
1 − �N
⃒
⃒
⃒
⃒ = � ⇒ |�| ≈ �−1/N
34

Let � denote the machine precision. Then
N ��,N
4 9741.98
8 98.70
16 9.93
32 3.15
64 1.78
128 1.33
256 1.15
35

36

filter function
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
−18
−16
−14
−12
−10
−8
−6
−4
−2
0
37

Define the circle Γ� as
Γ� = {� ∈ C : |�| = ��,N}
Assume that Γ� is inside � and suppose that no eigenvalue �k is
located on this circle.
Then we can split the set of eigenvalues into two subsets:
▶ the eigenvalues located inside Γ�
▶ the eigenvalues located outside Γ�
38

Inside Γ�
�1, . . . , �n(Γ�)
ordered such that
|b0(�1)| ≥ · · · ≥ |b0(�n(Γ�))|
Outside Γ�
�n(Γ�)+1, . . . , �n(�)
ordered such that
|b0(�n(Γ�)+1)| ≥ · · · ≥ |b0(�n(�))|
39

It follows that
S̃p =
n(Γ�)
︁
k=1
vk wH
k V̂ bp(�k )
+
n(�)
︁
k=n(Γ�)+1
vk wH
k V̂ bp(�k ) +
N−1
︁
j=0
�j zp
j R(zj )V̂
We know that
|b0(�k )| ≤ |b0(�n(Γ�)+1)| ≲ �
for k = n(Γ�) + 1, . . . , n(�).
40

The convergence radius r of R(z) satisfies
r ≥ ��,N
Let us examine the norm of the second and the third sum in the
expression for S̃p.
Define ˆ
� as
ˆ
� = �n(Γ�)+1
Then the norm of the dominant component of the second sum is
equal to
c1|bp(ˆ
�)| = c1
⃒
⃒
⃒
⃒
⃒
ˆ
�p
1 − ˆ
�N
⃒
⃒
⃒
⃒
⃒
≈ c1|ˆ
�|p−N
≤ c1�1− p
N
41

One can show that the norm of the third sum behaves as
c2rp−N
≤ c2�1− p
N
because
r ≥ ��,N = �−1/N
Note that it is best to keep p small compared to N.
42

In summary, we obtain that
S̃p =
n(Γ�)
︁
k=1
vk wH
k V̂ bp(�k ) + ∆1 + ∆2
with
‖∆1‖ ≈ c1|ˆ
�|p−N
≤ c1�1− p
N
and
‖∆2‖ ≈ c2rp−N
≤ c2�1− p
N
Numerical examples will show that, under certain conditions, we
can retrieve vk (2-norm 1) and �k with errors of magnitude
c3
�
b0(�k )
Further analysis is necessary.
43

Robust extraction of the eigenvalues �k and
the eigenvectors vk from the computed
moments S̃p
44

Tensor decomposition
Recall that
S̃p =
n(�)
︁
k=1
vk wH
k V̂ bp(�k ) +
N−1
︁
j=0
�j zp
j R(zj )V̂
and that
bp(�k ) = �p
k b0(�k )
in case of the unit circle and the trapedoizal rule.
45

It follows that
S̃p =
n(�)
︁
k=1
vk wH
k V̂ �p
k b0(�k ) +
N−1
︁
j=0
�j zp
j R(zj )V̂
= V Λp
Ŵ H
+
N−1
︁
j=0
�j zp
j R(zj )V̂
where
V =
︀
v1 · · · vn(�)
︀
Λ = diag(�1, . . . , �n(�))
W =
︀
w1 · · · wn(�)
︀
Ŵ H = diag
︀
b0(�1), . . . , b0(�n(�))
︀
W HV̂
46

Since R(z) is analytic, the second term of S̃p is small:
S̃p ≈ V Λp
Ŵ H
∈ Cm×q
where m is the problem size and n(�) ≤ q ≤ m.
Define H and H< as
H =
︀
Ŝ0
︀
≈ V Ŵ H
and
H<
=
︀
Ŝ1
︀
≈ V ΛŴ H
47

It follows that the canonical polyadic decomposition of the tensor
consisting of the two slices H and H< is given by
n(�)
︁
k=1
⎡
⎢
⎢
⎢
⎣
vk
vk �k
.
.
.
vk ��
k
⎤
⎥
⎥
⎥
⎦
⊙
⎡
⎢
⎢
⎢
⎣
ŵk
�k ŵk
.
.
.
��
k ŵk
⎤
⎥
⎥
⎥
⎦
⊙
︂
1
�k
︂
where ŵk denotes the kth column vector of Ŵ for
k = 1, . . . , n(�).
Tensorlab provides a robust algorithm for computing this
canonical polyadic decomposition.
48

Skeleton of our algorithm:
1. Choose a filter function, i.e., a quadrature formula,
and where to apply it in the complex plane based on
▶ T(z) and
▶ the domain in which the wanted eigenvalues are lying.
2. Choose V̂ ∈ Cm×M̃ of rank M̃.
3. Compute the Hankel matrices H and H< by contour
integration using the quadrature formula.
4. Compute the canonical polyadic decomposition of the tensor
consisting of the two slices H and H<.
5. The first and third factor matrix give the approximating
eigenvectors and corresponding eigenvalues.
49

Numerical experiments
We consider the gun problem from the NLEVP collection by
Betcke, Higham, Mehrmann, Schroeder and Tisseur
(2013).
▶ Nonlinear eigenvalue problem from model of a radio-frequency
gun cavity.
▶ Problem size m = 9956
▶ The function T has the following form:
T(z) =
︀
K M iW1 iW2
︀
⎡
⎢
⎢
⎣
1
−z
√
z
√
z − �
⎤
⎥
⎥
⎦
where K, M, W1 and W2 are sparse matrices, and
� = (108.8774)2.
51

We would like to approximate all eigenvalues located inside the
circle that
▶ is symmetric with respect to the real axis
▶ intersects the real axis at � = (108.8774)2 and � = 3402
We choose Γ as the unit circle and N = 32.
What is a good choice for the center � and the radius � such that
we can apply the theory on T(� + �z)?
52

Scenario 1
z |b0(z)| � + �z
−3.1 ≈ 10−16 �
2 ≈ 10−10 �
Because the convergence radius r of R(z) is equal to 3.1
and
|b0(r)| ≈ 10−16
the R(z) part does not influence (up to machine precision) the
computation of S̃0 and S̃1. In other words,
S̃0 =
n(�)
︁
k=1
vk wH
k V̂ b0(�k )
S̃1 =
n(�)
︁
k=1
vk wH
k V̂ �k b0(�k )
53

This is indicated by the singular values of S̃0 and S̃1 shown in the
next figure.
54

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
10−19
10−17
10−15
10−13
10−11
10−9
10−7
10−5
10−3
singular values of the 0th and 1th moment
0th moment
1st moment
55

By computing the canonical polyadic tensor decomposition with 23
terms, we obtain an approximation of 23 eigenvalues.
Relative residual for ˜
�k :
�k =
‖T(˜
�k )ṽk ‖1
‖T(˜
�k )‖1‖ṽk ‖1
Because the accuracy of ˜
�k depends on the extraction of the
corresponding information from S̃0 and S̃1, this accuracy is limited
to |b0(˜
�k )|. Hence, we expect that
�k |b0(˜
�k )| ≈ |b0(r)|
56

0 2 4 6 8 10 12 14 16 18 20 22 24
10−17
10−15
10−13
10−11
10−9
10−7
10−5
10−3
10−1
101 norm(residual)
abs(filter value)
product
57

Let us check that
‖R�‖ ≈ �
︂
1
r
︂�
where r = 3.1 is the convergence radius of
R(z) =
︀
�≥0 R�z�.
The results shown in the next figure imply that r̃ ≈ 2.55.
58

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140
10−20
10−19
10−18
10−17
10−16
10−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
coefficients of R(z)
59

We miss an eigenvalue �⋆ having the following properties:
|b0(�⋆
)| ≈ 10−13
and
|�⋆
| ≈ 2.58 ≈ r̃
60

filter function
2 4 6 8 10 12 14
x 10
4
−6
−4
−2
0
2
4
6
x 10
4
−18
−16
−14
−12
−10
−8
−6
−4
−2
0
61

Scenario 2
z |b0(z)| � + �z
−2 ≈ 10−10 �
2 ≈ 10−10 �
The convergence radius r of R(z) now equals r = 2 and
|b0(r)| ≈ 10−10
In the computation of the moments S̃0 and S̃1 the R(z) term now
comes in with size of order 10−10.
The next figure shows the singular values of S̃0 and S̃1.
62

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
10−16
10−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
singular values of the 0th and 1th moment
0th moment
1st moment
63

By computing the canonical polyadic tensor decomposition with 24
terms, we obtain an approximation of 24 eigenvalues.
Relative residual for ˜
�k :
�k =
‖T(˜
�k )ṽk ‖1
‖T(˜
�k )‖1‖ṽk ‖1
Because the accuracy of ˜
�k depends on the extraction of the
corresponding information from S̃0 and S̃1, this accuracy is limited
to |b0(˜
�k )|. Hence, we expect that
�k |b0(˜
�k )| ≈ |b0(r)|
64

0 2 4 6 8 10 12 14 16 18 20 22 24
10−15
10−13
10−11
10−9
10−7
10−5
10−3
10−1
101 norm(residual)
abs(filter value)
product
65

Let us check that
‖R�‖ ≈ �
︂
1
r
︂�
where r = 2 is the convergence radius of
R(z) =
︀
�≥0 R�z�.
The results shown in the next figure imply that r̃ ≈ 2.
66

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140
10−21
10−19
10−17
10−15
10−13
10−11
10−9
10−7
10−5
coefficients of R(z)
67

filter function
0 2 4 6 8 10 12 14
x 10
4
−8
−6
−4
−2
0
2
4
6
8
x 10
4
−18
−16
−14
−12
−10
−8
−6
−4
−2
0
68

Nonlinear Eigenvalue Problems Solved via Contour Integrals

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Nonlinear Eigenvalue Problems Solved via Contour Integrals