Applications of block Geršgorin sets arising from linearizations of quadratic matrix polynomials
4.2 Block Geršgorin sets arising from lineariza- tions of quadratic matrix polynomials
In this section we examine the block Geršgorin sets arising from the matrix pen- cils L(z) belonging to the vector spaces L1(P), L2(P) and DL(P) that are strong
linearizations of P(z) ∈ Pn as given in (4.0.1). The partition π is taken to be the canonical partitioning of blocks as stated in Theorems 4.1.1, 4.1.3 and 4.1.5. With this partitionπand induced operator normë.ëp,for the sake of simplicity, throughout this chapter we denote the block Geršgorin set Γπp(L) obtained via the linearization L(z) of P(z) by the notation Γ(P).
Theorems 4.2.1 and 4.2.3 provide the block Geršgorin sets Γ(P)corresponding to strong linearizations inL1(P)andL2(P)with ansatz vectorsαe1andαe2respectively.
Theorem 4.2.1. Let P(z) = A2z2 +A1z +A0 ∈ Pn with A0 Ó= 0. Then the block Geršgorin set Γ(P) containing the spectrum of P(z) is given by
Γ(P) = R1∪R2
where the sets R1 andR2 take different forms depending on the choice of linearization L(z). The details are as follows.
(i) IfL(z) is the first companion form linearization, then R1 = {z ∈C∞:...(zA2+A1)−1...−1
p ≤ ëA0ëp}, (4.2.1)
R2 = {z ∈C:|z| ≤1}. (4.2.2)
(ii) If L(z) is the second companion form linearization, then R1 ={z ∈C∞:...(zA2+A1)−1...−1
p ≤1}, R2 ={z ∈C:|z| ≤ ëA0ëp}.
(iii) If L(z)∈L1(P) with right ansatz vector v =αe1, α∈C\ {0} is as in (4.1.1), then
R1 =
;
z ∈C∞ :...(α(zA2+A1) +W1)−1...−1
p ≤ ëαA0−zW1ëp
<
, R2 = {z ∈C:|z| ≤κ}
where κ = ëW2ëp
.. .W2−1...
p is the condition number of the non-singular matrix W2 ∈Cn,n and W1 ∈Cn,n is arbitrary.
(iv) If L(z)∈L2(P) with left ansatz vector v =αe1, α∈C\ {0} as in (4.1.3) then R1 =
;
z ∈C∞ :...(α(zA2+A1) +W1)−1...−1
p ≤ ëW2ëp
<
, R2 =
;
z ∈C:|z| ≤...W2−1...
pëαA0−zW1ëp
<
where W2 ∈Cn,n is non-singular andW1 ∈Cn,n is arbitrary.
82 4.2. Block Geršgorin sets arising from linearizations of quadratic matrix . . . Proof. We first prove(iii).From (4.1.1), the linearization L(z)∈L1(P) with ansatz vector αe1 takes the form
L(z) =
α(zA2+A1) +W1 αA0−zW1
W2 −zW2
.
Applying the definition of block Geršgorin sets to L(z), we have Γ(P) = R1 ∪ R2
where R1 =
;
z ∈C∞:...(α(zA2+A1) +W1)−1...−1
p ≤ ëαA0−zW1ëp
<
, and R2 =
;
z ∈C∞:|z|...W2−1...−1
p ≤ ëW2ëp
<
=
;
z ∈C:|z| ≤ ëW2ëp
.. .W2−1...
p
<
. Using Theorem 4.0.1, Λ(L)⊆Γ(L).Consequently, Λ(P) = Λ(L)⊆Γ(L).
In order to determine whether the eigenvalue ∞ belongs to the set R1 orR2, we look at the block Geršgorin set arising from revL(z)given by
z
W1+αA1 αA0
W2 0
+
αA2 −W1
0 −W2
,
where W2 ∈ Cn,n is non-singular and W1 ∈ Cn,n is arbitrary. Clearly, 0 belongs to the set RrevL1 := {z ∈ C : ...(α(zA1+A2) +zW1)−1...−1
p ≤ ëαzA0−W1ëp} if A2 is singular. Also conversely if 0 ∈ RrevL1 for every W1 ∈ Cn,n, then A2 is a singular matrix. Therefore we associate ∞ with R1.
(iv) can be derived in a similar manner using the form ofL(z)from (4.1.3). Also since for the eigenvalue at∞, we have
revL(z) =z
W1+αA1 W2 αA0 0
+
αA2 0
−W1 −W2
and infinity is associated with R1 as 0∈RrevL1 :=
;
z ∈C:...(α(zA1+A2) +zW1)−1...−1
p ≤ |z| ëW2ëp
<
for all admissible choices of W1 and W2 if and only ifA2 is singular.
With α = 1, W1 = 0 and W2 =−In in (iii) and (iv) respectively, we get(i) and (ii).
Remark 4.2.2. Observe that in case A2 is singular, the set R1 corresponding to the first and second companion form is always unbounded.
Theorem 4.2.3. For a quadratic matrix polynomial P(z) =A2z2+A1z+A0 ∈Pn, the block Geršgorin sets corresponding to the linearizations in L1(P) andL2(P)with
ansatz vector v =αe2 are given by the union of sets R1 and R2 where R1 =
z ∈C∞:|z| ≥ 1 ëW2ëp
.. .W2−1...
p
, R2 =
;
z ∈C:...(αA0−zW1)−1...−1
p ≤ ëα(zA2+A1) +W1ëp
<
, if L(z)∈L1(P) is of the form (4.1.2), and
R1 =
;
z ∈C∞:...W2−1...−1
p ≤ ëα(zA2+A1) +W1ëp
<
, (4.2.3)
R2 =
;
z ∈C:...(αA0−zW1)−1...−1
p ≤ |z| ëW2ëp
<
, (4.2.4)
if L(z) ∈ L2(P) is of the form as given in (4.1.4). In either case W2 ∈ Cn,n is non-singular and W1 ∈Cn,n is arbitrary.
Proof. The proof follows by arguing as in the proof of Theorem 4.2.1. Note that here the sets R1 and R2 may be unbounded in each case. For both the linearizations, 0 belongs to the set R1 corresponding to revL(z) for all choices of W1 and W2. But this is not the case for R2. Hence ∞is associated with R1 in both instances.
It is evident from Theorem 4.2.1 that if the pencil zA2+A1 is singular, then the block Geršgorin sets corresponding to the first and second companion forms are the entire complex plane. We conjecture that the converse also holds.
The block Geršgorin sets in Theorems 4.2.1 and 4.2.3 are obtained via lineariza- tions with respect to ansatz vectorsαe1 andαe2,whereα∈C\{0}.Now we consider the general case of an ansatz vector v that is not a multiple of e1 ore2.
Theorem 4.2.4. Let P(z) =A2z2+A1z+A0 ∈Pn. Suppose L(z)is a linearization of P(z) in L1(P) with corresponding ansatz vector v ∈C2 which is not a multiple of e1 or e2. Let M =
m1 m2
m3 m4
∈C2,2 be non-singular.
(i) IfM v =αe1, α∈C\ {0} then the block Geršgorin setsΓ(P) = R1∪R2 arising from L(z) is such that
R1 =
;
z ∈C∞ :...(zA2+A1+W1)−1...−1
p ≤ ëA0−zW1ëp
<
, (4.2.5) R2 =
;
z ∈C:...(A0−zW2)−1...−1
p ≤ ëzA2+A1+W2ëp
<
. (4.2.6) Here W1 =W1− mm24W2 and W2 = W1 − mm13W2 where W1, W2 ∈ Cn,n with W2
non-singular such that L(z) = (αM−1⊗In)
z
A2 −W1
0 −W2
+
W1+A1 A0
W2 0
.
84 4.2. Block Geršgorin sets arising from linearizations of quadratic matrix . . . (ii) If M v =αe2, α∈C\ {0} thenΓ(P) =R1∪R2 is the block Geršgorin set asso-
ciated with L(z)where R1 andR2 are given by (4.2.5) and (4.2.6) respectively, with the only difference being that W1 = W1 − mm42W2 and W2 = W1 − mm31W2
where W1, W2 ∈Cn,n with W2 non-singular such that L(z) = (αM−1⊗In)
z
0 −W2
A2 −W1
+
W2 0
W1+A1 A0
.
Proof. Let v =
v1
v2
∈ C2. Then by the hypothesis v1 and v2 are both non-zero.
First we prove(i).By Theorem 4.1.1, 1Mα ⊗In
2L(z)∈L1(P) with ansatz vectore1. Therefore, there exists W1, W2 ∈Cn,n with W2 non-singular such that
L(z) = (αM−1⊗In) A
z
C A2 −W1 0 −W2
D +
C W1+A1 A0 W2 0
DB
= α
det(M)
C m4In −m2In
−m3In m1In D A
z
C A2 −W1 0 −W2
D +
C W1+A1 A0 W2 0
DB
= α
det(M)
C m4(zA2+A1+W1)−m2W2 m4(A0−zW1) +m2zW2
−m3(zA2+A1+W1) +m1W2 −m3(A0−zW1)−m1zW2 D
. Therefore, the block Geršgorin regionΓ(P) =R1∪R2 arising fromL(z)is such that
R1 =
;
z∈C∞:...(m4(zA2+A1+W1)−m2W2)−1...−1
p ≤ ëm4(A0−zW1) +m2zW2ëp
<
, R2 =
;
z∈C:...(m3(A0−zW1) +m1zW2)−1...−1
p ≤ ëm3(zA2+A1+W1)−m1W2ëp
<
as 0 belongs to the set R1 associated with revL(z) for all choices of W1 if and only if A2 is singular. Suppose m4 = 0. Then M v =e1 =⇒ m3v1 = 0 =⇒ m3 = 0 or v1 = 0. Both cases are impossible as M is non-singular and v1 Ó= 0. Hence, m4 Ó= 0.
Similarly, if m3 = 0 then M v =e1 =⇒ m4v4 = 0 =⇒ m3 = 0 or v2 = 0, which are both impossible as M is non-singular and v2 Ó= 0.Therefore,
R1=
z∈C∞: .. .. . 3
zA2+A1+W1−m2 m4W2
4−1.....
−1 p
≤ ..
..A0−zW1+zm2 m4W2
.. ..
p
,
R2=
z∈C: .. .. . 3
A0−zW1+m1 m3W2
4−1.....
−1 p
≤ ..
..zA2+A1+W1− m1 m3W2
.. ..p
. Now (4.2.5) and (4.2.6) follow by setting W1 =W1− mm24W2 and W2 =W1− mm13W2. The proof of(ii)follows by identical arguments as in this case by (4.1.2), there exists W1, W2 ∈Cn,n with W2 non-singular such that
3M α ⊗In
4
L(z) =z
0 −W2
A2 −W1
+
W2 0
W1+A1 A0
.
Theorem 4.2.5. Let P(z) =A2z2+A1z+A0 ∈Pn. Suppose L(z)is a linearization of P(z) in L2(P) with corresponding ansatz vector v =
v1 v2
∈ C2 which is not a multiple of e1 or e2. Let M =
m1 m2
m3 m4
∈C2,2 be non-singular.
(i) IfM v =αe1, α∈C\ {0} then the block Geršgorin set Γ(P) = R1∪R2 arising from L(z) is such that
R1 =
;
z∈C∞:...(zA2+A1+W1)−1...−1
p ≤
-- --
v2 v1 --
--ëzA2+A1+W2ëp
<
, (4.2.7) R2 =
;
z∈C:...(A0−zW2)−1...−1
p ≤
-- --
v1 v2 --
--ëA0−zW1ëp
<
. (4.2.8)
Here W1 =W1− mm24W2 and W2 = W1 − mm13W2 where W1, W2 ∈ Cn,n with W2
non-singular such that L(z) =
z
A2 0
−W1 −W2
+
W1+A1 W2
A0 0
(αM−t⊗In).
(ii) If M v = αe2, α ∈ C\ {0} then Γ(P) = R1 ∪ R2 is the block Geršgorin set associated with L(z) where R1 and R2 are as given in part (i) except for the fact that W1 = W1 − mm42W2 and W2 = W1 − mm31W2 where W1, W2 ∈ Cn,n with W2 non-singular such that
L(z) =
z
0 A2
−W2 −W1
+
W2 W1+A1
0 A0
(αM−t⊗In).
Proof. Note that, M v = αe1 ⇒ m3v1 +m4v2 = 0 ⇒ ---mm34--- = ---vv2
1
--
-. Also we have, M v = αe2 ⇒ m1v1 +m2v2 = 0 ⇒ ---mm12--- = ---vv21---. The rest of the proof now follows from that of Theorem 4.2.4 due to the fact that L2(P) = [(L1(Pt))]t. Also the point z = 0 belongs to the set R1 associated with revL(z) for all admissible choices of W1, W2 ∈Cn,n and ansatz vectorsv ∈C2 if and only if A2 is singular. Hence ∞ can belong to R1.
Remark 4.2.6. Observe that in Theorems 4.2.4 and 4.2.5, either
W1 W2
=
1 −mm24 1 −mm13
⊗In
W1
W2
, (4.2.9)
or
W1 W2
=
1 −mm42 1 −mm31
⊗In
W1 W2
(4.2.10)
86 4.2. Block Geršgorin sets arising from linearizations of quadratic matrix . . . where
1 −mm24 1 −mm13
and
1 −mm42 1 −mm31
are non-singular asM is non-singular. Also, as W2 is non-singular, therefore in each caseW2−W1 is non-singular. This has impor- tant implications for forming the setsR1andR2 in Theorem 4.2.4 and Theorem 4.2.5, because it means that if W1,W2 ∈ Cn,n are arbitrarily chosen such that W2 −W1 is non-singular, then infinitely many corresponding pairs of matrices W1, W2 ∈ Cn,n with W2 non-singular can be formed via the relations (4.2.9) and (4.2.10) by making different choices of M and right ansatz vectors v. Therefore, once R1 and R2 as in Theorem 4.2.4 and Theorem 4.2.5 are constructed with such choices of W1 and W2, their union form block Geršgorin sets Γ(P) that may be associated with infinitely many linearizations in L1(P) and L2(P) respectively.
The next theorem gives the eigenvalue localization sets for the spectra of a quadratic matrix polynomial P(z) arising from strong linearizations in DL(P) with respect to any ansatz vector v in C2.
Theorem 4.2.7. Consider a quadratic matrix polynomial P(z) ∈ Pn as in (4.0.1).
Let L(z) ∈ DL(P) with corresponding ansatz vector v =
v1
v2
∈ C2 be a strong linearization of P(z). The block Geršgorin set corresponding to L(z) is given by Γ(P) =R1 ∪R2 where
R1 =
;
z ∈C∞:...(v1(zA2+A1)−v2A2)−1...−1
p ≤ ëzv2A2+v1A0ëp
<
, R2 =
;
z ∈C:...(v2(zA1+A0)−zv1A0)−1...−1
p ≤ ëzv2A2+v1A0ëp
<
.
Proof. The proof follows immediately by using the definition of Γ(P) and Theorem 4.1.5. Note that 0 belongs to the set R1 corresponding to revP(z) for all choices of ansatz vectors v if and only if A2 is singular. Hence ∞ is associated with R1.
Remark 4.2.8. Since DL(P) = L1(P)∩L2(P), it is natural to expect that the block Geršgorin set R1∪R2 given in Theorem 4.2.7 is a particular case of the sets R1 and R2 obtained earlier with respect to strong linearizations in L1(P) and L2(P). Indeed it is easy to see that if v1 = 0, then R1 and R2 in Theorem 4.2.7 are obtained by choosing W1 = −v2A1 and W2 =A2 in the sets R1 and R2 in Theorem 4.2.3. The choice of W2 is justified by the fact that if L(z)∈DL(P)with ansatz vector v =
0 v2
is a strong linearization of P(z), then A2 is non-singular.
Similarly, if v2 = 0, thenR1 andR2 as in Theorem 4.2.7 are obtained by choosing W1 = 0 and W2 =v1A0 in the descriptions ofR1 and R2 as given in points (iii) and (iv) of Theorem 4.2.1.
If v1v2 Ó= 0, then the set R1∪R2 from Theorem 4.2.7 is the same set as the one obtained in Theorem 4.2.4 by choosing W1 =−vv21A2 in (4.2.5) and W2 = vv1
2A0−A1
in (4.2.6). It is also the same set as the one obtained from Theorem 4.2.5 by making the same choices ofW1 andW2 in (4.2.7) and (4.2.8). Observe that in both instances, the choices ofW1 andW2 are justified by the fact that if a pencil inDL(P)with ansatz vector v =
v1
v2
where v1v2 Ó= 0 is a strong linearization of P(z), then the difference W1−W2 =−vv12P1−vv212 has to be non-singular.
The remaining results in this section depict important properties of block Gerš- gorin sets for P(z) ∈ Pn arising from strong linearizations in L1(P) or DL(P). In the next lemma, we state a well known inequality which will be frequently used in the remaining sections of the chapter, a proof of which is included for the sake of completeness.
Lemma 4.2.9. For any A, B ∈Cn,n,
..
.A−1...−1
p − ëBëp ≤...(A+B)−1...−1
p ≤...A−1...−1
p +ëBëp.
Proof. For anyX, Y ∈Cn,n, Y−1−X−1 =Y−1(X−Y)X−1. Therefore, taking norm ë·ëp on both sides,
-- -- .. .Y−1...
p−...X−1...
p
--
--≤...Y−1−X−1...
p ≤...X−1...
p
.. .Y−1...
pëX−Yëp. Dividing both sides byëX−1ëpëY−1ëp, we obtain
-- -- ..
.X−1...−1
p −...Y−1...−1
p
--
--≤ ëX−Yëp.
The desired inequality now follows by replacing X byA and Y byA+B.
Theorem 4.2.10. LetP(z) =A2z2+A1z+A0 ∈Pn. Then for any strong lineariza- tion L(z) in L1(P) or DL(P),
{z ∈Λ(P) :|z| ≥1} ⊆R1 and {z∈Λ(P) :|z| ≤1} ⊆R2.
Proof. In view of Remark 4.2.8, the block Geršgorin sets arising from linearizations inDL(P)are particular cases of those arising from linearizations inL1(P).Thus it is enough to prove the theorem for strong linearization L(z) ∈L1(P). If λ =∞, then by (4.2.5), λ ∈ R1. So we assume without loss of generality that λ ∈ C. Then, we have
..
.(λA2+A1+W1)−1...−1
p = ...((P(λ)−A0)/λ+W1)−1...−1
p
≤ |λ|−13...(P(λ))−1...−1
p +ëA0−λW1ëp
4
= |λ|−1ëA0−λW1ëp ≤ ëA0 −λW1ëp (since |λ| ≥1)
88 4.2. Block Geršgorin sets arising from linearizations of quadratic matrix . . . Thus λ∈R1 where R1 is as in (4.2.5). Now suppose λ∈Λ(P) with |λ| ≤1. Then,
..
.(A0 −λW2)−1...−1
p = ...(P(λ)−λ(A2λ+A1+W2))−1...−1
p
≤ ...(P(λ))−1...−1
p +|λ| ëA2λ+A1+W2ëp
≤ ëA2λ+A1+W2ëp (as |λ| ≤1)
Thus λ ∈R2 where R2 is as in (4.2.6). Since ëSëpëS−1ëp ≥1, for any non-singular matrix S, the proofs for the case of the other linearizations in L1(P) with ansatz vectors αe1 or αe2 follow in a similar way.
Theorem 4.2.10 has the important consequence of making the inclusion regions containing the eigenvalues of P(z) ∈ Pn even more tight than the original block Geršgorin sets Γ(P).
Corollary 4.2.11. Let P(z) = A2z2+A1z+A0 ∈Pn andD be the closed unit disc.
Let Γ(P) = R1∪R2 be a block Geršgorin region whereR1 and R2 are associated with linearizations in L1(P) or DL(P). Then
Λ(P)⊆(R1∩Dc)∪(R2∩D). (4.2.11) Moreover, if R1 is given by (4.2.5), and the corresponding R2 is given by (4.2.6), then
Λ(P) = Ü{(R1∩Dc)∪(R2∩D) : (W1,W2)∈T} (4.2.12) where T={(X, Y)∈Cn,n×Cn,n :X−Y is non-singular }.
Proof. Since (4.2.11) is an immediate consequence of Theorem 4.2.10, we prove (4.2.12). So let,
R1 =
;
z ∈C∞:...(zA2+A1+W1)−1...−1
p ≤ ëA0−zW1ëp
<
, R2 =
;
z ∈C:...(A0−zW2)−1...−1
p ≤ ëzA2+A1+W2ëp
<
where(W1,W2)∈T,in view of Remark 4.2.6. Since Λ(P)⊆(R1∩Dc)∪(R2∩D)for all (W1,W2)∈T, to complete the proof, we show that
Ü
(W1,W2)∈T
(R1∩Dc)∪(R2∩D)⊆Λ(P).
Letz ∈(R1∩Dc)∪(R2∩D)for all(W1,W2)∈T. If|z|>1, thenz∈R1∩Dc.Let W1 =z−1A0 andW2 =W1+M for some non-singularM ∈Cn,n, so that(W1,W2)∈T. Since z ∈R1 for any (W1,W2)∈T, we have
..
.(P(z))−1...−1
p =|z|...(zA2+A1 +A0/z)−1...−1
p ≤ |z| ëA0−z(A0/z)ëp = 0.
If |z| ≤ 1, then z ∈ R2 ∩ D. Since z ∈ R2 for any (W1,W2) ∈ T, choosing (W2+M,W2)∈T where W2 =−(A1+zA2) and M ∈Cn,n is non-singular,
..
.(P(z))−1...−1
p =...(A0−zW2)−1...−1
p ≤ ëzA2 +A1+W2ëp = 0.
Therefore, z ∈ Λ(P). Hence, Ü
(W1,W2)∈T
(R1 ∩Dc)∪(R2 ∩D) ⊆ Λ(P) and the proof follows.
To conclude this section we illustrate some of the block Geršgorin regions formed above. In view of Corollary 4.2.11, when the underlying linearization is from L1(P) or DL(P), we will be plotting the inclusion regions as given by (4.2.11).
Example 4.2.12. Consider the quadratic matrix polynomialT(z) =A2z2+A1z+A0, where
A2 =
−1.61711.3891 1.86991.9302 −1.17651.1206 1.6517 0.0896 2.1788
, A1 =
−1.84632.9724 2.41043.3995 −2.04701.4926 2.2453 −0.7627 2.8893
,
A0 =
0.79220.9595 0.03570.8491 0.67870.7577 0.6557 0.9339 0.7431
.
Here we plot the localization sets with respect to the norm ë.ë2.Since the leading co- efficient matrixA2 is non-singular, all the eigenvalues ofT(z)are finite. Figure 4.2.1 (left) plots block Geršgorin sets associated with linearizations inL2(P)given by The- orem 4.2.5. In view of Remark 4.2.6, we choose W1 = 0 and W2 = −A1, and v =
v1
v2
∈ C2 as v1 = 2ëA2ë2, v2 = σmin(A2). Figure 4.2.1 (right) plots block Geršgorin sets associated with the second companion linearization given by Theorem 4.2.1(ii).
Figures 4.2.2 and 4.2.3 plot inclusion sets given by Corollary 4.2.11 for various forms of R1 and R2.
The choices of R1 and R2 in Figure 4.2.2 (left) are given by (4.2.5) and (4.2.6) respectively with W1 = 0 and W2 =−A1 (as A1 is non-singular).
The sets R1 and R2 in Figure 4.2.2 (right) arise from linearizations in DL(P) as given by Theorem 4.2.7. Here since σ −2
min(A2)ëA2ë2 is not an eigenvalue of T(z), we choose v1 =ëA2ë2 and v2 = σ 2
min(A2).
Finally, the R1 and R2 in Figure 4.2.3 are associated with the first companion linearization given by Theorem 4.2.1(i).
For comparison we also plot the Geršgorin-type localization sets for T(z) as ob- tained in Chapter 3 without going to its linearization. For this purpose consider a
90 4.2. Block Geršgorin sets arising from linearizations of quadratic matrix . . . partition π ={0,2,3} of T(z) and norm ë·ë2. Then the block Geršgorin set is illus- trated in Figure 4.2.4 (left). The block minimal Geršgorin set, the block Brualdi set and the block Brauer set are identical and are as shown in Figure 4.2.4 (right).
Figure 4.2.1: Block Geršgorin sets for T(z) corresponding to linearizations in L2(P) (left) and second companion form (right) for T(z). The eigenvalues are represented by •.
Figure 4.2.2: Inclusion sets given by Corollary 4.2.11 where R1 and R2 are associ- ated with linearizations in L1(P) (left) and with linearizations in DL(P) (right) for T(z). The symbol • denote the eigenvalues.
simple upper and lower bounds on the eigenvalues of the matrix polynomials.
These new localization sets constructed via the linearizations of the quadratic matrix polynomials also lead to simple conditions on the coefficient matrices of the polyno- mials so that their eigenvalues are located in particular regions of the complex plane.
These regions are chosen to be ones that are important in applications. The analysis results in upper bounds on the solutions of a number of distance problems associated with these polynomials.
Figure 4.2.3: Inclusion set given by Corollary 4.2.11 where R1 and R2 are associ- ated with the first companion linearization for T(z). The eigenvalues are denoted by
•.
-3 -2 -1 0 1 2
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Figure 4.2.4: Block Geršgorin set (left) and block minimal Geršgorin set (right) corresponding to the partitionπ={0,2,3}forT(z). The eigenvalues are represented by •.