2. On Numerical Radius of a Matrix

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Volume 2012, Article ID 129132,15pages doi:10.1155/2012/129132

Research Article

On Numerical Radius of a Matrix and Estimation of Bounds for Zeros of a Polynomial

Kallol Paul and Santanu Bag

Department of Mathematics, Jadavpur University, Kolkata 700032, India

Correspondence should be addressed to Kallol Paul,[email protected] Received 31 March 2012; Accepted 5 June 2012

Academic Editor: Teodor Bulboaca

Copyrightq2012 K. Paul and S. Bag. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We obtain inequalities involving numerical radius of a matrixA∈MnC. Using this result, we find upper bounds for zeros of a given polynomial. We also give a method to estimate the spectral radius of a given matrixA∈MnCup to the desired degree of accuracy.

1. Introduction

Suppose A ∈ MnC. LetWA,σAdenote respectively the numerical range, spectrum ofAandwA,rσAdenote respectively the numerical radius, spectral radius ofA, that is,

WA {Ax, x:x1}, wA sup{|λ|:λ∈WA},

σA

λ:λis an eigenvalue ofA , rσA sup{|λ|:λ∈σA}.

1.1

It is well known that

iA/2≤wA≤ A.

Kittaneh1improved on the second inequality to prove that.

iiwA≤1/2A1/2A²^1/2.

Clearly,1/2A 1/2A²^1/2 ≤ A so that inequality ii is sharper than the second inequality ofi.

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Letpz zⁿan−1zⁿ⁻¹· · ·a1za0be a monic polynomial wherea0, a1, . . . , an−1are complex numbers and let

C p

⎛

⎜⎜

⎜⎝

−an−1 −an−2 · · −a1 −a0

1 0 · · 0 0

0 1 · · 0 0

0 0 · · 0 0

· · · ·

· · · · 1 0

⎞

⎟⎟

⎟⎠

1.2

be the Frobenius companion matrix of the polynomialpz. Then, it is well known that zeros ofpare exactly the eigenvalues ofCp. ConsideringCpas an element ofMnC, we see that ifzis root of the polynomial equationpz 0, then

|z| ≤w C

p

, |z| ≤rσ

C p

. 1.3

Based on inequalityii, Kittaneh1 obtained an estimation for wCpwhich gives an upper bound for zeros of the polynomialpz.

InSection 1we find numerical radius of some special class of matrices and use the results obtained to give a better estimation of bounds for zeros of a polynomial.

2. On Numerical Radius of a Matrix

We first obtain bounds for numerical radius of a matrix in MnC and use it to obtain numerical radius for some special class of matrices.

Theorem 2.1. SupposeT ∈MnCand

T A B

C D

, 2.1

whereA∈MrC,B∈Mr,n−rC,C∈Mn−r,rCandD∈Mn−rC. Then, iwT≤1/2wA wD

wA−wD² BC²and

iiT² ≤ 1/2A² B² C² D² 1/2

A²C²− B²− D²²4ABCD². Proof. iLetZ∈Cⁿand

Z X

Y

, 2.2

whereX∈C^randY ∈C^n−rwithZ1.

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Then,

TZ, Z

AXBY CXDY

,

X Y

AX, XBY, XCX, YDY, Y 2.3

and so

|TZ, Z| ≤ |AX, X||DY, Y|BXYCXY. 2.4

Therefore, we have

wT≤ sup

X²Y²1

wAX²wDY² BCXY ,

sup

θ∈0,2π

wAcos²θwDsin²θ BCcosθsinθ ,

≤ 1 2

wA wD

wA−wD² BC²

.

2.5

This completes the first part of the proof.

iiProceeding as iniwe can prove the second part. This completes the proof of the theorem.

Remark 2.2. As an application ofiinTheorem 2.1,Thas another estimation by T² T^∗TwT^∗Tas follows:

T²≤ 1 2

wA^∗AC^∗C wB^∗BD^∗D

wA^∗AC^∗C−wB^∗BD^∗D²4A^∗BC^∗D²

.

2.6

Furuta2obtained numerical radius for a bounded linear operatorTof the above form with A aIr,B bA,C cA^∗,D dIn−r, and a, b, c, d ∈ R. If we consider A aIr,D dIn−r,C0_n−r,r wherea, d∈R, then we can exactly calculatewTandTas proved in the next theorem.

Theorem 2.3. SupposeB∈Mr,n−rCand

T

aIr B On−r,r dIn−r

. 2.7

Then

iwT 1/2|ad|

a−d²B²and iiT 1/√

2

a²d²B²

a²−d²− B²²4a²B².

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Proof. iFollowing the method employed in the previous theorem, we can show that

wT≤ 1 2

|ad|

a−d²B²

. 2.8

We only need to show that there existsz0, z0 1 such that|Tz0, z0|equals the quantity in the RHS.

SupposeBattains its norm atywithy1.

Letz By ky^twherekis a scalar. Then,z²B²|k|². Now

Tz, z

aIr B On−r,r dIn−r

By

ky

,

By

ky

2.9

so that

Tz, z · 1 z²

akB²dk²

B²k² . 2.10

Thus for all scalark, we get

wT≥

akB²dk²

B²k² . 2.11

Case 1da≥0. Define a functionφ:R → Rby

φk akB²dk²

B²k² . 2.12

Then using elementary calculus, we can show thatφkattains its maximum atk0 d−a d−a²B²so that forz0 1/

By²k0y²By k0y^twe get

|Tz0, z0| 1 2

ad

a−d²B²

. 2.13

Thus, we get

wT 1

2

|ad|

a−d²B²

. 2.14

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Case 2da≤0. As before we can show that there existsk0 d−a−

d−a²B²so that forz0 1/

By²k0y²By k0y^twe get

|Tz0, z0| 1 2

|ad|

a−d²B²

. 2.15

Thus in all cases, we get

wT 1

2

|ad|

a−d²B²

. 2.16

This completes the proof ofi.

iiThe proof is similar to the earlier one.

This completes the proof of the theorem.

UsingTheorem 2.3, we can find numerical radius of an idempotent matrixA, that is, a matrix for whichA²Aand also for a matrix for whichA²I.

Corollary 2.4. SupposeA∈MnCwithA²A. Then

wA A

2 1

2. 2.17

Proof. By Schur’s theorem,Ais unitarily equivalent to an upper triangular matrix. So without loss of generality, we can assume that

A

Ir Br,n−r

O_n−r,r 0_n−r

, 2.18

whereIris the identity matrix,B_r,n−r is any matrix. Using the last theorem, we get

wA A

2 1

2. 2.19

Corollary 2.5. SupposeA∈MnCandA²I. Then

wA 1

2

A 1

A

. 2.20

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Proof. Acan be expressed as

A

Ir Br,n−r

O_n−r,r −I_n−r

, 2.21

whereIris the identity matrix,Br,n−r is any matrix. ByTheorem 2.3, we have

A

11

2B² 1 2B

4B²,

wA 1

2

4B² .

2.22

Therefore

A²11

2B²1

2B

4B², 1

A² 1

1 1/2B² 1/2B

4B² 11

2B²−1

2B

4B².

2.23

By adding, we get

A² 1

A² 2B² ⇒

A 1

A 2

4B²

⇒wA 1 2

A 1

A

.

2.24

Corollary 2.6. SupposeA∈MnCwithA²ⁿI. Then

wA≥ 1

2

Aⁿ 1 Aⁿ

1/n

. 2.25

Proof. It follows from the fact thatAⁿ²IandwAⁿ≥wAⁿ.

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3. Bounds for Zeros of Polynomials

Letpz zⁿa_n−1zⁿ⁻¹· · ·a1za0be a monic polynomial wherea0, a1, . . . , a_n−1are complex numbers and let

C p

⎛

⎜⎜

⎜⎝

−a_n−1 −a_n−2 · · −a1 −a0

1 0 · · 0 0

0 1 · · 0 0

0 0 · · 0 0

· · · ·

· · · · 1 0

⎞

⎟⎟

⎟⎠

3.1

be the Frobenius companion matrix of the polynomialpz. Then, it is well known that zeros ofpare exactly the eigenvalues ofCp. ConsideringCpas a linear operator onCⁿ, we see that ifzis root of the polynomial equationpz 0 then

|z| ≤w C

p asσ

C p

⊂W C

p

, 3.2

where σCp is the spectrum of operator Cp. Estimation of the roots of zeros of the polynomialpzhas been done by many mathematicians over the years, some of them are mentioned below. Letλbe a root of the polynomial equationpz 0.

iCarmichael and Mason3proved that

|λ| ≤

1|a0|²|a1|²· · ·|a_n−1|²_1/2

. 3.3

iiMontel4,5proved that

|λ| ≤ |a0||a0−a1|· · ·|an−2−an−1||an−11|,

|λ| ≤n−1 |a0||a1|· · ·|a_n−1|. 3.4

iiiCauchy3proved that

|λ| ≤1max{|a0|,|a1|, . . . ,|an−1|}. 3.5

ivFujii and Kubo6,7proved that

|λ| ≤cos π n1 1

2

⎡

⎣ _n−2

i0

|ai|² 1/2

|an−1|

⎤

⎦. 3.6

vAlpin et al.8proved that

|λ| ≤max

1≤k≤n1|a_n−1|1|a_n−2|· · ·1|a_n−k|^1/k. 3.7

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viKittaneh1proved that

|λ| ≤ 1 2

!!C

p!!!!!C p₂!!!^1/2

. 3.8

We develop an inequality involving numerical radius with the help of which we estimate the zeros of the polynomialp. We show with examples that our estimation is better than the estimations mentioned above.

Theorem 3.1. Ifλis a zero of the polynomialpz, then

|λ| ≤ a_n−1

n 1

2

⎡

⎢⎢

⎢⎣cosπ n

#$

$$

$%cos²π n

⎛

⎝1

#$

$%ⁿ⁻²

r0|αr|²

⎞

⎠

2

⎤

⎥⎥

⎥⎦, 3.9

whereαr '_n−r

k0nCk−a_n−1/n^kan−k, r 0,1,2, . . . , n−2.

Proof. Puttingzξhin the polynomial equationpz zⁿanzⁿ⁻¹· · ·a2za10, we get ξhⁿan−1ξhⁿ⁻¹· · ·a1ξh a0 0. 3.10

Substitutingh−a_n−1/n, we get

ξⁿαn−2ξⁿ⁻²αn−3ξⁿ⁻³· · ·α1ξα00, 3.11

whereαr'_n−r

k0nCK−an−1/n^kan−k, r0,1,2, . . . , n−2.

LetCξbe the Frobenius companion matrix of the polynomialqξ ξⁿα_n−2ξⁿ⁻² αn−3ξⁿ⁻³· · ·α1ξα0.

ThenCξ _{A B}

C D

,whereA 0₁, B −αn−2,−αn−3, . . . ,−α0_1,n−1

C

⎛

⎜⎜

⎜⎝ 1 0

·

· 0

⎞

⎟⎟

⎟⎠

n−1,1

, D

⎛

⎜⎜

⎜⎝

0 0 · · 0 1 0 · · 0

· · · · 0

· · · · 0 0 0 · 1 0

⎞

⎟⎟

⎟⎠

n−1,n−1

. 3.12

UsingTheorem 2.1, we get wCξ≤ 1

2

wA wD

wA−wD² BC²

⇒wCξ≤ 1 2

⎡

⎢⎢

⎢⎣cosπ n

#$

$$

$%cos²π n

⎛

⎝1

#$

$%ⁿ⁻²

r0

|αr|²

⎞

⎠

2

⎤

⎥⎥

⎥⎦.

3.13

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This shows that ifξ0is a zero of the polynomialqξ, then

|ξ0| ≤ 1 2

⎡

⎢⎢

⎢⎣cosπ n

#$

$$

$%cos²π n

⎛

⎝1

#$

$%ⁿ⁻²

r0

|αr|²

⎞

⎠

2

⎤

⎥⎥

⎥⎦. 3.14

Thus ifλis a zero of the polynomialpz, then

|λ| ≤ an−1

n 1

2

⎡

⎢⎢

⎢⎣cosπ n

#$

$$

$%cos²π n

⎛

⎝1

#$

$%ⁿ⁻²

r0

|αr|²

⎞

⎠

2

⎤

⎥⎥

⎥⎦. 3.15

Example 3.2. Consider the polynomial equationpz z³−3z²2z 0. Then the bounds estimated by diﬀerent mathematicians are as shown inTable 1.

But our estimation shows that ifλis a zero of the polynomial then|λ| ≤ 2.280776406 which is much better than all the estimations mentioned above.

The companion matrix of the polynomial after removing the second term can be written as

⎛

⎝0 1 0 1 0 0 0 1 0

⎞

⎠

A2,2 B2,1

C1,1 D1,1

. 3.16

Then using the above theorems, it is easy to show that |λ| ≤ 2.207 which is even better estimation.

Example 3.3. Consider the polynomial equationpz z⁵−8z⁴25z³−38z²28z−8 0.

Then, the bounds estimated by diﬀerent mathematicians are as shown inTable 2.

But our estimation shows that ifλis a zero of the polynomial then|λ| ≤ 2.703669110 which is much better than all the estimations mentioned above.

Theorem 3.4. Letpz zⁿan−1zⁿ⁻¹· · ·a1za0havingαi i1,2, . . . , nas zeros and for each m∈N,pmz zⁿa^m_n−1zⁿ⁻¹· · ·a^m₂ za^m₁ za^m₀ is a polynomial havingα²_i^m i1,2, . . . , n as zeros. Ifλis a zero of the polynomialpz, then for allm

|λ| ≤

⎛

⎜⎜

⎝1 2

⎡

⎢⎢

⎣a^m_n−1cos π

n

#$

$$

%

a^m_n−1−cos π

n 2

⎛

⎝1

#$

$%ⁿ

k2

a^m_n−k²

⎞

⎠

2⎤

⎥⎥

⎦

⎞

⎟⎟

⎠

1/2^m

.

3.17

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Table 1

Carmichael and Mason 3.741657387

Montel 7

Cauchy 4

Fujii and Kubo 4.0098824

Yuri, Chien and Yeh 3.464101615

Kittaneh 3.44572894

Table 2

Carmichael and Mason 54.60769176

Montel 215

Cauchy 39

Fujii and Kubo 32.16529279

Yuri, Chien and Yeh 9

Proof. We first prove the lemma which shows that the coeﬃcients ofpmzcan be expressed in terms of coeﬃcients ofpz.

Lemma 3.5. Supposepz zⁿan−1zⁿ⁻¹· · ·a1za0is a monic polynomial, wherea0, a1, . . . , an−1

are complex numbers and αi, i 1,2, . . . , nare the zeros of this polynomial. If p1z zⁿ a¹_n−1zⁿ⁻¹· · ·a¹₁ za¹₀ is the polynomial havingα²_i i1,2, . . . , nas zeros, then forr 1,2, . . . , n:

a¹r −1^2n−r

a²_r2

n−r

k1

−1^ka_rka_r−k

, wherean1, a_nka_n−k0. 3.18

Proof. We have

det

z²I−C p₂

det zI−C

p det

zIC p ⇒p1

z²

pzp−z

⇒z²ⁿa¹_n−1z²ⁿ⁻¹· · ·a¹₁ z²a¹₀ −1ⁿ

zⁿa_n−1zⁿ⁻¹· · ·a1za0

×

zⁿ−a_n−1zⁿ⁻¹· · · −1ⁿ⁻¹a1z −1ⁿa0

.

3.19

Comparing the coeﬃcient ofz^2r, we get forr1,2, . . . n:

a¹r −1^2n−r

a²_r2 n−r k1

−1^karkar−k

, wherean1, ankan−k0. 3.20

This completes the proof of lemma.

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The companion matrix of the monic polynomialpmzis

C pm

⎛

⎜⎜

⎜⎝

−a^m_n−1 −a^m_n−2 · · −a^m₁ −a^m₀

1 0 · · 0 0

0 1 · · 0 0

0 0 · · 0 0

· · · ·

0 0 · · 1 0

⎞

⎟⎟

⎟⎠

. 3.21

We have

w C

pm

≥rσ

C pm

rσ

C p2^m

. 3.22

So

rσ

C p

≤ w

C pm

_1/2^m

. 3.23

UsingTheorem 2.1, we get

w C

pm

≤ 1 2

⎡

⎢⎢

⎣a^m_n−1cos π

n

#$

$$

%

a^m_n−1−cos π

n ₂

⎛

⎝1

#$

$%ⁿ

k2

a^m_n−k²

⎞

⎠

2⎤

⎥⎥

⎦. 3.24

Thus ifλis a zero of the polynomialpz, then

|λ| ≤

⎛

⎜⎜

⎝1 2

⎡

⎢⎢

⎣a^m_n−1cos π

n

#$

$$

%

a^m_n−1−cos π

n ₂

⎛

⎝1

#$

$%ⁿ

k2

a^m_n−k²

⎞

⎠

2⎤

⎥⎥

⎦

⎞

⎟⎟

⎠

1/2^m

.

3.25

We next prove the theorem.

Theorem 3.6. Supposepz zⁿan−1zⁿ⁻¹· · ·a1za0is a monic polynomial andαiare the roots of this equationi1,2, . . . , n, wherea0, a1, . . . , an−1are complex numbers with|α1|>1>|α2|>· · ·>

|αn|. If the equation having rootsα²_i^m fori1,2, . . . , nispmz zⁿa^m_n−1zⁿ⁻¹· · ·a^m₁ za^m₀ , then there existsm0∈Nsuch that

1|a^m_n−k| ≤ |a^m_n−1|whenever m≥m0 and for k2,3, . . . , n;

2 wCmp^1/2^m converges torσCp.

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Proof. 1We prove this fork2 and the rest are similar.

First observe that a^m_n−1

n k1

α²_k^m

≥ |α1|²^m−ⁿ

k2

|αk|²^m, a^m_n−2

n j /k1

α²_j^mα²_k^m ≤ |α1|²^m

_n

k2

|αk|²^m

ⁿ

j /k2

αj²^m|αk|²^m.

3.26

Now in order to have

a^m_n−2≤a^m_n−1. 3.27

We get

|α1|²^m −ⁿ

k2

|αk|²^m ≥ |α1|²^m _n

k2

|αk|²^m

ⁿ

j /k2

αj²^m|αk|²^m, 3.28

that is,

|α1|²^m ≥

1|α1|²^m(ⁿ

k2

|αk|²^m )

ⁿ

j /k2

αj²^m|αk|²^m, 3.29

that is,

|α1|²^m 1|α1|²^m ≥

'_n

k2|αk|²^m 1|α1|²^m

'_n

j /k2αj²^m|αk|²^m

1|α1|²^m . 3.30

Clearly, this inequality holds good as the left-hand side converges to 1, but the right-hand side converges to 0.

2We have

w Cm

p

≤ 1 2

⎡

⎢⎢

⎣a^m_n−1cos π

n

#$

$$

%

a^m_n−1−cos π

n ₂

⎛

⎝1

#$

$%ⁿ

k2

a^m_n−k²

⎞

⎠

2⎤

⎥⎥

⎦, 3.31

that is,

w Cm

p

≤ 1 2

⎡

⎣a^m_n−11

a^m_n−1² 1√

n−2a^m_n−1 ₂⎤

⎦, 3.32

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that is,

w Cm

p

≤ 1 2

⎡

⎣a^m_n−1a^m_n−1

a^m_n−1² a^m_n−1√

n−2a^m_n−1 ₂⎤

⎦. 3.33

So we get

w Cm

p

≤Ka^m_n−1≤K

n i1

α²_i^m

≤K|α1|²^m

1ⁿ

i2

αi

α1

²

m

. 3.34

Now

rσ

C p₂^m

rσ

Cm

p

≤w Cm

p

≤K|α1|²^m

1ⁿ

i2

αi

α1

²^m

. 3.35

Therefore,

rσ

C p

≤* w

Cm

p+1/2^m

≤ |α1|

K

1ⁿ

i2

αi

α1

²^m

_1/2^m

. 3.36

As the terms inside the bracket on the RHS converges to 1, we get the desired result.

Application. As an application we can exactly find the spectral radius of a given matrix.

Consider a given matrixAof ordern.

Step 1. We first find the characteristic polynomialqz zⁿbn−1zⁿ⁻¹· · ·b1zb0. Suppose αi, i1,2, . . . , rare the distinct roots ofqz 0 with|α1|>|α2|>· · ·>|αr|.

Step 2. Findpz qz/gcdqz, qz z^rar−1z^r−1· · ·a1za0. Then, roots ofqz 0 areαi, i 1,2, . . . , r without multiplicity. Letpmz zⁿa^m_n−1zⁿ⁻¹· · ·a^m₁ za^m₀ be the polynomial havingα²_i^m fori1,2, . . . , ras its zeros.

Step 3. Since|α2| < |α1|, taking < 1/4|α1| − |α2|, we can see that|α2| < |α1 −2 . Again using a result of9, we get|a^m_n−1|^1/2^m converging to|α1|. So for this there exists anm0 ∈N such that|α1| − <|a^m_n−1|^1/2^m <|α1| for allm≥m0. Therefore,

|α2| <|α1| − <a^m_n−1^1/2^m <|α1| ∀m≥m0. 3.37

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Table 3

No. of iterations a^m₄ a^m₃ a^m₂ a^m₁ a^m₀ rσCqz

0 −1 0 0 0 −2⁻⁵ 2.85

1 −1 0 −2⁻⁴ 0 −2⁻¹⁰ 2.40

2 −1 −2⁻³ −2⁻⁹ −2⁻¹³ −2⁻²⁰ 2.20

3 −1.25 31×2⁻¹² 13×2⁻¹⁹ 3×2⁻²⁸ −2⁻⁴⁰ 2.11

4 1.547363281 0.000119 0 0 0 2.07

5 2.394094544 0 0 0 0 2.05

Step 4. Lett|a^m_n−1⁰|^1/2^m⁰ − . Findsz z^r'_r

k1ar−k/t^kz^r−kz^rcr−1z^r−1· · ·c1zc0. If the roots ofsz 0 areβi, thenβiαi/t, i1,2, . . . , rand

β1α1

t

>1>α2

t

β2>β3>· · ·>βr. 3.38

Then,szsatisfies all the criterion ofTheorem 3.6.

Step 5. The required sequence is xm twCms^1/2^m, which converges to the spectral radius of matrixA.

Example 3.7. Consider the 5th-degree polynomialqz z⁵2z⁴1.

By Rouche’s theorem, it is easy to see that all the roots except one are enclosed by the simple closed curve|z|2.

Considersz z⁵z⁴ 1/2⁵and then iterating the coeﬃcints ofQzwe get the following.

The highest absolute value of the zeros of the polynomial is 2.055 and by 5th iteration we get 2.05. Continuing the above process, we can find the highest absolute value of the zeros of the polynomial up to the desired degree of accuracy. The previous best result for this is known to be 2.414 given by Alpin8. The iterations are shown inTable 3.

Acknowledgments

The authors would like to thank the referees for their suggestions, one of the referees suggested the inclusion ofRemark 2.2followingTheorem 2.1. The research of the first author is partially supported by PURSE-DST, Govt. of India and the research of the second author is supported by CSIR, India.

References

1 F. Kittaneh, “A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix,” Studia Mathematica, vol. 158, no. 1, pp. 11–17, 2003.

2 T. Furuta, “Applications of polar decompositions of idempotent and 2-nilpotent operators,” Linear and Multilinear Algebra, vol. 56, no. 1-2, pp. 69–79, 2008.

3 R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.

4 P. Montel, “Sur quelques limites pour les modules des z´eros des polynomes,” Commentarii Mathematici Helvetici, vol. 7, no. 1, pp. 178–200, 1934.

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5 P. Montel, “Sur les bornes des modules des zeros des polynomes,” Tohoku Mathematical Journal, vol. 41, 1936.

6 M. Fujii and F. Kubo, “Operator norms as bounds for roots of algebraic equations,” Proceedings of the Japan Academy, vol. 49, pp. 805–808, 1973.

7 M. Fujii and F. Kubo, “Buzano’s inequality and bounds for roots of algebraic equations,” Proceedings of the American Mathematical Society, vol. 117, no. 2, pp. 359–361, 1993.

8 Y. A. Alpin, M.-T. Chien, and L. Yeh, “The numerical radius and bounds for zeros of a polynomial,”

Proceedings of the American Mathematical Society, vol. 131, no. 3, pp. 725–730, 2003.

9 C. A. Hutchinson, “On Graeﬀe’s method for the numerical solution of algebraic equations,” The American Mathematical Monthly, vol. 42, no. 3, pp. 149–161, 1935.

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