Let P ( X ) be a polynomial:

P ( X ) = X n + a n − 1 X n − 1 + ⋯ + a 1 X + a 0 ∈ ℂ [ X ]

( a 0 ≠ 0 , because when a 0 = 0 , then we have an evident root which is 0, we exclude this case, to studie non obvious cases).

Let C ( P ) be a matrix:

C ( P ) : = ( 0 ⋯ 0 0 − a 0 1 0 ⋯ 0 − a 1 ⋮ ⋱ ⋱ ⋮ ⋮ 0 ⋯ 1 ⋱ − a n − 2 0 ⋯ 0 1 − a n − 1 ) (1)

Let P C ( P ) ( X ) be, the characteristic polynomial of the matrix C ( P ) , I n is the identity matrix of degree n. The characteristic polynomial P C ( P ) ( X ) of the matrix C ( P ) coincides with P ( X ) :

P C ( P ) ( X ) : = det ( X I n − C ( P ) ) = P ( X ) .

With the help of the expansion with respect to the first column it is easy to verify by induction that:

det ( X I n − C ( P ) ) = P ( X ) = X n + a n − 1 X n − 1 + ⋯ + a 1 X + a 0 .

The given n × n matrix C ( P ) is called “Frobenius matrix” or the “standard Frobenius companion matrix of the polynomial P ( X ) ”.

Definition 1. [3]

We define a “companion matrix” to be an n × n matrix A over F [ a 0 , a 1 , ⋯ , a n − 1 ] with ( n 2 − n ) entries constant in F and the remaining entries − a 0 , − a 1 , ⋯ , − a n − 1 such that the characteristic polynomial of A is P ( X ) . We say that two companion matrices A and B are equivalent if either A or A^T can be obtained from B via a permutation similarity.

There are many other Frobenius companion matrices. In this paper, we notice:

F ( P ) is the classical Frobenius companion matrix associated to the polynomial P ( X ) :

F ( P ) = ( 0 1 0 ⋯ 0 0 0 1 ⋯ 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 ⋯ 1 − a 0 − a 1 − a 2 ⋯ − a n − 1 ) (2)

C 1 ( P ) is the first Frobenius companion matrix associated to the polynomial P ( X ) :

C 1 ( P ) = ( − a n − 1 − a n − 2 ⋯ − a 1 − a 0 1 0 ⋯ 0 0 0 1 ⋱ 0 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 1 0 ) (3)

C 2 ( P ) is the second Frobenius companion matrix associated to the polynomial P ( X ) :

C 2 ( P ) = ( − a n − 1 1 0 ⋯ 0 − a n − 2 0 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 − a 1 0 0 ⋱ 1 − a 0 0 0 ⋯ 0 ) (4)

Companion matrices have a great interest in that they offer the possibility of elegant demonstrations.

A common way to obtain inclusion areas for the zeros of a polynomial is to find eigenvalue inclusion areas for a companion matrix of the polynomial, whose eigenvalues are precisely the zeros of the polynomial.

In this section, it is about locating zeros of polynomials from the Gershgorin’s disks. These disks are estimated by using the diagonal elements of the matrix. The radii are then calculated from the polynomial’s coefficients. Gershgorin’s theorem states that the union of these disks contains all eigenvalues.

We formally state here Gershgorin’s theorem as lemma and its application to companion matrices.

Lemma 1. For a given matrix A of dimension n × n with complex elements a i j , all the eigenvalues are located in the union of the following n disks so that: [1].

Considering the rows:

∪ i = 1 n { z ∈ ℂ : | z − a i i | ≤ ∑ j = 1 , j ≠ i n | a i j | } (5)

Considering the columns:

∪ j = 1 n { z ∈ ℂ : | z − a i i | ≤ ∑ i = 1 , i ≠ j n | a i j | } (6)

Proof. Let λ be an eigenvalue of a complex n × n matrix A and whose corresponding eigenvectors are x.

So we have A x = λ x .

If x is an eigenvector, it has at least one of its components that is non-zero. Let x k be the component of x having the greatest absolute value, such that | x k | ≥ | x i | for all i = 1 , 2 , ⋯ , n and x k ≠ 0 .

As ( A x ) k = ( λ . x ) k , we have:

λ . x k = a k k x k + ∑ j = 1 , j ≠ k n     a k j x j ,

hence ( λ − a k k ) x k = ∑ j = 1 , j ≠ k n     a k j x j .

We will take the absolute value on either side of the equality and use the triangular inequality while dividing by | x k | .

Since | x j | / | x k | < 1 for every j ≠ k , we become:

| λ − a k k | ≤ ∑ j = 1 , j ≠ k n | a k j | | x j | | x k | ≤ ∑ j = 1 , j ≠ k n | a k j |

λ is then located in the disk a k k .

Without knowing the eigenvectors, we do not know to which k each eigenvalue corresponds. We must therefore take the union of all these disks to obtain a region which guarantees to contain all the eigenvalues in a safe way.

The proof of the theorem doesn’t take into account any particular property of the matrix A. It can be then interesting to consider some properties of matrices and to study the advantages related to these particularities.

Since our study concerns Fiedler companion matrices, we will first consider the particular properties of companion matrices by using explicit forms.

We consider the classical companion matrix C ( P ) :

C ( P ) = ( 0 ⋯ 0 0 − a 0 1 0 ⋯ 0 − a 1 ⋮ ⋱ ⋱ ⋮ ⋮ 0 ⋯ 1 ⋱ − a n − 2 0 ⋯ 0 1 − a n − 1 ) (7)

It is already known that the eigenvalues of the matrix C ( P ) are roots of its characteristic polynomial P ( X ) ∈ ℂ [ X ] :

P ( X ) = X n + a n − 1 X n − 1 + ⋯ + a 1 X + a 0 , a 0 ≠ 0 .

Gershgorin’s theorem applies to the matrix C ( P ) guarantees that all the zeros of the polynomial P ( X ) belong to Γ ( 0 ) which is the union of n disks such as:

Γ ( 0 ) = ∪ j = 1 n     Γ j .

where:

Γ 1 = { X ∈ ℂ : | X | ≤ | a 0 | }                               ⋮ Γ j = { X ∈ ℂ : | X | ≤ 1 + | a j − 1 | }     for     2 ≤ j ≤ n − 1                               ⋮ Γ n = { X ∈ ℂ : | X + a n − 1 | ≤ 1 }

Let λ be a zero of the polynomial P ( X ) .

The inequalities allow us to say that for any zero λ of P ( X ) , we have:

| λ | ≤ | Γ ( 0 ) | = max { | a 0 | , 1 + | a 1 | , ⋯ , 1 + | a n − 1 | } (8)

The obtained bound is better than the bound of Cauchy namely:

| λ | ≤ 1 + max { | a 0 | , | a 1 | , ⋯ , | a n − 1 | }

Example 1:

Let P ( X ) = X 8 − X 7 − 3 X 6 − 2 X 5 − 2 X 4 − 4 X 3 − 3 X 2 − 5 X − 3 be a complex polynomial.

By applying Gershgorin’s theorem, we have:

Γ 1 = { X ∈ ℂ : | X | ≤ 3 } Γ 2 = { X ∈ ℂ : | X | ≤ 1 + 5 = 6 } Γ 3 = { X ∈ ℂ : | X | ≤ 1 + 3 = 4 } Γ 4 = { X ∈ ℂ : | X | ≤ 1 + 4 = 5 }

Γ 5 = { X ∈ ℂ : | X | ≤ 1 + 2 = 3 } Γ 6 = { X ∈ ℂ : | X | ≤ 1 + 2 = 3 } Γ 8 = { X ∈ ℂ : | X − 1 | ≤ 1 }

So Γ ( 0 ) = Γ 2 ∪ Γ 8 = Γ 2 = { X ∈ ℂ : | X | ≤ 6 } .

Then, all the zeros of the polynomial P ( X ) are therefore included in the disk: Γ ( 0 ) { X ∈ ℂ : | X | ≤ 6 } .

The general form of the Gershgorin’s theorem doesn’t exploit the specific structure of the matrix. By rewriting the equations from which we obtain the Gershgorin’s disks and by exploiting them, we can obtain improved forms of those disks. More generally, we consider x and λ respectively eigenvalue and eigenvector of the companion matrix C ( P ) , we have:

λ x = C ( P ) x .

From this equality follows the equations:

λ x 1 = − a 0 x n λ x 2 = x 1 − a 1 x n λ x j = x j − 1 − a j − 1 x n   where     ( 3 ≤ j ≤ n − 2 ) λ x n − 1 = x n − 2 − a n − 2 x n λ x n = x n − 1 − a n − 1 x n

This set of equations is used to modify the Γ ( 0 ) disks of Gershgorin in order to obtain Γ ( 1 ) which allows to obtain an improvement of the bounds of the roots for polynomials. For this, we first make changes on Γ 1 to Ω 1 and then Γ 2 to Ω 2 respectively. Each step produces a new set of zero inclusions that are summarized in the following theorem [1].

Theorem 2. Let P ( X ) = X n + a n − 1 X n − 1 + ⋯ + a 1 X + a 0 , a 0 ≠ 0 , be a polynomial ∈ ℂ [ X ] . All the zeros λ of P ( X ) can be found in the set Γ ( 1 ) where:

Γ ( 1 ) = Ω 1 ∪ ( ∪ j = 2 n     Γ j ) .

We then have: | λ | ≤ | Γ ( 1 ) | = max { γ , 1 + | a 1 | , ⋯ , 1 + | a n − 1 | } where | Ω 1 | = γ .

Likewise, all the zeros λ of P ( X ) can be found in the set Γ ( 2 ) where:

Γ ( 2 ) = Ω 1 ∪ Ω 2 ∪ ( ∪ j = 3 n     Γ j ) .

We also then have: | λ | ≤ | Γ ( 2 ) | = max { γ , δ , ⋯ , 1 + | a n − 1 | } where | Ω 2 | = δ .

Proof. In order to change respectively Γ 1 to Ω 1 and then Γ 2 to Ω 2 , some steps are required. Each step allows to calculate a bound on the moduli of the zeros for the obtained modified set.

First modified disk Ω 1

We consider the vector x to be the eigenvector with the greatest absolute value. About this modification, we start from the equation λ x 1 = − a 0 x n . We eliminate x n in the equation λ x n = x n − 1 − a n − 1 x n .

It gives: λ ( λ + a n − 1 ) x 1 = − a 0 x n − 1 . So if x 1 has the greatest coefficient’s absolute value of the eigenvector x, we obtain | λ | ≤ | a 0 | in addition to the inequality.

We obtain another inequality that must be satisfied by λ precisely | λ | | λ + a n − 1 | ≤ | a 0 | .

We then define:

K 1 = { z ∈ ℂ : | z | | z + a n − 1 | ≤ | a 0 | } .

Ω 1 = Γ 1 ∩ K 1 .

The limit of K 1 is a quartic curve known as the oval of Cassini. It consists of either one or two loops. Ovals of Cassini also appear in a different and slightly more complicated eigenvalue inclusion set. We can therefore choose to replace Γ 1 by Ω 1 in Gershgorin’s theorem.

Since Ω 1 ⊆ Γ 1 , we clearly have Γ ( 1 ) ⊆ Γ ( 0 ) . The new obtained set of inclusions is smaller than Γ ( 0 ) . We can expect this to happen when | a 0 | is large enough for Γ 1 to dominate Γ ( 0 ) .

By changing Ω 0 to Ω 1 , from Γ ( 0 ) = ( ∪ j = 1 n     Γ j ) , we then get:

Γ ( 1 ) = Ω 1 ∪ ( ∪ j = 2 n     Γ j ) .

Let γ ∈ ℝ , so that | Ω 1 | = γ . It follows that from equation 8 we can conclude that:

| λ | ≤ | Γ ( 1 ) | = max { | Ω 1 | , 1 + | a 1 | , ⋯ , 1 + | a n − 1 | } = max { γ , 1 + | a 1 | , ⋯ , 1 + | a n − 1 | } .

Hence the first part of the theorem.

Second modified disk Ω 2

In the same way, we use the equation λ x 1 = − a 0 x n . We eliminate x n in the equation λ x n = x n − 1 − a n − 1 x n . Then, we use it to replace x n by − λ x 1 α 0 in the equation λ x 2 = x 1 − a 1 x n .

It gives λ x 2 = ( 1 + a 1 a 0 λ ) x 1 .

Since | x 2 | ≥ | x j | for every j, then we have | λ | | x 2 | = | 1 + a 1 a 0 λ | | x x | .

When we divide by | x 2 | , it gives | λ | ≤ | 1 + a 1 a 0 λ | .

This helps to define a new set K 2 = { z ∈ ℂ : | z | ≤ | 1 + a 1 a 0 λ | } and λ ∈ K 2 .

However, since we have | x 2 | ≥ | x 1 | for every j, we have from λ x 2 = x 1 − a 1 x n that λ ∈ Γ 2 . It is a disk centered at the origin with radius 1 + a 2 .

We conclude that, in this case, λ ∈ Γ 2 ∩ K 2 . We define a set Ω 2 so that Ω 2 = λ ∈ Γ 2 ∩ K 2 . Similarly to the set Γ ( 1 ) , from the equality Γ ( 2 ) = Ω 1 ∪ Ω 2 ∪ ( ∪ j = 3 n     Γ j ) , we can deduce that:

| λ | ≤ | Γ ( 2 ) | = max { | Ω 1 | , | Ω 2 | , 1 + | a 2 | , ⋯ , 1 + | a n − 1 | } = max { γ , δ , 1 + | a 2 | , ⋯ , 1 + | a n − 1 | } ,

where | Ω 1 | = γ and | Ω 2 | = δ .

The theorem is proved.

From this theorem, we consider estimated values of the litteratur for our illustrative examples.

We announce the following theorem as principal result of this work. The first part of the theorem was already part of former issued paper [2]. The originality of the theorem comes from the fact that, by using the Hessenberg form, in the special case that ∑ i = 0 n − 1 | a i | < 1 , we can automatically deduce a general constant bound for this kind of polynomials.

Theorem 3. (Hessenberg [2], [4] ).

Let P ( X ) = X n + a n − 1 X n − 1 + ⋯ + a 1 X + a 0 , a 0 ≠ 0 , be a polynomial ∈ ℂ [ X ] . Any zero λ of P ( X ) satisfies the relation:

| λ | ≤ ‖ L r ‖ ∞ where L r = ( 0 I r 0 − a n − 1 0 ⋮ I n − r − 1 − a r + 1 − a 0 ⋯ − a r 0 )

r = max { k | ∑ i = 0 k − 1 | a i | < 1 } , r = 0 if | a 0 | ≥ 1 .

When: ∑ i = 0 n − 1 | a i | < 1 then | λ | ≤ 1 .

Proof. The estimation of the general bound for | λ | can be found in [2].

What to be proved is the case: ∑ i = 0 n − 1 | a i | < 1 then | λ | ≤ 1 ?

In this case, since ∑ i = 0 n − 1 | a i | < 1 , from the first part of the theorem, we have for the matrix L n − 1 : L n − 1 = F ( P ) .

It follows that: | λ | ≤ max { 1 ; | a 1 | + | a 2 | + ⋯ + | a n − 1 | } = 1 .

We introduce the following theorem as a specific case of the modified disks of Gershgorin in which γ and δ are computed. It will be followed by other theorems of the litteratur that will be used for comparison to estimate bounds for roots of some given polynomials.

Theorem 4. (Ovals of Cassini [1] ).

Let P ( X ) = X n + a n − 1 X n − 1 + ⋯ + a 1 X + a 0 , a 0 ≠ 0 , be a polynomial ∈ ℂ [ X ] . Any zero λ of P ( X ) satisfies the relation λ ≤ | Γ | where:

Γ = max { γ , δ , 1 + | a 2 | , ⋯ , 1 + | a n − 1 | } (9)

γ = { 1 2 ( | a n − 1 | + | a n − 1 | 2 + 4 | a 0 | ) ( | a 0 | > 1 + | a n − 1 | ) 0 ( | a 0 | ≤ 1 + | a n − 1 | )

δ = { 1 + a 1 | a 0 | − | a 1 | ( | a 0 | > 1 + | a 1 | ) 1 + | a 1 | ( | a 0 | ≤ 1 + | a 1 | )

Proof. The Equation (9) comes from the theorem 2. Values of Γ and δ have been taken from [1] for our illustrative examples.

Theorem 5. (Dehmer [5] ).

Let P ( X ) = X n + a n − 1 X n − 1 + ⋯ + a 1 X + a 0 , a 0 ≠ 0 , be a polynomial ∈ ℂ [ X ] . Any zero λ of P ( X ) satisfies the relation:

| λ | ≤ 1 + ϕ 2 2 + ( ϕ 2 − 1 ) 2 + 4 M 1 2 .

M 1 : = max 0 ≤ i ≤ n − 2 | a j |     et     ϕ 2 : = | a n − 1 |

Proof. This bound is the usual bound found in the litteratur as Dehmer bound. It can be found in [5].

Theorem 6. (Cauchy, Montel and Carmichel-Mason [6] ).

Let P ( X ) = X n + a n − 1 X n − 1 + ⋯ + a 1 X + a 0 , a 0 ≠ 0 , be a polynomial ∈ ℂ [ X ] . Any zero λ of P ( X ) satisfies the relation:

• Cauchy’s bound:

| λ | ≤ max { | a 0 | , 1 + | a 1 | , ⋯ , 1 + | a n − 1 | } .

• Montel’s bound:

| λ | ≤ max { 1 , | a 0 | + | a 1 | + ⋯ + | a n − 1 | } .

• Carmichel-Mason’s bound:

| λ | ≤ 1 + | a 0 | 2 + ⋯ + | a n − 1 | 2 .

Proof. Cauchy’s bound, Montel’s bound and Carmichel-Mason’s bound are classical bounds that are usually found in the litteratur. We do not prove them in this paper [6].

In this section we will take some examples to illustrate the sharpness of the theorems on modified disks of Gershgorin for companion matrices. We will make a particular study on polynomial for degree n = 5 by using following bounds: Ovals of Cassini, Hessenberg, Dehmer, Cauchy, Montel and Carmichel-Mason. All the bounds that are going to be used are listed in Figure 1. Illustrative examples have been given in this section.

Example 2. Let P 1 ( X ) = X 5 + 4 X 4 + 3 X 3 + 2 X 2 + X + 5 ∈ ℂ [ X ] . We have: a 0 = 5 ⇒ | a 0 | > 1 .

The roots and their bounds of polynomial P 1 ( X ) are illustrated in Table 1 and Table 2.

Example 3. Soit P 2 ( X ) = X 5 − 10 X 4 + 30 X 3 − 20 X 2 − 31 X + 30 ∈ ℂ [ X ] . The roots and their bounds of the polynomial P 2 ( X ) are illustrated in Table 3 and Table 4.

Example 4. Soit P 3 ( X ) = X 5 − 19 X 4 + 126 X 3 − 346 X 2 + 385 X − 147 ∈ ℂ [ X ] . The roots and their bounds of polynomial P 3 ( X ) are illustrated in Table 5 and Table 6.

We can see that the sparse companion matrices become interesting for use, in case of small values of the coefficients of polynomials. This is why we make a closer look at these cases.

Example 5. Let

P 4 ( X ) = X 5 + 0.04 X 4 + 0.03 X 3 + 0.02 X 2 + 0.01 X + 0.05 ∈ ℂ [ X ] .

if a 0 = 0.05 , then | a 0 | < 1 , | a 0 | ≠ 0 and

| a 0 | + | a 1 | + | a 2 | + | a 3 | + | a 4 | = 0.05 + 0.01 + 0.02 + 0.03 + 0.04 = 0.15 .

The roots and their bounds of the polynomial P 4 ( x ) are illustrated in Table 7 and Table 8.

Example 6. Let

P 5 ( X ) = X 5 − 10 ( − 5 ) X 4 + 10 ( − 6 ) X 3 − 10 ( − 7 ) X 2 + 10 ( − 8 ) X − 10 ( − 9 ) ∈ ℂ [ X ] .

We have:

| a 0 | + | a 1 | + | a 2 | + | a 3 | + | a 4 | = 10 ( − 9 ) + 10 ( − 8 ) + 10 ( − 7 ) + 10 ( − 6 ) + 10 ( − 5 ) < 1 .

The roots and their bounds of the polynomial P 3 ( X ) are illustrate in Table 9 and Table 10.