Let P be a polynomial ∈ ℂ [ X ] , P ( X ) = X n + a n − 1 X n − 1 + ⋯ + a 1 X + a 0 .

Let A be the associated companion matrix of P ( X ) .

Definition 1: A Fiedler companion matrix is the pattern of 2 n − 1 different matrices which have exactly the zeros of polynomial P ( X ) as eigenvalue [2] [4] [7].

The first and the second Frobenius companion matrices, C 1 ( P ) and C 2 ( P ) of polynomial P ( X ) , are Fiedler matrices:

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

Some important properties of Fiedler matrices:

1) All Fiedler matrices are similar and so all of them have P ( X ) as charactéristic polynomial [2];

2) Due to a symmetric property of the basis factors of Fiedler matrices, the transpose of any Fiedler matrix is another Fiedler matrix [7].

The method of providing bounds for roots of polynomial P ( X ) consist in applying the following Gershgorin’s theorem to Fiedler companion matrices, for any λ ∈ ℂ , root of P ( X ) , to determine the nonnegative numbers L ( P ) and U ( P ) depending on the coefficients of P ( X ) such as [7]:

L ( P ) ≤ | λ | ≤ U ( P ) . (2)

We formally state the Gershgorin’s theorems and its application to companion matrices, in order to introduce the 1-norm and the ∞-norm.

Theorem 1. (Gershgorin’s Theorem and the 1-norm and ∞-norm) [6].

Let: A = ( a i j ) be an-by-n complex matrix, λ ∈ ℂ an eigenvalue of matrix A.

C j = ∑ i = 1 , i ≠ j n | a i j | the modulus of non diagonal elements of the column j.

All the eigenvalues λ of A are located in the union of the disks:

| λ − a j j | ≤ C j ,   ( j = 1 , 2 , ⋯ , n ) .

For each eigenvalue λ we have:

| λ | ≤ | a j j | + C j = max 1 ≤ j ≤ n ∑ i = 1 n | a i j | . (3)

This is equivalent to the 1-norm: ‖ A ‖ 1 . In the same way, we obtain the ∞-norm, ‖ A ‖ ∞ [1]:

| λ | ≤ | a i i | + R i = max 1 ≤ i ≤ n ∑ j = 1 n | a i j | (4)

The inequalities of theorem 1 have been used to get following classical bounds. [5]

Theorem 2. Let P ( X ) = X n + a n − 1 X n − 1 + ⋯ + a 1 X + a 0 be a monic polynomial, P ( X ) ∈ ℂ [ X ] , a 0 ≠ 0 and λ any root of P ( X ) . Then λ satisfies the following classical inequalities [7]:

1) Cauchy’s bounds: Gerschgorin’s theorem, applied to the second compagnon matrix C 2 ( P ) with the ∞-norm, gives:

| λ | ≤ max { | a 0 | , 1 + | a 1 | , ⋯ , 1 + | a n − 1 | } . (5)

2) Montel’s bounds: Gerschgorin’s theorem, applied to the second compagnon matrix C 2 ( P ) with ‖   .   ‖ 1 , gives:

| λ | ≤ max { 1 , | a 0 | + | a 1 | + ⋯ + | a n − 1 | } . (6)

3) Carmichel-Mason’s bounds: Gerschgorin’s theorem, applied to the second compagnon matrix C 2 ( P ) with ‖   .   ‖ 2 , gives:

| λ | ≤ 1 + | a 0 | 2 + ⋯ + | a n − 1 | 2 . (7)

The 2-norm is: ‖ A ‖ 2 = σ max ( A ) .

4) Frobenius’s bounds: Gerschgorin’s theorem, applied to the second compagnon matrix C 2 ( P ) with ‖   .   ‖ F , gives:

| λ | ≤ ( n − 1 ) + | a 0 | 2 + ⋯ + | a n − 1 | 2 . (8)

Some properties of the norms:

1) From Gerschgorin’s theorem, we can easily see that, bounds for roots of polynomials depend on the selected companion matrix for ‖   .   ‖ 1 and ‖   .   ‖ ∞ : a good choice of a Fiedler Matrix can give improved values of bounds;

2) By using C 1 ( P ) instead of C 2 ( P ) = ( C 1 ( P ) ) T , we obtain:

Cauchy’s bounds for the ‖   .   ‖ 1 ;

Montel’s bounds for the ‖   .   ‖ ∞ .

3) From the symmetric property of M i factors we can conclude that for determination of the sharpest bounds of the 1-norm and the ∞-norm, we need to analyze only the ∞-norm of all Fiedler matrices and their inverses because this includes also the bounds coming from the 1-norm as far as ‖   .   ‖ 1 = ‖   . T ‖ ∞ .

As far as the value of the bound for roots of a polynomial depends on the seleted companion matrix, let us introduced types of companion matrices that can hopefully improved the bound’s value.

We’ve seen that the Frobenius matrix is itself a Fiedler matrix. Sparse companion matrices has been recently characterized and they includes the Fiedler matrices as a special case.

Definition 2: the Frobenius companion matrix has exactly n − 1 of the constant entries set to 1, the remaining ( n − 1 ) 2 constant entries are zero. As noted in [1], while a companion matrix must have at least 2 n − 1 nonzero entries, there are companion matrices that have more than 2 n − 1 nonzero entries. As such, we will say that a companion matrix is Sparse if it has exactly 2 n − 1 nonzero entries. Sparse companion matrices have also been called intercyclic companion matrices because of an associated digraph structure as noted in [1].

If the ( n − 1 ) nonzero constant entries of a sparse companion are set to 1, then this companion matrix is called unit sparse.

Example 1. For the Frobenius companion matrix C 1 ( P ) :

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 ) (9)

We have:

● C 1 ( P ) has n − 1 entries set to 1 et n entries with the value a i ;

● ( n − 1 ) 2 entries of C 1 ( P ) are set to 0;

● ( 2 n − 1 ) entries are nonzeros in case a i ≠ 0, ∀ i .

Theorem 3. Let A be a ( n × n ) complex matrix. A is a unit sparse companion matrix if and only if A is equivalent to a unit lower Hessenberg matrix C for some ( n − m ) × ( m + 1 ) matrices K with m ( n − 1 − m ) zero entries, such that C has − a n − 1 − k on its k^th subdiagonal, for 0 ≤ k ≤ n − 1 [1].

C = ( 0 I m 0 K I n − m − 1 0 ) (10)

This form of matrix will be used to determin new bounds for roots of polynomials ( I m is the m × m identity matrix).

Due to the particular form of the unit lower Hessenberg matrix, that gives good possibilities for operations, it is important to check for a matrix in this form that provides good bounds. Thus, Vander Meulen establishes in [2] that there is a particular Fiedler matrix which provides the best upper bound for over all Fiedler matrices for the ∞-norm. This is obtained by using the following Hessenberg matrix.

Let L b be the following matrix:

L b = ( 0 I b 0 − a n − 1 0 ⋮ I n − b − 1 − a b + 1 − a 0 … − a b 0 ) (11)

L b is a Fiedler companion matrix in the form of Hessenberg [1]. This matrix is supposed to provide a good majoration of the eigenvalue for the infiny norm.

Theorem 4. Let P be a polynomial, P ( X ) = X n + a n − 1 X n − 1 + ⋯ + a 1 X + a 0 ∈ ℂ [ X ] , a 0 ≠ 0 .

Let F b be a Fiedler companion matrix of P in the Hessenberg form with a b appearing in row n for some b ∈ { 0,1,2, ⋯ , n − 2 } but a b + 1 is in row a b + 1 .

Let L b be a Fiedler companion matrix in the form of Hessenberg.

Let r ∈ ℕ so that:

r = max { k | ∑ i = 0 k − 1 | a i | < 1 } ,     r = 0     if     | a 0 | ≥ 1. (12)

Then for all b ∈ { 0,1,2, ⋯ , n − 2 } :

‖ L r ‖ ∞ ≤ ‖ L b ‖ ∞ ≤ ‖ F b ‖ ∞ . (13)

Thus for submultiplicative matrix norms, we have:

| λ | ≤ ‖ L r ‖ ∞ (14)

Proof.:

Equation (12) and (13) are respectively from theorems 4.1 and 4.2 in [1].

For K = max { | a i | , | i > b } and g = max { i | , | a i | = K } . Then ‖ L b ‖ ∞ = max { 1 + K , ∑ j = 0 b | a j | } .

There are no other coefficients of P in the same row as a g in L b but there may be more coefficients in the same row as a g in F b . Therefore ‖ F b ‖ ∞ ≥ 1 + K and ‖ F b ‖ ∞ ≥ ∑ j = 0 b | a j | . Thus ‖ F b ‖ ∞ ≥ ‖ L b ‖ ∞ .

Suppose b > r . Then

‖ L b ‖ ∞ = max { ∑ i = 0 b | a i | , 1 + | a b + 1 | , 1 + | a b + 2 | , ⋯ , 1 + | a n − 1 | } = max { ∑ i = 0 r | a i | + ∑ i = r + 1 b | a i | , 1 + | a b + 1 | , 1 + | a b + 2 | , ⋯ , 1 + | a n − 1 | } ≥ max { ∑ i = 0 r | a i | , 1 + | a r + 1 | , 1 + | a r + 2 | , ⋯ , 1 + | a n − 1 | } = ‖ L r ‖ ∞

Suppose b < r . Then

‖ L b ‖ ∞ = max { ∑ i = 0 b | a i | , 1 + | a b + 1 | , 1 + | a b + 2 | , ⋯ , 1 + | a n − 1 | } = max { 1 + | a b + 1 | , 1 + | a b + 2 | , ⋯ , 1 + | a n − 1 | } ≥ max { 1 + | a r | , 1 + | a r + 1 | , ⋯ , 1 + | a n − 1 | } ≥ max { ∑ i = 0 r | a i | , 1 + | a r + 1 | , 1 + | a r + 2 | , ⋯ , 1 + | a n − 1 | } = ‖ L r ‖ ∞

Since the ∞-norm is a matrix norm, we have ‖ A ⋅ B ‖ ∞ ≤ ‖ A ‖ ∞ ⋅ ‖ B ‖ ∞ for any matrix. [1]

That means:

If A = a i j , B = b i j and A B = ( a b ) i j .

‖ A B ‖ ∞ = max 1 ≤ i ≤ n ∑ j = 1 n | ( a b ) i j | ≤ ( max 1 ≤ i ≤ n ∑ j = 1 n | a i j | ) ⋅ ( max 1 ≤ i ≤ n ∑ j = 1 n | b i j | ) = ‖ A ‖ ∞ ⋅ ‖ B ‖ ∞

⇒ ‖   .   ‖ ∞ is submultiplicative.

Consequently we have: | λ | ≤ ‖ C ‖ for every eigenvalue λ of matrix C. [1]

Since the eigenvalues of a companion matrix C ( P ) are the same as the roots of P, this method can be used to find bounds for the roots of P.

So that: | λ | ≤ ‖ L r ‖ ∞ ≤ ‖ L b ‖ ∞ ≤ ‖ F b ‖ ∞ . □

Through illustrative examples in case of polynomial by degree n = 5 , we can try to see how sharp this theorem is.

In this section we make a particular study on polynomial for degree n = 5 by using theorem 4 about sparse companion matrices. The result will be compared to bounds from Cauchy, Montel nd Carmichel-Mason’s theorems [3].

Let P ( X ) = X 5 + a 4 X 4 + a 3 X 3 + a 2 X 2 + a 1 X + a 0 ∈ ℂ [ X ] , with a 0 ≠ 0 .

It is about to use theorems 4 to determin a best formula which gives good bounds.

● First case: | a 0 | ≥ 1 .

By application of theorem 4, we obtain: r = 0 then | λ | ≤ ‖ L 0 ‖ ∞ .

L 0 = C 2 ( P ) = ( − a 4 1 0 0 0 − a 3 0 1 0 0 − a 2 0 0 1 0 − a 1 0 0 0 1 − a 0 0 0 0 0 )

Then | λ | ≤ ‖ L 0 ‖ ∞ = max { 1 + | a 4 | ; 1 + | a 3 | ; 1 + | a 2 | ; 1 + | a 1 | ; | a 0 | } .

This is Cauchy’s bound applied to C 2 ( P ) with the ∞-norm.

● Second case: | a 0 | < 1 but | a 0 | + | a 1 | ≥ 1 .

By application of theorem 4, we obtain: r = 1 then | λ | ≤ ‖ L 1 ‖ ∞ .

L 1 = ( 0 1 0 0 0 0 − a 4 1 0 0 0 − a 3 0 1 0 0 − a 2 0 0 1 − a 0 − a 1 0 0 0 )

Then | λ | ≤ ‖ L 1 ‖ ∞ = max { 1 ; 1 + | a 4 | ; 1 + | a 3 | ; 1 + | a 2 | ; | a 0 | + | a 1 | } .

⇒ | λ | ≤ max { 1 + | a 4 | ; 1 + | a 3 | ; 1 + | a 2 | ; | a 0 | + | a 1 | } .

● Third case: | a 0 | + | a 1 | < 1 but | a 0 | + | a 1 | + | a 2 | ≥ 1 .

By application of theorem 4, we obtain: r = 2 then | λ | ≤ ‖ L 2 ‖ ∞

L 2 = ( 0 1 0 0 0 0 0 1 0 0 0 0 − a 4 1 0 0 0 − a 3 0 1 − a 0 − a 1 − a 2 0 0 )

Then | λ | ≤ ‖ L 2 ‖ ∞ = max { 1 ; 1 + | a 4 | ; 1 + | a 3 | ; | a 0 | + | a 1 | + | a 2 | } .

⇒ | λ | ≤ max { 1 + | a 4 | ; 1 + | a 3 | ; | a 0 | + | a 1 | + | a 2 | } .

● Fourth case: | a 0 | + | a 1 | + | a 2 | < 1 but | a 0 | + | a 1 | + | a 2 | + | a 3 | ≥ 1 .

By application of theorem 4, we obtain: r = 3 then | λ | ≤ ‖ L 3 ‖ ∞

L 3 = ( 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 − a 4 1 − a 0 − a 1 − a 2 − a 3 0 )

Then | λ | ≤ ‖ L 3 ‖ ∞ = max { 1 + | a 4 | ; | a 0 | + | a 1 | + | a 2 | + | a 3 | } .

⇒ | λ | ≤ max { 1 + | a 4 | ; | a 0 | + | a 1 | + | a 2 | + | a 3 | } .

● Fifth case: | a 0 | + | a 1 | + | a 2 | + | a 3 | < 1

By application of 4, we obtain: r = 4 then | λ | ≤ ‖ L 4 ‖ ∞

L 4 = ( 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 − a 0 − a 1 − a 2 − a 3 − a 4 )

Then | λ | ≤ ‖ L 4 ‖ ∞ = max { 1 ; | a 0 | + | a 1 | + | a 2 | + | a 3 | + | a 4 | } : this is Montel’s bounds applied to C 2 ( P ) with the 1-norm.

Table 1 presentes the different cases of bounds for degree n = 5 .

Example 2:

Let P ( X ) = X 5 + 4 X 4 + 3 X 3 + 2 X 2 + X + 5 ∈ ℂ [ X ] ,

We have: a 0 = 5 ⇒ | a 0 | > 1 .

Then:

1) By application of theorem 4, we obtain:

| λ | ≤ ‖ L 0 ‖ ∞ = max { 1 + 4 ; 1 + 3 ; 1 + 2 ; 1 + 1 ; 5 } = 5 .

2) Cauchy’s bounds: Gerschgorin’s theorem, applied to the second compagnon matrix C 2 ( P ) with the ∞-norm, gives:

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

⇒ | λ | ≤ max { | 5 | , 1 + | 1 | , 1 + | 2 | , 1 + | 3 | , 1 + | 4 | }

⇒ | λ | ≤ max { 5 , 2 , 3 , 4 , 5 }

⇒ | λ | ≤ 5

3) Montel’s bound: Gerschgorin’s theorem, applied to the second compagnon matrix C 2 ( P ) with ‖   .   ‖ 1 , gives:

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

⇒ | λ | ≤ max { 1 , 5 + 1 + 2 + 3 + 4 }

⇒ | λ | ≤ 15

4) Carmichel-Mason’s bound: Gerschgorin’s theorem, applied to the second compagnon matrix C 2 ( P ) with ‖   .   ‖ 2 , gives:

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

⇒ | λ | ≤ 1 + | a 0 | 2 + ⋯ + | a 4 | 2

⇒ | λ | ≤ 1 + 5 2 + 1 2 + 2 2 + 3 2 + 4 2

⇒ | λ | ≤ 56 ≈ 7.48

The roots and their bounds for example 2 are illustrated in Table 2, Table 3 and Figure 1 [6].

We can see that, the Sparse companion matrix becomes interesting in case of small value of polynomial’s coefficients. Let us then choose a polynomial with small values.

Example 3:

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

We have: a 0 = 0.05 ⇒ | a 0 | < 1 , | a 0 | ≠ 0 .

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

Then:

1) By application of theorem 4, we have the fifth case of Sparse and obtain | λ | ≤ max { 1 ; | a 0 | + ⋯ + | a 4 | } = 1 .

2) Cauchy’s bound: Gerschgorin’s theorem, applied to the second compagnon matrix C 2 ( P ) with the ∞-norm, gives:

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

⇒ | λ | ≤ max { | 0.05 | , 1 + | 0.01 | , 1 + | 0.02 | , 1 + | 0.03 | , 1 + | 0.04 | }

⇒ | λ | ≤ max { 0.05 , 1.01 , 1.02 , 1.03 , 1.04 }

⇒ | λ | ≤ 1.04

3) Montel’s bound: Gerschgorin’s theorem, applied to the second compagnon matrix C 2 ( P ) with ‖   .   ‖ 1 , gives:

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

⇒ | λ | ≤ max { 1 , 0.05 + 0.01 + 0.02 + 0.03 + 0.04 }

⇒ | λ | ≤ 1

4) Carmichel-Mason’s bound: Gerschgorin’s theorem, applied to the second compagnon matrix C 2 ( P ) with ‖   .   ‖ 2 , gives:

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

⇒ | λ | ≤ 1 + | a 0 | 2 + ⋯ + | a 4 | 2

⇒ | λ | ≤ 1 + 0.05 2 + 0.01 2 + 0.02 2 + 0.03 2 + 0.04 2

⇒ | λ | ≤ 1.0055 ≈ 1.0027462291

The roots and their bounds for example 2 are illustrated in Table 4, Table 5 and Figure 2 [6].

In this section, we investigate about the Sparse method of providing improved bounds of roots for polynomials. We compare bounds of roots of polynomial obtained by the Sparse method, to those found in the literature. According to the starting formel given in (12), we saw that the more we have coefficients closed to zero with a norm less than 1, the more the Sparse method is useful. Indeed for

r = max { k | ∑ i = 0 k − 1 | a i | < 1 } :

1) When r = 0 , the obtained Sparse matrices L 0 are similar to a Frobenius matrix. This occurs when | a 0 | ≥ 1 . In this case, the theorem gives the same result as Cauchy’s bounds;

2) When r = n − 1 , the obtained matrices L n − 1 are also similar to a Frobenius matrix. But in this case, the theorem become very useful and brings very interesting bounds with high effectiveness of use. This occurs when ∑ i = 0 n − 1 | a i | < 1 .

Examples of bounds have been evaluated in case of polynomials by degree n = 5 to illustrate how sharp the Sparse method is. In those examples, we saw that:

1) For high value of given polynomial’s coefficients, Cauchy gives similar bounds as the Sparse form. Those obtained bounds are better than Montel and Carmichel-Mason’s bounds;

2) For very minor values of the polynomial’s coefficients, the sharpest bounds are the Sparse and Montel’s ones.

In the above studied cases, each considered classical method couldn’t give itself sharpest bounds for high and minor values of the given polynomial’s coefficients. Those methods are appropriated for high values or minor values of the coefficients but not for both of them. By using the Sparse matrices, the obtained bounds belong each time to the sharpest ones in the cases of high and minor coefficient’s values of a the given polynomials.