^{1}

^{*}

^{1}

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In this paper, we combine Graeffe matrices with the classical numerical method of Dandelin-Graeffe to estimate bounds for the moduli of the zeros of polynomials. Furthermore, we give some examples showing significant gain for the convergence towards the polynomials dominant zeros moduli.

Deriving zero bounds for polynomials is a classical problem that has been proven essential in various disciplines such as controling engineering problems, eigenvalue problems in mathematical physics, and digital audio signal processing problems―to name just a few [

An estimated value of the largest moduli of the roots of a polynomial can be obtained as a limit of a sum of power of these roots using resultants [

The rest of the paper is organized as follows. In Section 2, we introduce the Graeffe matrix of a complex polynomial P and derive its trace from the classical Jordan normal form. In Section 3, we extend the Dandelin- Graeffe’s method to find the maximum moduli of the zeros of P, relying on the limit of Graeffe matrix’s trace of P. Through some examples, Section 3 illustrates that a few iterations of Graeffe’s matrix trace can give very tight bounds for the absolute value of the zeros of polynomials, with comparatively fast convergence.

For an n-dimensional matrix A, we call the spectrum of A the set of all its eigenvalues, denoted by

Lemma 1. A square complex matrix A of order n is similar to a block diagonal matrix

for

For

Remark: The spectrum of a matrix

Lemma 2. If

Proof. For arbitrary indices i and j, it is direct to verify the claim by using a proof by induction.

From the previous result, we deduce the two following corollaries:

Corollary 1. The m-th power of an n-dimensional matrix A is similar to the m-th power of its Jordan normal form as a direct sum of upper triangular matrices. Besides, each triangular block will consist of m-th power of its associate eigenvalue on the main diagonal.

Corollary 2. The trace of the m-th power of A is the sum of the m-th powers of the eigenvalues of A.

Definition 1. For

Let

Definition 2. The characteristic polynomial of

Remarks:

1) It can be computed also using resultants, as quoted in the following lemma ( [

2) Relying on Proposition 1, it follows that for all

m | 2^{13} | 2^{15} | 2^{20} | 2^{25} | 2^{27} |
---|---|---|---|---|---|

Graeffegen | 0.234 s | 1.216 s | 262.284 s | 3971.235 s | 4854.183 s |

GraeffeMatrix | 0.093 s | 0.171 s | 0.592 s | 28.126 s | 137.421 s |

Proposition 1. Let P be a polynomial of degree d with complex coefficients, then the trace of

Proof. It is well-known that the eigenvalues are the zeros of the polynomial P; therefore, the result follows immediately from Corollary 2.

Proposition 2. Let P be a polynomial of degree d with complex coefficients and

There exists a sequence of polynomials

the zeros of

Proof. The basic technique to write down the proclaimed sequence of polynomials is done through an iteration refered to as the Dandelin-Graeffe method (see [

Starting with

a polynomial of degree

For the induction process, one performs the following iteration

which transforms

of the zeros of

Lemma 3.

Let

Proof. One can refer to [

As a special case of Lemma 3, we can consider sequences of traces of

Proposition 3. Let

where

Remark: It is important to note that the existence of an unique dominant zero is essential to the validity of the previous result, as shown in ( [

Lemma 4. Let

Remark: Bearing in mind that there is no need to check the unicity of the largest modulus of the zero of a given ploynomial, a few iterations of d consecutive values of the traces

Proposition 4.

Let

where

1) We consider the polynomial

with an unique absolute value of dominant zero, namely 5. The application of the Graeffe’s zero-squaring method (Proposition 3) to the Graeffe’s polynomial of order

k | 3 | 4 | 5 | 6 |
---|---|---|---|---|

5.003067358 | 5.000280728 | 5.000000012 | 5.000000000 |

The previous result turns out as a considerably better bound, comparing with some classical explicit bounds gathered from Dehmer ( [

Refered after | Cauchy | Joyal | Mohammad | Kojima | Jain |
---|---|---|---|---|---|

Bound | 32 | 17.827723451 | 20 | 20 | 33.219280948 |

2) Let

By Proposition 4, few iterations of sequences of the maximum of

yield some sharp estimations of the real modulus of this double zero, for small values of m (

10^{2} | 7.048688850 | 7.048205127 | 7.047730920 | 7.047265952 | 7.046809955 | 7.048688850 |

10^{3} | 7.004853712 | 7.004848862 | 7.004844021 | 7.004839190 | 7.004834368 | 7.004853712 |

10^{4} | 7.000485220 | 7.000485171 | 7.000485123 | 7.000485074 | 7.000485026 | 7.000485220 |

10^{5} | 7.000048520 | 7.000048520 | 7.000048519 | 7.000048519 | 7.000048519 | 7.000048520 |

Moreover, the method leads to better results when locating explicit bounds for zeros of polynomials as shown in

Refered after | Cauchy | Joyal | Mohammad | Kojima | Jain |
---|---|---|---|---|---|

Bound | 386 | 85.165518364 | 38 | 38 | 363.116633802 |

3) Let’s now consider the polynomial

10^{2} | 6.066280152 | 5.999999999 | 6.064973530 | 6.000000000 | 6.063717428 | 6.066280152 |

10^{3} | 6.008323534 | 6.004156167 | 6.008306909 | 6.004147877 | 6.008290350 | 6.008323534 |

10^{4} | 6.000831834 | 6.000415861 | 6.000831668 | 6.000415778 | 6.000831502 | 6.000831834 |

10^{5} | 6.000083178 | 6.000041589 | 6.000083177 | 6.000041588 | 6.000083175 | 6.000083178 |

It’s important to notice that the convergence seems to become a bit slower than that in the previous case of the existence of a double dominant zero.

Moreover, the method leads to better results when locating explicit bounds for the zeros of polynomials (see

Refered after | Cauchy | Joyal | Mohammad | Kojima | Jain |
---|---|---|---|---|---|

Bound | 7 | 3.854101966 | 12 | 12 | 39.863137138 |

We would like to thank Professor Maurice Mignotte for his valuable suggestions and comments. We thank also the Editor and the anonymous referees for their comments.

Graeffegen := proc (

local

if

else G := sort(resultant

end if

end proc:

GraeffeMatrix := proc (

local

sort(CharacteristicPolynomial

end proc:

・ In Graeffegen, P is a polynomial in

・ In GraeffeMatrix, B = CompanionMatrix(P, X) stands for the CompanionMatrix of the polynomial ^{k}.