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The Quality of Zero Bounds for Complex Polynomials

The Quality of Zero Bounds for Complex Polynomials

  • Matthias Dehmer, 
  • Yury Robertovich Tsoy
PLOS
x

Abstract

In this paper, we evaluate the quality of zero bounds on the moduli of univariate complex polynomials. We select classical and recently developed bounds and evaluate their quality by using several sets of complex polynomials. As the quality of priori bounds has not been investigated thoroughly, our results can be useful to find optimal bounds to locate the zeros of complex polynomials.

Introduction

Deriving zero bounds for real and complex zeros of polynomials is a classical problem that has been proven essential in various disciplines such as engineering, mathematics, and mathematical chemistry [1][6]. As indicated, there is a large body of literature dealing with the problem of providing disks in the complex plane representing so-called inclusion radii (bounds) where all zeros of an univariate complex polynomial are situated. Let be a an univariate complex polynomial. Then, a crucial question is to investigate how accurate an inclusion radius is, i.e., how well does the bound reflect the real location of the zeros of by determining the quantity where is the bound under consideration and . It is clear that the more optimal a zero bound is, the better the value can serve as an estimate to start a numerical procedure such as Newton’s or Sturm’s method [7].

Starting from a set of complex polynomials, it is often difficult to find an optimal bound, i.e., for which either vanishes or is very little. Another problem is that for many bounds, sharpness results do not exist. Sharpness means there exists a polynomial possessing a zero that lies on the circle which includes all zeros of the polynomial in question. This problem calls for a systematic treatment namely to study the optimality of zero bounds for particular classes of polynomials numerically. To our best knowledge, this problem has not yet been explored properly; see, e.g., [8]. A reason for this is surely the vast amount of existing bounds for locating the zeros of real and complex polynomials [9][13]. The only attempt in this direction we got aware of is due McNamee and Olhovsky [8]. They implemented several zero bounds by using 1200 polynomials with random real or complex roots and calculated their values numerically [8]. Among other calculated zero bounds they did not state explicitly in [8], the bounds ue to Deutsch [14] and Kalantari [13] have been evaluated and found to be optimal by using the mentioned set of polynomials [8].

The main contribution of this paper is as follows: In contrast to [8], we evaluate classical and more recently developed zero bounds by using different classes of complex polynomials numerically. Among these classes are also lacunary polynomials and those, whose coefficients satisfy certain conditions by means of inequalities. We calculate several bounds for complex polynomials due to Cauchy [3], Dehmer [1], [9], Kalantari [13], Jain [15], Joyal [10] etc., see Table 1Table 6. As a result, we find that some of the bounds due to Dehmer, Joyal and Cauchy outperform Kalantari’s bounds by using particular classes of polynomials. This result triggers the hypothesis that it may be worthwhile to further develop bounds for special polynomials (e.g., lacunary polynomials or complex polynomials with special conditions for the polynomials coefficients) which are more optimal than by using general zero bounds. For instance, Theorem (10) developed by Dehmer et al. [9] will prove this hypothesis.

Methods

In the following, we state the most important zero bounds for locating the zeros of complex polynomials as theorems we are going to use in this study. The quality of these statements will be evaluated in the section ‘Results’. We distinguish two classes of bounds, namely explicit and implicit zero bounds, see [1], [9].

Explicit Bounds for Complex Polynomials: Classical and Recent Results

The following bounds [4], [16], [17] represent functions of all coefficients of a given polynomial. In fact, this type of zero bound has been called explicit bound [1], [9] as the value of the bound can be calculated explicitly by using quantities based on the moduli of the polynomial coefficients.

Theorem 1 (Cauchy [3]) Let

be a complex polynomial. All zeros of lie in the closed disk , where

Theorem 2 (Joyal [10]) Let

be a complex polynomial. All zeros lie in (1) where

(2)

Theorem 3 (Mohammad [15]) Let

be a complex polynomial. All zeros lie in

(3)

Theorem 4 (Kojima [12]) Let

be a complex polynomial. All zeros lie in

(4)

Theorem 5 (Jain [15]) Let

be a complex polynomial. All zeros lie in

(5)

Theorem 6 (Kuniyeda [11]) Let mit . All zeros oflie in(6)

Theorem 7 (Kuniyeda [11]) For , all zeros of

lie in (7)

Theorem 8 (Joyal [10]) For and , all zeros oflie in(8)where

(9)The following explicit bounds have been recently proven by Dehmer and Mowshowitz [9]. It has been shown that they often lead to considerably better values than by using classical ones. Clearly, this depends on the underlying class of polynomials. See also section ‘Results’.

Theorem 9 (Dehmer [9]) Letbe a complex polynomial. All zeros of lie in the closed disk(10)where

(11)The next theorem gives a bound for polynomials with restrictions on the coefficients. Dehmer [1] has shown that such bounds can be more precise and often lead to better results when locating the zeros of polynomials. See also Table 3.

Theorem 10 (Dehmer [9]) Letbe a complex polynomial. Suppose that and

(12) All zeros of lie in the closed disk (13)

The bound is sharp for all polynomials of the form (14)

Further recent results when proving upper bounds have been found by Kalantari [13]. He has found a family of zeros bounds for analytic functions that has been proven powerful when comparing the resulting bounds with classical ones by using complex polynomials [8].

Theorem 11 (Kalantari [13]) Let and let be the positive root of the polynomial(15)

For and , all zeros of the complex polynomiallie in the closed disk

(16)Theorem 12 (Kalantari [13]) Let and let be the positive root of the polynomial

For and , all zeros of the complex polynomiallie in the closed disk

(17).

Implicit Bounds for Complex Polynomials: Classical and Recent Results

The bound value of an implicit zero bounds depends on determining the root of a so-called concomitant polynomial [9]. This polynomial can often be obtained from the proof of the underlying theorem. An example thereof is Equation (18).

Theorem 13 (Cauchy [3]) Letbe a complex polynomial. All zeros of lie in the closed disk , where denotes the positive zero of

(18)The following implicit zero bounds might be easier to determine (e.g., by hand) when applying this apparatus in practice.

Theorem 14 (Dehmer [1]) Letbe a complex polynomial. All zeros of lie in the closed disk where denotes the positive root of the equation(19)and

(20) The bound is sharp for all polynomials of the form (21)

Theorem 15 (Dehmer [9]) Letbe a complex polynomial. All zeros of lie in the closed disk where denotes the positive root of the equation(22)and

(23) The bound is sharp for all polynomials of the form (24)

In particular, the concomitant polynomial of the next theorem is cubical, see Equation (27). This can be beneficial for practical applications as we only have to determine the positive root of a polynomial whose degree equals three.

Theorem 16 (Dehmer [9]) Let(25)and

(26) In addition, let be a complex polynomial. All zeros of lie in the closed disk where is the largest positive root of the equation

(27) Moreover, (28)

Bounds for Special Lacunary Polynomials

In this section, we state bounds [9] for lacunary polynomials, i.e., polynomials in which some coefficients vanish. The hypothesis is that special bounds for lacunary polynomials might lead to better results than by using general zero bounds, see the statements in the previous section.

Theorem 17 (Dehmer [9]) Letbe a complex polynomial. All zeros of lie in , where is the unique positive root of the equation

(29)Using the same method of proof we establish the following.

Theorem 18 (Dehmer [9]) Letbe a polynomial with arbitrary coefficients. All zeros of lie in , where is the unique positive root of the equation

(30)We conclude this section with the following theorem.

Theorem 19 (Dehmer [9]) Letbe a complex polynomial. All zeros of lie in

(31)

Data: Classes of Complex Polynomials

We define the classes of polynomials used in this study as follows (GD stands for Gaussian Distribution).

Definition 1 (32)

Definition 2 (33)

Definition 3

(34)

Definition 4 (35)

Definition 5 (36)

Definition 6 (37)

Software

We developed a C# program for calculating 19 zero bounds by using complex polynomials. These bounds have been calculated by generating random polynomials based on the following distributions: Gaussian, Poisson, Geometric, and Uniform in [−1, 1]. We used the batch mode of this software to compute all available bounds for a specified number of polynomials having certain degrees. For each class (see section ‘Data: Classes of Complex Polynomials’), we calculate the average by performing 1000 independent runs. To calculate the zeros of the random polynomials, one has to select.

  • the degree of a polynomial.
  • the type of distribution.
  • whether the polynomial is complex or real-valued.

We emphasize that in this study, we only used complex polynomials. After each batch run, the following information is available:

  • the type of distribution and distribution parameters.
  • the parameter required for some bounds, for example see Equations (6) or (7).
  • tables with mean and standard deviation of ranks of the bounds in terms of their optimality for different polynomial orders.
  • tables with mean and standard deviation of bound values for different degrees of the polynomials.

Moreover for each polynomial degree, a brief summary of the results for a batch run is available, which includes:

  • the type of distribution, distribution parameters and the polynomial degree.
  • number of runs.
  • the parameter , see Equations (6) or (7).
  • bound name with the best (worst) sum rank.
  • bound name with the worst (largest) sum rank.
  • list of the used bounds sorted by their sum rank in ascending order (from best to worst) with information on their sum ranks, minimal and maximal rank achieved during all runs.

Apart from the batch mode, the program can also be used in a single mode. By doing so, the program creates a log file containing the following items:

  • the degree of a polynomial.
  • the type of distribution.
  • the generated complex or real coefficients of the polynomial.
  • the parameter , see Equations (6) or (7).
  • names of the most sharp and weak bounds with their corresponding values.
  • a ranking of the bounds in terms of their optimality in ascending order.

Results

In this section, we evaluate the quality of the zero bounds presented in the previous sections. We start by observing that Theorem (13) due to Cauchy is often quite sharp. See, for example, the results in Tables 14. This is not surprising as this bound is known to be optimal for its class of implicit zero bounds, see [6]. As the numerical results show, this does not mean that other bounds (based on another paradigm) outperform this bound by using special classes of polynomials. This proves our hypothesis that special bounds (e.g., Theorem (10), (19)) may be more suitable and optimal for special classes of polynomials than general bounds (i.e., bounds where no restrictions for the polynomial coefficients are used). In particular, we see that by using the polynomials of Definition (3), the bound due to Dehmer, Theorem (10) outperforms Cauchy’s bound (Theorem (13)) if . Again, Theorem (10) is based on inequalities involving the polynomial coefficients and leads to a better mean value than by using a general zero bound.

Also, the results by using lacunary polynomials (see Definition (6)) support this hypothesis too. By considering Table 6, we observe that the special bounds for lacunary polynomials due to Dehmer, Theorem (17), (18) perform very similar to Theorem (13) due to Cauchy based on the mean values. In contrast, the explicit bound also developed by Cauchy, Theorem (1) does not give feasible values. Also, the bound due to Jain, Theorem (5) and Kojima, Theorem (4) are not feasible by using the here presented classes of polynomials. This holds for all classes of polynomials used in this study, see Table 16. More generally, it has been shown that the inclusion radii given by Theorem (1), (5) are often useless in terms of the real location of the zeros of an underlying polynomial, see [9].

In the following, we discuss some particular cases to find classes of polynomials where the bounds due to Dehmer perform well. Let be the zeros of a complex polynomial and let . Also, we define the quantity , where is the corresponding bound value for . Now, consider the polynomial(38)

We yield that Kalantaris bound, Theorem (12) is best, and . Particularly Table 1 supports the fact that Kalantaris bound, Theorem (12) performs well for among the used zero bounds. Second best is the bound due to Joyal, Theorem (2), and . Third best is Dehmer’s bound, Theorem (14), and . Particularly, Theorem (14) outperforms the classical bound due to Cauchy, Theorem (11), and . But note that for many other polynomials of , Joyal’s bound, Theorem (2) often was often the best one and Theorem (14) due to Dehmer the second best one. Theorem (14) has the advantage that the positive root of the concomitant polynomial (see Equation (19)) might be easier to determine than by using other bounds which rely on more complex concomitant polynomials, e.g., see Theorem (13).

We already mentioned above that by using special polynomials, some special bounds (based on conditions for the polynomial coefficients) are better suited than by using general zero bounds, e.g., Theorem (1), (2), (11) etc. A positive example for this is the polynomial(39) has the property that its coefficients are sampled from a Gaussian Distribution (GD) and it holds We yield that the special bound due to Dehmer, Theorem (10) is best and . The general zero bounds due to Cauchy and Joyal (Theorem (13) and Theorem (8); for ) are second best, , and third best , . Thus, the special zero bound, Theorem (10) outperforms two classical and general zero bounds by using . It is clear that the coefficients of the underlying polynomials have a strong impact on the values of the bounds as it be seen by the comparison of and . This can be also seen by comparing the quantities and . Finally, the chosen bounds are more optimal for in the sense that the values are much smaller.

Apart from Dehmer’s bounds, we also discuss the quality of the bounds due to Kalantari [13] in a more general context. Note that the Kalantari bounds [8] have already been evaluated and compared with others by McNamee and Olhovsky [8]. In particular, these bounds have been proven efficient for 1200 polynomials with random real or complex roots. We start with general polynomials given by Definition (1). For , some of Dehmer’s bounds, e.g., Theorem (9), (15) outperform the bound of Kalantari, Theorem (12). If grows, we see that by using other classes the Kalantari bound, Theorem (12) is quite sharp compared to other bounds (except the Cauchy bound, Theorem (13)). The second bound developed by Kalantari, Theorem (11) is worse than almost all Dehmer bounds and others for all . Note that we only evaluated the Kalantari bounds for ; see the underlying concomitant polynomial represented by Equation (15). As a conclusive remark, we find (Table 1Table 6) that other zero bounds due to Dehmer, Cauchy and Joyal often outperform these bounds. This interesting finding is in contrast to the result due to McNamee and Olhovsky [8], who identified the Kalantari bounds as best when being compared to other classical bounds such as the ones due to Deutsch [14].

Summary and Conclusion

In this paper, we investigated the quality of zero bounds for complex polynomials numerically. By knowing that the bound values surely depend on the underlying coefficients, we generated several classes of complex polynomials (see section ‘Data: Classes of Complex Polynomials’) to apply the bounds. The set of bounds we have applied consists of (i) classical bounds due to Cauchy [3], Joyal [10], Kuniyeda [11], Kojima [12] etc. and (ii) recently developed bounds due to Dehmer [1], [9] and Kalantari [13]. Note that the just mentioned zero bounds are different to the ones used by McNamee and Olhovsky [8]. Our findings based on the used classes of complex polynomials show that some of the classical results, e.g., Kuniyeda, Kojima and Mohammad are not suitable to locate the zeros optimally. This does not mean that for some other classes or special polynomials, these bounds could perform better. As shown by Rahman and Schmeisser [6], the classical (implicit) zero bound due to Cauchy, Theorem (13) is often optimal within this class of bounds. Thus, it is not surprising that this bound often performs best for our classes. Anyway, we have found other zero bounds which outperformed this bound for particular classes of polynomials. Hence, it would be valuable to derive further special bounds for special classes of polynomials, see also [1], [9].

This study has illustrated some strong and weak points of the used bounds. As conclusion, it seems that there exist only a few zero bounds which give optimal bound values for a variety of complex polynomials. A reason for this is that in view of the vast amount of existing bounds, their quality has only been very little investigated. Also the result where some of the Dehmer bounds (Theorem (17), (18)) outperform the classical (and sharp) Cauchy bound for lacunary polynomials make us conclude that it will be useful to derive further novel bounds for special cases. In fact, special polynomials, i.e., whose coefficients fulfill special conditions often occur in control engineering, algebraic biology and mathematical chemistry.

Author Contributions

Analyzed the data: YT. Wrote the paper: MD YT. Performed the mathematical analysis and interpreted the results: MD.

References

  1. 1. Dehmer M On the location of zeros of complex polynomials. Journal of Inequalities in Pure and Applied Mathematics, Vol 7 (1) (2006)
  2. 2. Heitzinger W, Troch WI, Valentin G (1985) Praxis nichtlinearer Gleichungen. Carl Hanser Verlag. München, Wien, Germany, Austria.
  3. 3. Marden M (1966) Geometry of polynomials. Mathematical Surveys of the American Mathematical Society, Vol. 3. Rhode Island, USA.
  4. 4. Mignotte M, Stefanescu D (1999) Polynomials: An Algorithmic Approach. Discrete Mathematics and Theoretical Computer Science. Springer. Singapore.
  5. 5. Prasolov VV (2004) Polynomials. Springer.
  6. 6. Rahman QI, Schmeisser G (2002) Analytic Theory of Polynomials. Critical Points, Zeros and Extremal Properties. Clarendon Press. Oxford, UK.
  7. 7. Obreschkoff N (1963) Verteilung und Berechnung der Nullstellen reeller Polynome. Hochschulbücher für Mathematik, Vol. 55. VEB Deutscher Verlag der Wissenschaften. Berlin, Germany.
  8. 8. McNamee JM, Olhovsky M (2005) A comparison of a priori bounds on (real or complex) roots of polynomials. In: Proceedings of 17th IMACS World Congress, Paris, France.
  9. 9. Dehmer M, Mowshowitz A (2011) Bounds on the moduli of polynomial zeros. Applied Mathematics and Computation, accepted 218: 4128–4137.
  10. 10. Joyal A, Labelle G, Rahman QI (1967) On the location of polynomials. Canadian Mathematical Bulletin 10: 53–63.
  11. 11. Kuniyeda M (1916) Note on the roots of algebraic equations. Tôhoku Mathematical Journal 8: 167–173.
  12. 12. Kojima J (1914) On a theorem of Hadamard and its applications. Tôhoku Mathematical Journal 5: 54–60.
  13. 13. Kalantari B (2005) An infinite family of bounds on zeros of analytic functions and relationship to smale’s bound. Mathematics of Computation 74: 841–852.
  14. 14. Deutsch E (1981) Bounds for the zeros of polynomials. American Mathematical Monthly 88: 205–206.
  15. 15. Jain VK (1986) On the zeros of polynomials II. Journal of Mathematical and Physical Sciences 20: 259–267.
  16. 16. Dehmer M (2004) Die analytische Theorie der Polynome. Nullstellenschranken für komplexwertige Polynome. Weissensee-Verlag. Berlin, Germany.
  17. 17. Milovanovic GV, Mitrinovic DS, Rassias TM (1994) Topics in Polynomials: Extremal Problems, Inequalities, Zeros. World Scientific.