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The authors confirm that co-author Matthias Dehmer is a PLOS ONE Editorial Board member. This does not alter the authors' adherence to PLOS ONE Editorial policies and criteria.

Analyzed the data: YT. Wrote the paper: MD YT.

In this paper, we investigate the performance of zero bounds due to Kalantari and Dehmer by using special classes of polynomials. Our findings are evidenced by numerical as well as analytical results.

The problem of calculating the zeros of polynomials has been at the core of various algorithmic problems in engineering, computer science, mathematics, and mathematical chemistry

We emphasize that numerous papers and books have been contributed dealing with the problem of locating the zeros of complex polynomials, see, e.g.,

In this paper, we deal with the problem of evaluating the quality of zero bounds numerically. A successor of this paper is

The main contribution of this paper is as follows: We focus on evaluating zero bounds developed by Kalantari

In the following, we state the zero bounds for locating the zeros of complex polynomials as theorems we will explore in this paper. The numerical results will be presented in the section ‘

The next theorem gives a bound for polynomials with restrictions on the coefficients. Dehmer

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

Kalantari, Th. (1) | 1.739731 | 1.676344 | 1.633205 | 1.613733 | 1.618263 | 1.616748 | 1.623021 | 1.620896 |

Kalantari, Th. (2) | 1.44096 | 1.392649 | 1.365094 | 1.348481 | 1.351714 | 1.354525 | 1.350528 | 1.351485 |

Dehmer, Th. (3) | 1.627128 | 1.463874 | 1.424159 | 1.411194 | 1.418361 | 1.425269 | 1.441339 | 1.446148 |

Dehmer, Th. (4) | 1.394486 | 1.495732 | 1.571549 | 1.630066 | 1.665533 | 1.696626 | 1.739386 | 1.777438 |

In

By using the finite geometric series, we obtain

In order to get an inequality for

Determining the zeros of the latter function gives

As

If we can prove that the positive zero of

Applying the Descartes' rule of signs to

By performing elementary calculations, we get

From this inequality, we also infer

We finally show that the right hand side of this inequality is less than 4. That means claiming

Yields

But by performing elementary calculations we find that this inequality is valid for

As in

In order to perform a statistical analysis, we have generated 1000 complex polynomials for each of the Definitions 1–6 and

If _{1} is tighter than the bound of Theorem _{2}.

To compare different bounds averaged values of

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

Kalantari, Th. (1) | 1.740618 | 1.674687 | 1.640986 | 1.615829 | 1.622275 | 1.62177 | 1.615899 | 1.625303 |

Kalantari, Th. (2) | 1.444768 | 1.39455 | 1.36902 | 1.352937 | 1.354811 | 1.353599 | 1.349038 | 1.354005 |

Dehmer, Th. (3) | 1.500152 | 1.436359 | 1.420059 | 1.411527 | 1.42618 | 1.432393 | 1.445271 | 1.45789 |

Dehmer, Th. (4) | 1.449222 | 1.566359 | 1.634031 | 1.673176 | 1.732903 | 1.770081 | 1.807097 | 1.831334 |

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

Kalantari, Th. (1) | 1.704931 | 1.625397 | 1.582924 | 1.572831 | 1.554937 | 1.556792 | 1.553397 | 1.566236 |

Kalantari, Th. (2) | 1.429949 | 1.37537 | 1.357003 | 1.342865 | 1.340357 | 1.335035 | 1.330336 | 1.33636 |

Dehmer, Th. (3) | 1.483377 | 1.432982 | 1.398882 | 1.39568 | 1.381683 | 1.390759 | 1.400006 | 1.413897 |

Dehmer, Th. (4) | 1.369496 | 1.483964 | 1.517359 | 1.548124 | 1.551111 | 1.563088 | 1.596057 | 1.609 |

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

Kalantari, Th. (1) | 1.750715 | 2.012142 | 1.837759 | 1.838256 | 1.823758 | 1.820873 | 1.810426 | 1.83248 |

Kalantari, Th. (2) | 1.45379 | 1.572551 | 1.459837 | 1.460034 | 1.454432 | 1.461645 | 1.445665 | 1.457515 |

Dehmer, Th. (3) | 1.609873 | 1.749532 | 1.566365 | 1.562589 | 1.56622 | 1.581601 | 1.58269 | 1.629539 |

Dehmer, Th. (4) | 1.388793 | 1.773258 | 1.722926 | 1.783884 | 1.859031 | 1.979154 | 2.005109 | 2.168481 |

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

Kalantari, Th. (1) | 1.776129 | 2.05439 | 2.000812 | 1.953628 | 1.926602 | 1.908316 | 1.900434 | 1.911558 |

Kalantari, Th. (2) | 1.455061 | 1.592503 | 1.553934 | 1.528648 | 1.508396 | 1.499813 | 1.500933 | 1.500791 |

Dehmer, Th. (3) | 1.519732 | 1.646073 | 1.614704 | 1.609604 | 1.625371 | 1.633324 | 1.677303 | 1.703949 |

Dehmer, Th. (4) | 1.449809 | 1.841836 | 1.90772 | 2.015138 | 2.123091 | 2.222602 | 2.429363 | 2.520859 |

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

Kalantari, Th. (1) | 1.779496 | 1.730165 | 1.782391 | 1.822496 | 1.853114 | 1.872392 | 1.889178 | 1.902321 |

Kalantari, Th. (2) | 1.467053 | 1.399733 | 1.441984 | 1.47443 | 1.499201 | 1.514797 | 1.528377 | 1.53901 |

Dehmer, Th. (5) | 1.185785 | 1.03211 | 1.011294 | 1.00614 | 1.004918 | 1.002322 | 1.002466 | 1.001667 |

Dehmer, Th. (6) | 1.429314 | 1.146872 | 1.088454 | 1.064075 | 1.052297 | 1.041634 | 1.037083 | 1.032652 |

The pairwise comparison of the averaged values

We restrict our analysis to evaluate the performance of the bounds due to Kalantari and Dehmer only, see, section ‘

We start by interpreting the

The analytical comparison of the bounds has been intricate. That means it might be difficult to compare bounds which rely on different concepts (e.g., explicit vs. implicit bounds, see

In case of using the explicit zero bounds Theorem 1 and Theorem 3, it is straightforward to derive an analytical expression (condition) to compare the bounds by means of inequalities. If we start with the inequality (i.e., we assume that Theorem 1 is better than Theorem 3),

Otherwise, we yield

To get an inequality for the assumption that Kalantari's bound given by Theorem 2 is better than Dehmer's bound given by Theorem 3, we start with assuming

We yield

If

The results of the evaluation for lacunary polynomials (see Definition 6) can be seen in

According to Theorem 7, an upper bound for _{1}<1. Similar arguments can be applied when considering Theorem 6.

In this paper, we explored the performance of zero bounds due to Kalantari and Dehmer. In earlier contributions, it has been claimed

The main result of this paper is that some of the bounds due to Dehmer outperform the bounds due to Kalantari for special classes of polynomials. In particular when using lacunary polynomials (i.e., many coefficients equal zero) Dehmer's bounds showed excellent performance. We have underpinned our discussion to interpret the numerical results by analytical results. In particular, we have proved an upper bound for lacunary polynomials (see Theorem 7) and obtained conditions for some special cases to check whether one bound is better (or worse) than another by means of inequalities.

Another interesting line of research is to study the zeros of graph polynomials. Some recent related work dealing with applications on graph polynomials are