Perturbation Approach to Polynomial Root Estimation and Expected Maximum Modulus of Zeros with Uniform Perturbations

Abstract

This paper presents a significant extension of perturbation theory techniques for estimating the roots of polynomials. Building upon foundational results and recent work by Pakdemirli and Yurtsever, as well as taking inspiration from the concept of probabilistic bounds introduced by Sheikh et al., we develop and prove several novel theorems that address a wide range of polynomial structures. These include polynomials with multiple large coefficients, coefficients of different orders, alternating coefficient orders, large linear and constant terms, and exponentially decreasing coefficients. Among the key contributions is a theorem that establishes an upper bound on the expected maximum modulus of the zeros of polynomials with uniformly distributed perturbations in their coefficients. The theorem considers the case where all but the leading coefficient receive a uniformly and independently distributed perturbation in the interval $[-1,1]$. Our approach provides a comprehensive framework for estimating the order of magnitude of polynomial roots based on the structure and magnitude of their coefficients without the need for explicit root-finding algorithms. The results offer valuable insights into the relationship between coefficient patterns and root behavior, extending the applicability of perturbation-based root estimation to a broader class of polynomials. This work has potential applications in various fields, including random polynomials, control systems design, signal processing, and numerical analysis, where quick and reliable estimation of polynomial roots is crucial. Our findings contribute to the theoretical understanding of polynomial properties and provide practical tools for engineers and scientists dealing with polynomial equations in diverse contexts.

1. Introduction

The study of polynomial roots has been a fundamental area of mathematics for centuries, with applications ranging from pure algebra to various fields of science and engineering [1,2,3,4]. In recent years, the use of perturbation theory in estimating polynomial roots has gained significant attention, providing valuable insights into the behavior of roots under small parameter changes [5]. Perturbation theory is also crucial in studying random polynomials, where coefficients are drawn from probability distributions. The analysis of how small random perturbations affect the roots of polynomials provides insights into the stability and sensitivity of polynomial systems under uncertainty [6]. This approach has found applications in various fields, including the study of eigenvalues of random matrices and the behavior of complex systems [7].

Galántai and Hegedüs [8] present perturbation bounds on polynomial zeros that improve the results of Ostrowski, Elsner et al., and Beauzamy while also enhancing the backward stability result of Edelman and Murakami. Pakdemirli and Yurtsever [9] offer theorems to estimate the order of magnitude of roots in polynomials, aiding in a priori estimation before solving the equation. Choque-Rivero and Garza [10] study moment perturbation of matrix orthogonal polynomials, establishing an explicit relation between perturbed and original polynomials. In a significant study, Maiz [11] applies perturbation theory to derive polynomial solutions for real quantum potential systems, providing a useful tool for verifying and improving numerical and approximate methods in quantum theory. Recently, Dmytryshyn [12] developed an algorithm to recover a perturbation of a matrix polynomial from a perturbation of its first companion linearization. The accuracy and stability of numerical algorithms used in perturbation analysis are critical considerations, as highlighted by Higham [13]. These aspects become particularly important when dealing with high-degree polynomials or those with closely spaced roots, where small perturbations can lead to significant changes in root locations. The advancement of computational power has enabled researchers to employ sophisticated statistical models and simulations to study the behavior of polynomial roots under various perturbation scenarios. These computational approaches complement theoretical results and provide valuable insights into the complex dynamics of perturbed polynomial systems.

Ostrowski [14] provided a fundamental result on the continuity of polynomial roots with respect to their coefficients and provided bounds on the distance between the origins of two polynomials based on the differences in their coefficients. His theorem gives bounds on the distance between the roots of two polynomials based on the differences in their coefficients.

Theorem 1 

(Ostrowski [14]). Let $p\left(z\right)=z^n+a_1{z}^{n-1}+\cdots +a_n$ and $q\left(z\right)=z^n+b_1{z}^{n-1}$

$+\cdots +b_n$. For any root $x_i$ of $p\left(z\right)$, there exists a root $y_j$ of $q\left(z\right)$ such that

$$|x_i-y_j|\le {\left(\sum _{k=1}^n|a_k-b_k|{\gamma }^{n-k}\right)}^{1/n},$$

where $\gamma =2{max}_{1\le k\le n}\{|a_k{|}^{1/k},|b_k{|}^{1/k}\}$. Furthermore, the roots of p and q can be enumerated as ${\alpha }_1,\dots ,{\alpha }_n$ and ${\beta }_1,\dots ,{\beta }_n$, respectively, in such a way that ${max}_{1\le i\le n}|{\alpha }_i-{\beta }_i|\le (2n-1)$ ${\left({\sum }_{k=1}^n|a_k-b_k|{\gamma }^{n-k}\right)}^{1/n}.$

Research in this direction has been further developed and refined by subsequent researchers [15,16,17,18], contributing to a more comprehensive understanding of polynomial root perturbation. Bhatia et al. [17] extended the study of root perturbations to matrix eigenvalues. They established an upper bound for the variation of eigenvalues between two matrices.

Definition 1.

Let $A,B\in {\mathbb{C}}^{n\times n}$. Assume that $\sigma \left(A\right)=\{{\lambda }_1,\dots ,{\lambda }_n\}$ and $\sigma \left(B\right)=\{{\mu }_1,\dots ,{\mu }_n\}$. Let $S_n$ be the set of all permutations of $\{1,2,\dots ,n\}$. The eigenvalue variation of A and B is defined by $v(A,B)={min}_{\pi \in S_n}{max}_i|{\mu }_{\pi \left(i\right)}-{\lambda }_i|.$ The quantity $v(A,B)$ is also called the (optimal) matching distance between the eigenvalues of A and $B.$

Bhatia et al. improved Ostrowski’s matrix perturbation theorem in several aspects.

Theorem 2

(Bhatia et al. [17]). Let $A,B\in {\mathbb{C}}^{n\times n}$. Then $v(A,B)\le 4\times 2^{-1/n}(\parallel A\parallel +{\parallel B\parallel )}^{1-1/n}{\parallel A-B\parallel }^{1/n}.$

Beauzamy further refined the analysis of polynomial root perturbations by considering the multiplicity of roots. For their result, we first need the following definition.

Definition 2.

For the polynomial $p\left(z\right)={\sum }_{j=0}^n{a}_j{z}^{n-j}$, the Bombieri norm is defined by

$${\left[p\right]}_B={\left(\sum _{j=0}^n{\displaystyle \frac{|a_j{|}^2}{\left(\genfrac{}{}{0pt}{}{n}{j}\right)}}\right)}^{1/2}$$

The theorem provides a more precise bound that takes into account the derivative of the polynomial at the root.

Theorem 3

(Beauzamy [18]). Let $k\ge 1$ be an integer and $p\left(z\right)$ and $q\left(z\right)$ be two polynomials of degree n, with ${[p-q]}_B\le \epsilon$. If $x_i$ is any zero of $p\left(z\right)$ with multiplicity k, there exists a zero $y_j$ of $q\left(z\right)$, with $|x_i-y_j|\le {\left({\displaystyle \frac{n!}{(n-k)!}}\right)}^{1/k}{\displaystyle \frac{{\epsilon }^{1/k}}{|q^{\left(k\right)}\left(x_i\right){|}^{1/k}}}(1+|x_i{{|}^2)}^{n/2}.$ Further, if $\epsilon \le {\displaystyle \frac{(n-k)!}{2n!}}{|p^{\left(k\right)}\left(x_i\right)|}^2,$ then

$$|x_i-y_j|\le {\left({\displaystyle \frac{2n!}{(n-k)!}}\right)}^{1/k}{\displaystyle \frac{{\epsilon }^{1/k}}{|p^{\left(k\right)}\left(x_i\right){|}^{1/k}}}(1+|x_i{{|}^2)}^{(n-k)/2}$$

(1)

The work of Pakdemirli and Yurtsever [9] introduced a novel approach to estimating roots of polynomials using perturbation theory. In [9], Pakdemirli and Yurtsever, among other things, established the following theorems.

Theorem 4.

For a polynomial equation of the form

$a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0=0$

If all coefficients $a_i$$(i=0,\dots ,n)$ are of the same order of magnitude, then the roots are of $O\left(1\right)$.

Theorem 5.

For a polynomial equation of the form

$a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_m{x}^m+\cdots +a_{1}x+a_0=0$

If $a_m\sim O(1/{ϵ}^k)$$(k>0)$ with all other coefficients being of $O\left(1\right)$, then the possible roots may be of either of the following:

• $O\left({ϵ}^{k/m}\right)$ (for $m\ne 0$).
• $O(1/{ϵ}^{k/(n-m)})$ (for $m\ne n$).

Their method, based on the order of magnitude of coefficients, provided a framework for predicting the approximate magnitude of roots before solving the equation explicitly. This approach has proven particularly useful in cases where traditional numerical methods may struggle, such as when dealing with ill-conditioned polynomials or when a quick estimate is needed before applying more computationally intensive techniques [19]. These examples illustrate the continued interest and wide interface of perturbation theory, especially in the context of polynomials. The applications of this theory extend to various fields, including control systems [1], signal processing [20], and electrical engineering [3]. Numerical analysis techniques, crucial for implementing perturbation methods, are extensively covered in works such as [2,19,21].

The interdisciplinary nature of polynomial perturbation theory is further evidenced by its applications in nonlinear dynamics [22], computer-aided geometric design [23], and quantum chemistry [24].

In this paper, we aim to further expand the scope of perturbation-based root estimation techniques. We present several new theorems that address previously unexplored polynomial structures, including the following:

• Polynomials with multiple large coefficients.
• Polynomials with coefficients of different orders.
• Polynomials with alternating coefficient orders.
• Polynomials with large linear and constant terms.
• Polynomials with exponentially decreasing coefficients.
• Bounds of zeros of a randomly perturbed polynomial.

These theorems provide a comprehensive framework for estimating root magnitudes across a wide range of polynomial types, significantly extending the applicability of perturbation theory in root-finding problems. Moreover, we provide rigorous proofs for each theorem, along with illustrative examples that demonstrate their practical application.

Our work not only contributes to the theoretical understanding of polynomial root behavior but also has potential applications in various fields where quick root estimation is crucial. These include control systems design [1], signal processing [20], and numerical analysis [13], among others.

1.1. Key Concepts

In perturbation theory, the order of a coefficient refers to its magnitude relative to a small parameter $ϵ\ll 1$. Several examples follow:

• $O\left(1\right)$ means the coefficient is of order 1.
• $O\left({\displaystyle \frac{1}{ϵ}}\right)$ means the coefficient is much larger than 1, approximately ${\displaystyle \frac{1}{ϵ}}$.
• $O\left(ϵ\right)$ means the coefficient is much smaller than 1, approximately $ϵ$.

1.2. Limitations

Despite the advancements presented in this paper, several limitations should be acknowledged. The perturbation-based approach provides estimates of root magnitudes rather than exact root values, which may not be sufficient for all applications. The accuracy of the estimates may decrease for polynomials with very closely spaced roots or in cases where multiple perturbation effects interact in complex ways. It is important to note that the theorems presented are primarily applicable to scalar polynomials and may not directly extend to matrix polynomials or multivariate cases without further development; while the method offers quick estimates, it does not replace the need for rigorous numerical methods in applications requiring high precision. Additionally, the approach may have limited effectiveness for polynomials with highly oscillatory coefficient patterns or those arising from certain transcendental equations. These limitations present opportunities for future research and refinement of the perturbation-based root estimation techniques presented in this paper. Addressing these challenges could further enhance the applicability and reliability of these methods across a broader range of mathematical and engineering problems.

The rest of this paper is organized as follows: Section 2 presents the main theorems and their proofs. Section 3 provides numerical simulations and discussions. Section 4 explores the potential applications and implications of our results. Finally, Section 5 concludes this paper and suggests directions for future research.

1.3. Motivation

The motivation for this research stems from several key factors. Primarily, the ubiquity of polynomial equations in various scientific and engineering fields, including control systems [1], signal processing [20], and electrical engineering [3], underscores the importance of efficient root estimation techniques. There is a pressing need for quick and reliable estimation of polynomial roots, especially in cases where traditional numerical methods may struggle, such as with ill-conditioned polynomials [19].

2. Main Theorems

Theorem 6

(multiple large coefficients). Consider a polynomial equation of the form

$$a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0=0$$

Assume there are multiple coefficients of order $O\left({\displaystyle \frac{1}{{ϵ}^k}}\right)$ for some $k>0$, where these coefficients correspond to indices p and q satisfying $1\le p\le q\le n-1$. The remaining coefficients, including the constant term $a_0$ and the leading term $a_n$, are of order $O\left(1\right)$. Then, the roots are of order $O\left({ϵ}^{{\displaystyle \frac{k}{p}}}\right)$ or $O\left({\displaystyle \frac{1}{{ϵ}^{\frac{k}{n-q}}}}\right)$, where p is the smallest index and q is the largest index among the large coefficients.

Proof. 

Let p be the smallest index and q be the largest index such that $a_p,a_q\sim O\left({\displaystyle \frac{1}{{ϵ}^k}}\right)$. We will consider two cases for the possible order of magnitude of the roots.

Case 1: Assume $|x|\sim O\left({ϵ}^r\right)$, where $r>0$.

In this case, the term $a_p{x}^p$ will be the dominant term among those with large coefficients, as it has the smallest power of x. We need to balance this term with the constant term $a_0$:

$$a_p{x}^p\sim a_0$$

Substituting the orders of magnitude:

$${\displaystyle \frac{1}{{ϵ}^k}}·{\left({ϵ}^r\right)}^p\sim O\left(1\right)$$

$${\displaystyle \frac{{ϵ}^{pr}}{{ϵ}^k}}\sim O\left(1\right)$$

$${ϵ}^{pr-k}\sim O\left(1\right)$$

(2)

For relation (2) to hold, the exponent must satisfy

$$pr-k=0$$

$$r={\displaystyle \frac{k}{p}}$$

Therefore, one possible order for the roots is $O\left({ϵ}^{{\displaystyle \frac{k}{p}}}\right)$.

Case 2: Assume $|x|\sim O\left({\displaystyle \frac{1}{{ϵ}^r}}\right)$, where $r>0$.

In this case, the term $a_n{x}^n$ will be the dominant term among those with $O\left(1\right)$ coefficients. We need to balance this term with $a_q{x}^q$, which is the dominant term among those with large coefficients:

$$a_n{x}^n\sim a_q{x}^q$$

Substituting the orders of magnitude:

$$O\left(1\right)·{\left({\displaystyle \frac{1}{{ϵ}^r}}\right)}^n\sim {\displaystyle \frac{1}{{ϵ}^k}}·{\left({\displaystyle \frac{1}{{ϵ}^r}}\right)}^q$$

$${\displaystyle \frac{1}{{ϵ}^{nr}}}\sim {\displaystyle \frac{1}{{ϵ}^{qr+k}}}$$

(3)

For Equation (3) to hold, the exponents must satisfy

$$nr=qr+k$$

$$r(n-q)=k$$

$$r={\displaystyle \frac{k}{n-q}}$$

Therefore, another possible order for the roots is $O\left({\displaystyle \frac{1}{{ϵ}^{\frac{k}{n-q}}}}\right)$.

Thus, we have shown that the roots are of order $O\left({ϵ}^{{\displaystyle \frac{k}{p}}}\right)$ or $O\left({\displaystyle \frac{1}{{ϵ}^{\frac{k}{n-q}}}}\right)$, where p is the smallest index and q is the largest index among the large coefficients. □

Theorem 7

(coefficients of different orders). Consider a polynomial equation where the coefficients have different orders of magnitude:

$$a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0=0$$

If $a_i\sim O\left({\displaystyle \frac{1}{{ϵ}^{k_i}}}\right)$ for $i=0,1,\dots ,n$, where some $k_i$ may be zero or positive, then the possible orders of the roots are given by

$$O\left({ϵ}^{{\displaystyle \frac{k_i-k_j}{i-j}}}\right)$$

for any pair of distinct indices $i\ne j$.

Proof. 

We will prove this theorem by considering the balance between any two terms in the polynomial and showing that this leads to the stated order of magnitude for the roots.

Let i and j be any two distinct indices from the set $\{0,1,\dots ,n\}$. Consider balancing the i-th and j-th terms of the polynomial:

$$a_i{x}^i\sim a_j{x}^j$$

Substituting the orders of magnitude for the coefficients:

$${\displaystyle \frac{1}{{ϵ}^{k_i}}}·x^i\sim {\displaystyle \frac{1}{{ϵ}^{k_j}}}·x^j$$

Assume that $x\sim {ϵ}^r$ for some real number r. Substituting this into the above equation:

$${\displaystyle \frac{1}{{ϵ}^{k_i}}}·{\left({ϵ}^r\right)}^i\sim {\displaystyle \frac{1}{{ϵ}^{k_j}}}·{\left({ϵ}^r\right)}^j$$

$${\displaystyle \frac{{ϵ}^{ri}}{{ϵ}^{k_i}}}\sim {\displaystyle \frac{{ϵ}^{rj}}{{ϵ}^{k_j}}}$$

$${ϵ}^{ri-k_i}\sim {ϵ}^{rj-k_j}$$

(4)

For the relation (4) balance to hold, the exponents must be equal:

$$ri-k_i=rj-k_j$$

Solving for r:

$$ri-rj=k_i-k_j$$

$$r(i-j)=k_i-k_j$$

$$r={\displaystyle \frac{k_i-k_j}{i-j}}$$

Therefore, one possible order for the roots is $O\left({ϵ}^{{\displaystyle \frac{k_i-k_j}{i-j}}}\right)$.

Since this argument holds for any pair of distinct indices i and j, we have multiple possible orders for the roots, each corresponding to a different pair of terms being balanced.

Thus, we have proven that the possible orders of the roots are given by $O\left({ϵ}^{{\displaystyle \frac{k_i-k_j}{j-i}}}\right)$ for any pair of distinct indices $i\ne j$. □

Theorem 8

(alternating coefficient orders). Consider a polynomial equation of the form

$$a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0=0$$

(5)

where the coefficients alternate between two orders of magnitude:

$$a_i\sim \left\{\begin{array}{cc}O(1/{ϵ}^k)\hfill & ifiiseven\hfill \\ O\left(1\right)\hfill & ifiisodd\hfill \end{array}\right.$$

with $k>0$ and $ϵ\ll 1$. Then the roots of this polynomial are of order $O(1/{ϵ}^k)$.

Proof. 

Without loss of generality, let n be even. We consider two cases:

Case 1: Assume $|x|\sim O\left({ϵ}^r\right)$, where $r>0$.

Balancing the constant term with the first-order term:

$$a_0\sim a_{1}x$$

Substituting the orders of magnitude:

$${\displaystyle \frac{1}{{ϵ}^k}}\sim O\left(1\right)·{ϵ}^r$$

(6)

The expression (6):

$${ϵ}^{-k}\sim {ϵ}^r\Rightarrow r=-k$$

Case 2: Assume $|x|\sim O(1/{ϵ}^r)$, where $r>0$.

Balancing the n-th term with the $(n-1)$-th term:

$$a_n{x}^n\sim a_{n-1}{x}^{n-1}$$

Substituting the orders of magnitude:

$${\displaystyle \frac{1}{{ϵ}^k}}·{\left({\displaystyle \frac{1}{{ϵ}^r}}\right)}^n\sim O\left(1\right)·{\left({\displaystyle \frac{1}{{ϵ}^r}}\right)}^{n-1}$$

(7)

Expression (7) simplifies to

$${\displaystyle \frac{1}{{ϵ}^{k+nr}}}\sim {\displaystyle \frac{1}{{ϵ}^{(n-1)r}}}$$

(8)

For Equation (8) to hold, the exponents must be equal:

$$k+nr=(n-1)r\Rightarrow k=-r$$

giving us

$$r=-k$$

In both cases, we find $r=-k$. Therefore, the roots are of order $O(1/{ϵ}^k)$. □

Theorem 9

(polynomial with large linear and constant terms). Consider a polynomial equation of the form

$$x^n+a_{n-1}{x}^{n-1}+\cdots +a_2{x}^2+a_{1}x+a_0=0$$

where $a_1\sim O\left({\displaystyle \frac{1}{{ϵ}^k}}\right)$ and $a_0\sim O\left({\displaystyle \frac{1}{{ϵ}^m}}\right)$ with $m>k>0$ and $ϵ\ll 1$. All other coefficients are $O\left(1\right)$. Then, the roots of this polynomial are of order $O\left({ϵ}^{m-k}\right)$, $O\left({ϵ}^{{\displaystyle \frac{k}{n-1}}}\right)$, or $O\left(1\right)$.

Proof. 

We consider three cases based on the assumed order of the roots.

Case 1: Assume $|x|\sim O\left({ϵ}^r\right)$, where $r>0$.

Balancing the constant term with the linear term:

$$a_0\sim a_{1}x$$

Substituting the orders of magnitude:

$${\displaystyle \frac{1}{{ϵ}^m}}\sim {\displaystyle \frac{1}{{ϵ}^k}}·{ϵ}^r$$

(9)

Expression (9) gives

$${ϵ}^{-m}\sim {ϵ}^{r-k}\Rightarrow -m=r-k$$

$$r=m-k$$

For r to be positive, we require $m>k$. Therefore, the roots are of order $O\left({ϵ}^{m-k}\right)$.

Case 2: Assume $|x|\sim O\left({\displaystyle \frac{1}{{ϵ}^r}}\right)$, where $r>0$.

Balancing the n-th term with the linear term:

$$x^n\sim a_{1}x$$

Substituting the orders of magnitude:

$${\left({\displaystyle \frac{1}{{ϵ}^r}}\right)}^n\sim {\displaystyle \frac{1}{{ϵ}^k}}·{\displaystyle \frac{1}{{ϵ}^r}}$$

(10)

The expression (10) simplifies to

$${\displaystyle \frac{1}{{ϵ}^{nr}}}\sim {\displaystyle \frac{1}{{ϵ}^{k+r}}}$$

(11)

For Equation (11) to hold, the exponents must satisfy

$$nr=k+r$$

$$r(n-1)=k$$

$$r={\displaystyle \frac{k}{n-1}}$$

Thus, the roots are of order $O\left({ϵ}^{{\displaystyle \frac{k}{n-1}}}\right)$.

Case 3: Assume $|x|\sim O\left(1\right)$.

In this case, all terms are $O\left(1\right)$, and thus, the roots can be of order $O\left(1\right)$.

Therefore, the roots are of order $O\left({ϵ}^{m-k}\right)$ (from Case 1), $O\left({ϵ}^{{\displaystyle \frac{k}{n-1}}}\right)$ (from Case 2), or $O\left(1\right)$ (from Case 3). □

Theorem 10

(exponentially decreasing coefficients). Consider a polynomial equation of the form

$$a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0=0$$

where the coefficients decrease exponentially:

$$a_i\sim O\left({\displaystyle \frac{1}{{ϵ}^{k·2^i}}}\right)$$

with $k>0$, $ϵ\ll 1$, and $i=0,1,\dots ,n$. Then, the roots of this polynomial are of order $O\left({ϵ}^{k·2^i}\right)$ for each $i=0,1,\dots ,n-1$.

Proof. 

To determine the order of the roots, we consider the balance between consecutive terms in the polynomial.

Consider balancing the i-th and $(i+1)$-th terms of the polynomial:

$$a_i{x}^i\sim a_{i+1}{x}^{i+1}$$

Substituting the orders of magnitude:

$${\displaystyle \frac{1}{{ϵ}^{k·2^i}}}·x^i\sim {\displaystyle \frac{1}{{ϵ}^{k·2^{i+1}}}}·x^{i+1}$$

Assume $x\sim {ϵ}^r$ for some real number r. Substituting this into the above equation:

$${\displaystyle \frac{1}{{ϵ}^{k·2^i}}}·{\left({ϵ}^r\right)}^i\sim {\displaystyle \frac{1}{{ϵ}^{k·2^{i+1}}}}·{\left({ϵ}^r\right)}^{i+1}$$

Simplifying:

$${ϵ}^{ri-k·2^i}\sim {ϵ}^{r(i+1)-k·2^{i+1}}$$

Equate the exponents:

$$ri-k·2^i=r(i+1)-k·2^{i+1}$$

Simplifying:

$$ri-k·2^i=ri+r-k·2^{i+1}$$

Subtract $ri$ from both sides:

$$-k·2^i=r-k·2^{i+1}$$

Rearranging terms:

$$r=k·2^{i+1}-k·2^i$$

$$r=k·2^i(2-1)$$

$$r=k·2^i$$

Therefore, the order of the roots corresponding to the balance between the i-th and $(i+1)$-th terms is $O\left({ϵ}^{k·2^i}\right)$.

Since this argument holds for each pair of consecutive indices i and $i+1$ where $i=0,1,\dots ,n-1$, we conclude that the roots of the polynomial are of order $O\left({ϵ}^{k·2^i}\right)$ for each $i=0,1,\dots ,n-1$. That proves the theorem. □

Applications in Finding Bounds of Zeros

As an extension of our results to the study of bounds of zeros of random polynomials, taking inspiration from the work of Sheikh and Mir [25] about the zero bounds of random polynomials, we present the following theorem that concerns the roots of a randomly perturbed monic polynomial with uniformly perturbed coefficients. In the study of polynomials with perturbed coefficients, understanding the impact of these perturbations on the roots is crucial for both theoretical and practical applications. The following theorem addresses this by considering two classes of perturbed polynomials by employing the Cauchy bound for roots. Specifically, we consider the case when all but the leading coefficient receives a uniformly distributed perturbation in the interval $[-1,1],$ each modeled as independent uniform random variables. By deriving an upper bound on the expected maximum modulus of the polynomial’s zeros, the theorem provides insights into how these specific perturbations influence the distribution of the roots.

Theorem 11

(expected maximum modulus of zeros with uniform perturbations). Let $q\left(z\right)=z^n+(a_1+{\delta }_1)z^{n-1}+\cdots +(a_n+{\delta }_n)$ be a polynomial where each ${\delta }_k$ is a random variable representing a small perturbation, modeled as ${\delta }_k\sim Uniform(-1,1)$ where $Uniform(a,b)$ means being uniformly distributed over the interval $[a,b]$. The expected maximum modulus of the zeros of $q\left(z\right)$ is bounded above by

$$\mathbb{E}\left[\underset{i}{max}|z_i|\right]\le R+{\displaystyle \frac{n-1}{n+1}}$$

(12)

where $z_i$ are the zeros of $q\left(z\right),$ where $R=1+{max}_{0\le i\le n-1|a_i|}$ is the Cauchy bound of the zeros of polynomial

$$p\left(z\right)=z^n+a_{n-1}{z}^{n-1}+\cdots +a_0$$

(13)

Before proceeding to the proof, we need a lemma which we state and prove as follows:

Lemma 1 (expected maximum of uniform variables).

For ${\delta }_k\sim Uniform(-1,1)$, the expected maximum of n such variables is

$$\mathbb{E}\left[M\right]={\displaystyle \frac{n-1}{n+1}}$$

(14)

Proof of Lemma 1.

Let $M=max({\delta }_1,{\delta }_2,\dots ,{\delta }_n)$, where each ${\delta }_k$ is independently and identically distributed as $\mathrm{Uniform}(-1,1)$.

The cumulative distribution function (CDF) of a single uniform variable ${\delta }_k$ is given by

$$F\left(x\right)={\displaystyle \frac{x+1}{2}}\mathrm{for}-1\le x\le 1$$

(15)

The CDF of the maximum M of n such independent variables is

$$F_M\left(x\right)={\left({\displaystyle \frac{x+1}{2}}\right)}^n$$

(16)

The probability density function (PDF) of M is the derivative of the CDF:

$$f_M\left(x\right)={\displaystyle \frac{d}{dx}}{\left({\displaystyle \frac{x+1}{2}}\right)}^n=n{\left({\displaystyle \frac{x+1}{2}}\right)}^{n-1}·{\displaystyle \frac{1}{2}}$$

(17)

Thus, the PDF is

$$f_M\left(x\right)={\displaystyle \frac{n}{2}}{\left({\displaystyle \frac{x+1}{2}}\right)}^{n-1}$$

(18)

The expected value of M is

$$\mathbb{E}\left[M\right]={\int }_{-1}^{1}x·f_M\left(x\right)dx$$

(19)

Substituting the PDF:

$$\mathbb{E}\left[M\right]={\int }_{-1}^{1}x·{\displaystyle \frac{n}{2}}{\left({\displaystyle \frac{x+1}{2}}\right)}^{n-1}dx$$

(20)

Change variables: let $u={\displaystyle \frac{x+1}{2}}$, then $du={\displaystyle \frac{1}{2}}dx$ and $dx=2du$. The limits of integration change from $x=-1$ to $x=1$ to $u=0$ to $u=1$.

$$\begin{array}{cc}\hfill \mathbb{E}\left[M\right]& ={\int }_0^1(2u-1)·n·u^{n-1}du\hfill \\ \hfill & =n{\int }_0^1(2u^n-u^{n-1})du\hfill \\ \hfill & =n{\left[{\displaystyle \frac{2u^{n+1}}{n+1}}-{\displaystyle \frac{u^n}{n}}\right]}_0^1\hfill \\ \hfill & =n\left({\displaystyle \frac{2}{n+1}}-{\displaystyle \frac{1}{n}}\right)\hfill \\ \hfill & ={\displaystyle \frac{n-1}{n+1}}\hfill \end{array}$$

Thus, the expected maximum of n uniformly distributed variables is

$$\mathbb{E}\left[M\right]={\displaystyle \frac{n-1}{n+1}}$$

(21)

□

Proof of Theorem 11.

Let $M={max}_{1\le k\le n}|{\delta }_k|$. We claim that the maximum modulus of the zeros of $q\left(z\right)$ is bounded by $R+M$, where R is the bound on the maximum modulus of the zeros of the unperturbed polynomial $z^n+a_1{z}^{n-1}+\cdots +a_n$.

Consider the perturbed polynomial:

$$q\left(z\right)=z^n+(a_{n-1}+{\delta }_{n-1})z^{n-1}+\cdots +(a_0+{\delta }_0)$$

(22)

where each ${\delta }_k$ is a random variable representing a small perturbation, modeled as ${\delta }_k\sim \mathrm{Uniform}(-1,1)$.

For the unperturbed polynomial:

$$p\left(z\right)=z^n+a_{n-1}{z}^{n-1}+\cdots +a_0$$

(23)

Cauchy’s bound provides an upper limit on the magnitude of the roots:

$$|z_i|\le 1+max(|a_{n-1}|,|a_{n-2}|,\dots ,|a_0|)$$

(24)

Define this bound as R:

$$R=1+max(|a_{n-1}|,|a_{n-2}|,\dots ,|a_0|)$$

(25)

For the perturbed polynomial $q\left(z\right)$, the coefficients are $a_k+{\delta }_k$. The bound on the magnitude of the roots becomes

$$|z_i|\le 1+max(|a_{n-1}+{\delta }_{n-1}|,|a_{n-2}+{\delta }_{n-2}|,\dots ,|a_0+{\delta }_0|)$$

(26)

The maximum can be decomposed as

$$\begin{array}{c}max(|a_{n-1}+{\delta }_{n-1}|,|a_{n-2}+{\delta }_{n-2}|,\dots ,|a_0+{\delta }_0|)\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le max(|a_{n-1}|,|a_{n-2}|,\dots ,|a_0|)+max(|{\delta }_{n-1}|,|{\delta }_{n-2}|,\dots ,|{\delta }_0|)\end{array}$$

(27)

Define the maximum perturbation as M:

$$M=max(|{\delta }_{n-1}|,|{\delta }_{n-2}|,\dots ,|{\delta }_0|)$$

(28)

Substituting the decomposed maximum into the bound for the perturbed polynomial, we have

$$|z_i|\le 1+max(|a_{n-1}|,|a_{n-2}|,\dots ,|a_0|)+M$$

(29)

Thus, the bound for the perturbed polynomial is

$$|z_i|\le R+M$$

(30)

This derivation shows that the bound $R+M$ accounts for both the original bound R and the maximum effect of the perturbations M, ensuring that the magnitude of the roots is appropriately bounded.

Thus, the expected maximum modulus of the zeros is

$$\mathbb{E}\left[\underset{i}{max}|z_i|\right]\le \mathbb{E}[R+M]$$

(31)

Since R is a constant, we can separate the expectation:

$$\mathbb{E}[R+M]=R+\mathbb{E}\left[M\right]$$

(32)

Substituting the bound for $\mathbb{E}\left[M\right]$ using Lemma 1, we obtain:

$$\mathbb{E}\left[\underset{i}{max}|z_i|\right]\le R+{\displaystyle \frac{n-1}{n+1}}$$

(33)

$$\mathbb{E}\left[\underset{i}{max}|z_i|\right]\le \mathbb{E}[R+M]$$

(34)

That establishes the theorem. □

3. Discussion of Simulation Results of Uniform Perturbations

Our numerical simulations and visualizations provide valuable insights into the behavior of polynomial zeros under different perturbation scenarios, as described in Theorem 11. In our simulations depicted in Figure 1 for Theorem 11, we considered polynomials of degree $n=10$ with coefficients perturbed with the Cauchy bound $R=2$ for the underlying polynomial $p\left(z\right)$. We conducted a total of 10,000 trials to ensure statistical significance. The results demonstrate the effectiveness of the derived upper bound for the expected maximum modulus of zeros under uniform perturbations. The average maximum modulus of zeros obtained from our simulations was approximately 1.7011, which is consistently below the theoretical upper bound of 2.8182, validating the theorem’s prediction. The simulation results for Theorem 11 demonstrate the effectiveness of the derived upper bound for the expected maximum modulus of zeros under uniform perturbations. We observed that the average maximum modulus of zeros obtained from our simulations is consistently below the theoretical upper bound, validating the theorem’s prediction. Additionally, the distribution of maximum moduli shows a right-skewed shape, indicating that while most perturbed polynomials have zeros with relatively small moduli, there are occasional instances of larger maximum moduli. Furthermore, the theoretical bound provides a reliable estimate of the worst-case scenario, as it consistently lies to the right of the bulk of the distribution.

4. Conclusions

In this paper, we have significantly extended the application of perturbation theory to the estimation of polynomial roots. Building upon the foundational work of Pakdemirli and Yurtsever [9], we developed and proved several novel theorems that address a wide range of polynomial structures while developing a theorem illustrating how the work can be extended to randomly perturbed polynomials. The results we study include polynomials with multiple large coefficients, coefficients of different orders, alternating coefficient orders, large linear and constant terms, and exponentially decreasing coefficients. These theorems provide a comprehensive framework for estimating the order of magnitude of polynomial roots based on the structure and magnitude of their coefficients. Our results offer several key contributions to the field. First, they extend the applicability of perturbation-based root estimation to a broader class of polynomials, including those with complex coefficient patterns. Second, they provide quick and reliable estimates of root magnitudes without the need for explicit root-finding algorithms, which can be particularly valuable in time-sensitive applications or as a precursor to more detailed numerical analysis. Third, they offer insights into the relationship between coefficient patterns and root behavior, deepening our theoretical understanding of polynomial properties.

The numerical examples presented for the randomly perturbed polynomials with uniformly and identically distributed perturbation errors in this paper demonstrate the utility of our theorems across various scenarios that might arise in practical situations. These results suggest that our approach could be particularly valuable in fields such as control systems, signal processing, and numerical analysis, where quick estimation of polynomial roots is often necessary.

However, it is important to note the limitations of our approach. While these theorems provide excellent estimates for the order of magnitude of roots, they do not give exact root values. Additionally, in some cases with very close root magnitudes, the method may not distinguish between individual roots.

Future research directions could include extending these techniques to multivariate polynomials, investigating the application of these methods to transcendental equations, developing algorithms that combine these estimation techniques with traditional root-finding methods for improved efficiency, and exploring the implications of these results in specific application areas, such as filter design or stability analysis of dynamical systems.

In conclusion, this work represents a significant advancement in the field of perturbation-based polynomial root estimation. By providing a more comprehensive set of tools for analyzing polynomial roots, we hope to facilitate advancements in both theoretical mathematics and practical applications across various scientific and engineering disciplines.

5. Potential Applications in Various Scientific Fields

The novel theorems and techniques presented in this paper for estimating polynomial roots using perturbation theory have potential applications across a wide range of scientific and engineering disciplines. These applications stem from the ubiquity of polynomial equations in modeling complex systems and phenomena. In the field of random polynomials, where coefficients are drawn from probability distributions, perturbation analysis helps in understanding the stability and sensitivity of roots to small random variations [6]. This approach is particularly useful in fields such as statistical physics, where it aids in studying phase transitions and critical phenomena. Furthermore, perturbation techniques applied to random polynomials contribute to the analysis of eigenvalue distributions in random matrix theory, which has far-reaching implications in quantum chaos, wireless communications, and financial mathematics In the field of control systems engineering, our results can significantly impact the analysis and design of feedback control systems. The characteristic equation of a linear time-invariant system is often represented as a polynomial, and its roots determine the system’s stability and transient response [1]. Our quick estimation methods can provide engineers with rapid insights into system behavior during the design phase, potentially streamlining the iterative process of controller tuning.

Signal processing is another area that could benefit from these advancements. Many digital filters are designed based on polynomial equations, where the roots of these polynomials correspond to the filter’s poles and zeros [20]. Our theorems could offer a more intuitive understanding of how coefficient changes affect filter characteristics, possibly leading to more efficient filter design algorithms.

In the realm of numerical analysis, our work has implications for improving root-finding algorithms. By providing good initial estimates of root magnitudes, our methods could enhance the convergence speed and reliability of iterative root-finding techniques such as the Newton–Raphson method or Muller’s method [21]. This could be particularly valuable when dealing with high-degree polynomials or ill-conditioned systems.

The field of computational physics often involves solving polynomial equations that arise from discretized differential equations. Our perturbation-based approach could offer physicists quick insights into the behavior of these discretized systems, potentially guiding the choice of numerical methods or helping to identify potential instabilities in simulations [2].

In electrical engineering, particularly in circuit analysis and design, polynomial equations frequently appear in transfer functions and network analysis. Our methods could provide engineers with rapid estimates of circuit behavior, assisting in the early stages of design or in troubleshooting complex systems [3].

The study of dynamical systems, which spans multiple disciplines including mathematics, physics, and biology, often involves analyzing polynomial equations. Our theorems could offer new tools for understanding bifurcations and stability changes in these systems, potentially leading to new insights in areas such as population dynamics, chemical kinetics, and climate modeling [22].

In the field of computer graphics and geometric modeling, polynomial equations are used extensively in curve and surface representation. Our quick estimation techniques could potentially optimize algorithms for intersection detection or shape manipulation, leading to more efficient rendering and modeling software [23].

Quantum chemistry calculations often involve solving polynomial equations that arise from approximations to the Schrödinger equation. Our methods could potentially assist in the rapid estimation of molecular orbital energies or in the analysis of potential energy surfaces [24].

While these potential applications are diverse, they all share a common thread: the need for quick, reliable estimates of polynomial roots. Our work provides a theoretical foundation that could be adapted and refined for these specific contexts, potentially leading to advancements across multiple scientific disciplines. As researchers and practitioners in these fields begin to apply and extend our results, we anticipate a rich interplay between theory and application, driving further innovations in both perturbation theory and its practical implementations.