Perturbation Stability in Evaluating Modified Stieltjes Continued Fractions for Modern Applications

Abstract

Due to the demands of modern applications, such as forecasting using ordinal continued fraction regression models in economics, the study of structural stability to data perturbations has become an urgent need. While continued fractions demonstrate superior effectiveness over traditional polynomial models due to their inherent structural stability, their sensitivity to coefficient perturbations can lead to accuracy loss under finite-precision machine arithmetic. This paper investigates the stability to perturbations of the coefficients of modified Stieltjes continued fractions (modified S-fractions). We develop a rigorous theoretical framework based on a fundamental recursive formula for the relative error of the nth approximant, expressing error propagation directly in terms of coefficient perturbations. Using this geometric and analytical approach, we establish three distinct uniform stability sets in the complex plane: the half-plane ${\mathrm{\Omega }}_C$, the angular set ${\mathrm{\Omega }}_{\alpha ,C}$ under weaker geometric conditions, and the exterior of the disk ${\mathrm{\Omega }}_R$, which extends verified stability directly to points on and near the imaginary axis. Furthermore, we derive explicit, a priori error bounds for each stability set and prove pointwise stability on a significantly expanded domain, demonstrating that the modified S-fraction remains stable on $\mathbb{R}\setminus \left\{0\right\}$ for any bounded coefficients. These findings establish explicit boundaries for safe computation using standard double-precision arithmetic, providing a mathematical guarantee of algorithmic stability for high-demand contemporary applied models.

1. Introduction

The rapid advancement of computational modeling has established continued fractions as one of the most effective analytical tools for approximating special functions across various fields of science and engineering [1,2]. Traditionally valued for their rapid convergence properties, these mathematical structures have recently found innovative and critical applications in fields driven by empirical data, such as economics [3,4]. For instance, recent developments in forecasting using ordinal continued fraction regression models have demonstrated significantly greater predictive accuracy and stability than traditional polynomial-based models. This operational superiority is largely due to the inherent structural stability of continued fractions to unstable or imperfect real-world inputs.

However, the transition of continued fractions into high-demand contemporary applications exposes them to a critical operational challenge: numerical instability under finite-precision machine arithmetic. In practical implementations, input data are frequently subject to rounding errors, measurement uncertainties, or systemic perturbations. If a computing algorithm lacks sufficient stability, these microscopic perturbations can propagate through successive iterations, leading to a catastrophic loss of significant digits and rendering the model ineffective. Consequently, the demands of modern applied modeling necessitate a rigorous theoretical foundation that can guarantee the structural integrity of specific functional continued fraction classes against complex computational anomalies. To achieve this level of mathematical rigor, it is essential to clearly isolate four interconnected yet distinct concepts: convergence, perturbation stability, numerical stability, and rounding-error analysis.

• Convergence remains a classical, purely analytical property that dictates whether a continued fraction theoretically approaches a precise mathematical limit as the approximant order n approaches infinity under conditions of idealized, infinite precision.
• Conversely, perturbation stability evaluates the inherent capacity of the functional architecture to preserve its output value when its underlying coefficients are permanently altered or corrupted by external data noise, measurement uncertainties, or systemic inputs.
• Numerical stability characterizes the broader operational susceptibility of a specific evaluation algorithm to cumulative mathematical deviations during execution, often dictating the structural choice between forward and backward recurrence mechanisms.
• Finally, rounding-error analysis isolates the specific microscopic precision loss and digit decay triggered by finite-precision machine arithmetic—such as the standard IEEE 754 double-precision unit roundoff constraints—during iterative loops.

By explicitly separating these factors, this paper shifts the analytical focus away from unconstrained convergence metrics toward localized boundary mapping. Among these classes, Stieltjes continued fractions [5] (or S-fractions) occupy a prominent role due to their classic mathematical connections with orthogonal polynomials [6] and the moment problem [7], as well as their wide range of applications in applied problems (see, for example, [8,9,10]).

While their convergence properties are well documented (see [5] or [11] (Theorem 4.58) and [12] (Corollary 5.1), as well as [13,14,15]), existing research often falls short of defining explicit, localized boundaries of stability in the complex plane for their modified variants. Given these practical constraints, this paper theoretically investigates the stability to perturbations of modified S-fractions [2]:

$${\displaystyle \frac{a_0}{z+\frac{a_1}{z + \frac{a_2}{z + \genfrac{}{}{0pt}{}{}{\ddots }}}}},a_k>0,k\ge 0.$$

(1)

By deriving a new recursive error formula, we map the exact behavioral boundaries of these continued fractions, establishing explicit uniform and pointwise stability sets that ensure computational stability under modern application demands.

1.1. Literature Review

The necessity of evaluating functional continued fractions under nonideal computational conditions has sparked a robust lineage of analytical research. In practical data environments, understanding mere convergence is insufficient; isolating numerical stability, which characterizes the sensitivity of approximant values to rounding and precision loss, is paramount [16]. Early foundations established by Blanch [17] demonstrated that while forward recurrence algorithms (FR-algorithms) are highly susceptible to rapid error accumulation, backward recurrence algorithms (BR-algorithms) exhibit fundamentally superior stability profiles. Building on this operational distinction, Jones and Thron [11,18] pioneered methods to bound the relative error accumulated during backward recurrence by mapping errors directly to the underlying element sets of the continued fractions. These structural error-estimating techniques of continued fractions were subsequently refined [19] and extended to branched continued fraction expansions [20,21,22].

Modern applied modeling, however, places demands on continued fractions that extend beyond standard algorithmic rounding. Since real-world inputs in empirical regression systems are inherently noisy, contemporary research focuses heavily on stability to perturbations—the capacity of a continued fraction to preserve its output value when its functional coefficients are permanently altered by external data errors. For continued fractions with unit partial numerators, these stability zones were mapped by Hladun and Dmytryshyn [23]. Subsequent expansions by these authors established complex stability sets for continued fractions with unified partial denominators, which were successfully applied to map the boundaries of g-fractions [24].

Despite these advancements, traditional S-fractions, despite their deep classical utility in moment problems and orthogonal polynomials, have only recently been evaluated under the lens of perturbation stability. Specifically, Hladun and Dmytryshyn [25] mapped the perturbation stability of S-fractions with complex elements, where the complex variable z acts as a multiplier in the partial numerators ($a_{n}z$) against unit denominators. However, because their error propagation is modeled implicitly as a continued fraction, their framework restricts verified stability to compact, localized interior zones near the origin, producing bounded semi-circular and circular sets.

This paper directly addresses this gap by investigating the modified Stieltjes continued fraction variant, where the variable z forms the mathematical engine of the architecture by acting directly as the partial denominator ($a_n/z$). By shifting this underlying functional structure and deriving a fundamental recursive formula for the relative error, we map exact behavioral boundaries that fundamentally expand the geometric scope of perturbation theory. Rather than being confined to bounded regions, our approach establishes vast, unbounded uniform stability sets in the complex plane, including the half-plane ${\mathrm{\Omega }}_C,$ the angular set ${\mathrm{\Omega }}_{\alpha ,C},$ and the exterior of the disk ${\mathrm{\Omega }}_R.$ Most notably, while existing classical frameworks encounter mathematical volatility near the imaginary axis, our results extend verified stability directly onto the imaginary axis and its immediate neighborhoods via ${\mathrm{\Omega }}_R,$ while guaranteeing pointwise stability across the entire punctured real axis ($\mathbb{R}\setminus \left\{0\right\}$) for any bounded coefficients. This comprehensive mapping provides the precise mathematical assurances required to meet the algorithmic demands of modern applied computational frameworks.

1.2. Computational Experiment and Empirical Stability Stress Test

To evaluate the empirical robustness of the modified S-fraction under conditions that simulate high-demand modern data applications, we conducted a rigorous numerical stress test. In empirical modeling, algorithms must maintain structural integrity when processing noisy inputs or running under finite machine precision. For this experiment, we utilized a modified S-fraction representing the expansion of the square of the ratio of Euler gamma functions [26]:

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{\mathrm{\Gamma }((z+1)/4)}{\mathrm{\Gamma }((z+3)/4)}}\right)}^2={\displaystyle \frac{4}{z+\frac{1/2}{z + \frac{9/4}{z + \frac{25/4}{z + \frac{49/4}{z + \genfrac{}{}{0pt}{}{}{\ddots }}}}}}}.\end{array}$$

(2)

This specific functional architecture serves as an ideal baseline for testing error propagation because its coefficients grow systematically, mirroring the behavioral complexity found in advanced regression forecasting models.The experiment was structured to observe how micro-level coefficient perturbations—modeled here by standard double-precision rounding errors—accumulate as the complexity of the model increases. Approximants ${\widehat{f}}_n(z)$ of orders ranging from 1 to 2000 were computed using standard double-precision arithmetic compliant with the IEEE 754 standard (unit roundoff $u\approx 1.11\times {10}^{-16}$). To isolate the exact magnitude of accumulated error, these values were compared against highly accurate reference values $f_n(z)$ generated via software-based 128-bit extended precision arithmetic. The relative error $|({\widehat{f}}_n(z)-f_n(z))/f_n(z)|$ was systematically evaluated across a dense grid of test points z in the complex plane, bounded by $\mathrm{Re}(z),\mathrm{Im}(z)\in [-7.5,7.5]$.

A pointwise analysis of the relative error relative to the approximant order n reveals distinct behavioral zones that dictate the continued fraction’s real-world operational boundaries (see Figure 1a–h). For coordinate points positioned safely away from the origin and the imaginary axis, the modified S-fraction exhibits a remarkable, self-stabilizing structural robustness. In these secure domains, the relative error flatlines at the level of machine precision (≈${10}^{-16}$) and shows absolute resistance to error accumulation, even as the order of the continued fraction scales up to 2000 terms. This profile confirms the continued fraction’s supreme viability for stable execution in standard hardware environments. Conversely, as the test variable z migrates toward the origin, the system encounters a localized volatile zone. Here, the micro-errors accumulate dynamically over successive iterations, driving the relative error up by one to four orders of magnitude (${10}^{-15}$ to ${10}^{-12}$) and resulting in a measurable decay of significant digits.

Figure 1. Relative errors for continued fraction (2) at selected points z.

To map this behavior globally, the total distribution of precision loss was rendered into a comprehensive heat map (see Figure 2). The color gradients chart the exact spatial limits of algorithmic robustness under 64-bit rounding constraints. The visual data demonstrate that the vast majority of the complex plane comprises a completely safe computational field where the fraction preserves its structural integrity effortlessly.

Figure 2. Distribution of relative error for continued fraction (2).

Crucially for modern application design, the zone of accuracy loss does not bleed indefinitely across the complex plane; rather, it is tightly confined to a bounded, symmetrical corridor directly flanking the imaginary axis. This empirical layout proves that numerical instability is not an unmanageable trait of the evaluation algorithm itself but a predictable, spatially bounded phenomenon. For practical software engineering, this implies that standard, high-speed double-precision arithmetic is entirely sufficient for executing these models across major regions of the complex plane, while computationally expensive extended precision frameworks can be strictly rationed for instances where the variable z enters the mathematically verified volatile zone.

In this paper, we investigate this phenomenon theoretically and provide a priori estimates for the relative errors of the approximants. Our main goal is to analyze the stability of the modified S-fraction (1) under perturbations of its coefficients and to establish explicit sets of uniform and pointwise stability in the complex plane. This paper is organized as follows. Section 2 introduces the basic notions and derives a fundamental recursive formula for the relative error of the approximants. Section 3 establishes the sets of uniform stability (Theorems 1–3) and pointwise stability (Corollaries 1, 2, 4) along with explicit error bounds. Section 4 presents the application of the obtained results to the empirical analysis of the stability of the modified S-fraction for the function $\arctan (1/z)$ and to polynomial and continued fraction models for experimental studies and forecasting the dynamics of patent activity in Ukraine. Section 5 contains concluding remarks and outlines directions for further research.

2. Formulas for Relative Rounding Error

Let n be an arbitrary natural number. By perturbing the coefficients $a_k$, $k\ge 0$, of the continued fraction (1), we obtain the perturbed modified S-fraction

$${\displaystyle \frac{{\widehat{a}}_0}{z+\frac{{\widehat{a}}_1}{z + \frac{{\widehat{a}}_2}{z + \genfrac{}{}{0pt}{}{}{\ddots }}}}},{\widehat{a}}_k>0,k\ge 0.$$

(3)

Denote by

$$f_n(z)={\displaystyle \frac{a_0}{z+\frac{a_1}{z + \frac{a_2}{z + \genfrac{}{}{0pt}{}{}{{\displaystyle \ddots +\frac{a_n}{z}}}}}}}$$

(4)

and

$${\widehat{f}}_n(z)={\displaystyle \frac{{\widehat{a}}_0}{z+\frac{{\widehat{a}}_1}{z + \frac{{\widehat{a}}_2}{z + \genfrac{}{}{0pt}{}{}{{\displaystyle \ddots +\frac{{\widehat{a}}_n}{z}}}}}}}$$

(5)

the nth approximants of the modified S-fraction (1) and the perturbed modified S-fraction (3), respectively. Assume that $f_n(z)\ne 0$ for all $z\in \mathrm{\Omega }$, $\mathrm{\Omega }\subseteq \mathbb{C}.$

Definition 1.

We call the modified S-fraction (1) stable to perturbations at a point $z_0\in \mathbb{C}$ if for every $\epsilon >0$ there exists ${\delta }_{\epsilon }(z_0)>0$ such that for all ${\widehat{a}}_k>0$, $k\ge 0$, satisfying the condition

$$\left|{\displaystyle \frac{{\widehat{a}}_k-a_k}{a_k}}\right|<{\delta }_{\epsilon }(z_0),k\ge 0,$$

the inequality

$$\left|{\displaystyle \frac{{\widehat{f}}_n(z_0)-f_n(z_0)}{f_n(z_0)}}\right|<\epsilon$$

holds for all $n\ge 1$.

Definition 2.

We call the set $\mathrm{\Omega },$ $\mathrm{\Omega }\subseteq \mathbb{C},$ a set of stability to perturbations of the modified S-fraction (1) if it is stable to perturbations at every point of this set.

Definition 3.

We call the set $\mathrm{\Omega },$ $\mathrm{\Omega }\subseteq \mathbb{C},$ a set of uniform stability to perturbations of the modified S-fraction (1) if for every $\epsilon >0$ there exists ${\delta }_{\epsilon }>0$ such that for all $z\in \mathrm{\Omega }$ and all ${\widehat{a}}_k>0$, $k\ge 0$, satisfying the condition

$$\left|{\displaystyle \frac{{\widehat{a}}_k-a_k}{a_k}}\right|<{\delta }_{\epsilon },k\ge 0,$$

the inequality

$$\left|{\displaystyle \frac{{\widehat{f}}_n(z)-f_n(z)}{f_n(z)}}\right|<\epsilon$$

holds for all $n\ge 1$.

Now, denote by

$$Q_k^{(n)}(z)=z+{\displaystyle \frac{a_{k+1}}{z + \frac{a_{k+2}}{z + \frac{a_{k+3}}{z + \genfrac{}{}{0pt}{}{}{{\displaystyle \ddots +\frac{a_n}{z}}}}}}},0\le k\le n-1,Q_n^{(n)}(z)=z,$$

(6)

$${\widehat{Q}}_k^{(n)}(z)=z+{\displaystyle \frac{{\widehat{a}}_{k+1}}{z + \frac{{\widehat{a}}_{k+2}}{z + \frac{{\widehat{a}}_{k+3}}{z + \genfrac{}{}{0pt}{}{}{{\displaystyle \ddots +\frac{{\widehat{a}}_n}{z}}}}}}},0\le k\le n-1,{\widehat{Q}}_n^{(n)}(z)=z.$$

(7)

the tails of the nth approximant (4) and the perturbed nth approximant (5), respectively. Assume that $Q_k^{(n)}(z)\ne 0$, ${\widehat{Q}}_k^{(n)}(z)\ne 0$, for all $0\le k\le n$ and $z\in \mathrm{\Omega }$, $\mathrm{\Omega }\subseteq \mathbb{C}.$

Next, we establish a formula for the relative error of the approximant (4). For this purpose, we consider the relative errors of the coefficients ${\widehat{a}}_k$ of the modified S-fraction (3)

$${\widehat{a}}_k=a_k(1+{\epsilon }_k^{(a,1)}),k\ge 0,$$

the relative error of the nth approximant (5)

$${\widehat{f}}_n=f_n(1+{\epsilon }_n^{(f)}),$$

and the relative errors of the tails (7)

$${\widehat{Q}}_k^{(n)}(z)=Q_k^{(n)}(z)(1+{\epsilon }_{k,n}^{(Q,1)}),0\le k\le n.$$

We also consider the relative errors with respect to the perturbed values

$$a_k={\widehat{a}}_k(1+{\epsilon }_k^{(a,-1)}),k\ge 0,$$

and

$$Q_k^{(n)}(z)={\widehat{Q}}_k^{(n)}(z)(1+{\epsilon }_{k,n}^{(Q,-1)}),0\le k\le n.$$

Note that the relative errors ${\epsilon }_k^{(a,1)}$, ${\epsilon }_k^{(a,-1)}$ and ${\epsilon }_{k,n}^{(Q,1)}$, ${\epsilon }_{k,n}^{(Q,-1)}$ satisfy the following relations

$$(1+{\epsilon }_k^{(a,1)})(1+{\epsilon }_k^{(a,-1)})=1,k\ge 0,$$

and

$$(1+{\epsilon }_{k,n}^{(Q,1)})(1+{\epsilon }_{k,n}^{(Q,-1)})=1,0\le k\le n.$$

The following lemmas hold.

Lemma 1.

Let $Q_k^{(n)}(z)$ and ${\widehat{Q}}_k^{(n)}(z)$, $0\le k\le n$, $n\ge 1,$ be the tails of the approximants $f_n(z)$ and ${\widehat{f}}_n(z)$, defined by (6) and (7), respectively. Let $Q_k^{(n)}(z)\ne 0$ and ${\widehat{Q}}_k^{(n)}(z)\ne 0$ for all $0\le k\le n$, $n\ge 1,$ and $z\in \mathrm{\Omega },$$\mathrm{\Omega }\subseteq \mathbb{C}$. Then, for $0\le k\le n,$$n\ge 1$, and $\sigma \in \{-1,1\}$,

$${\epsilon }_{k,n}^{(Q,\sigma )}={\displaystyle \sum _{p=1}^{n-k}}(1+{\epsilon }_{k+p,n}^{(Q,{(-1)}^p\sigma )}){\epsilon }_{k+p}^{(a,{(-1)}^{p+1}\sigma )}{\displaystyle \prod _{m=1}^p}{G}_{k+m}^{(n,{(-1)}^{m+1}\sigma )}(z),$$

(8)

where

$$G_k^{(n,\sigma )}(z)=\left\{\begin{array}{c}{\displaystyle \frac{{\widehat{a}}_k}{{\widehat{Q}}_{k-1}^{(n)}{\widehat{Q}}_k^{(n)}}}if\sigma =-1,\hfill \\ {\displaystyle \frac{a_k}{Q_{k-1}^{(n)}{Q}_k^{(n)}}}if\sigma =1.\hfill \end{array}\right.$$

Proof. 

Let n be an arbitrary natural number. From formulas (6) and (7), which define the tails $Q_k^{(n)}(z)$ and ${\widehat{Q}}_k^{(n)}(z)$, respectively, the following recurrence relations follow:

$$Q_k^{(n)}(z)=z+{\displaystyle \frac{a_{k+1}}{Q_{k+1}^{(n)}(z)}},0\le k\le n-1,Q_n^{(n)}(z)=z,$$

and

$${\widehat{Q}}_k^{(n)}(z)=z+{\displaystyle \frac{{\widehat{a}}_{k+1}}{{\widehat{Q}}_{k+1}^{(n)}(z)}},0\le k\le n-1,{\widehat{Q}}_n^{(n)}(z)=z.$$

We will prove that for $0\le k\le n-1$ and $\sigma \in \{-1,1\}$ the following formula holds

$${\epsilon }_{k,n}^{(Q,\sigma )}=G_{k+1}^{(n,\sigma )}(z)((1+{\epsilon }_{k+1,n}^{(Q,-\sigma )}){\epsilon }_{k+1}^{(a,\sigma )}+{\epsilon }_{k+1,n}^{(Q,-\sigma )}).$$

(9)

For $\sigma =-1$, we have

$$\begin{array}{cc}\hfill {\epsilon }_{k,n}^{(Q,-1)}& ={\displaystyle \frac{Q_k^{(n)}(z)}{{\widehat{Q}}_k^{(n)}(z)}}-1\hfill \\ & ={\displaystyle \frac{z+a_{k+1}/Q_{k+1}^{(n)}(z)}{{\widehat{Q}}_k^{(n)}(z)}}-1\hfill \\ & ={\displaystyle \frac{z}{{\widehat{Q}}_k^{(n)}(z)}}-1+{\displaystyle \frac{{\widehat{a}}_{k+1}(1+{\epsilon }_{k+1}^{(a,-1)})}{{\widehat{Q}}_k^{(n)}(z){\widehat{Q}}_{k+1}^{(n)}(z)(1+{\epsilon }_{k+1,n}^{(Q,-1)})}}.\hfill \end{array}$$

Taking into account that

$$1-{\displaystyle \frac{{\widehat{a}}_{k+1}}{{\widehat{Q}}_k^{(n)}(z){\widehat{Q}}_{k+1}^{(n)}(z)}}={\displaystyle \frac{z}{{\widehat{Q}}_k^{(n)}(z)}}$$

and

$$1+{\epsilon }_{k+1,n}^{(Q,1)}={\displaystyle \frac{1}{1+{\epsilon }_{k+1,n}^{(Q,-1)}}},$$

we have

$$\begin{array}{cc}\hfill {\epsilon }_{k,n}^{(Q,-1)}& =-{\displaystyle \frac{{\widehat{a}}_{k+1}}{{\widehat{Q}}_k^{(n)}(z){\widehat{Q}}_{k+1}^{(n)}(z)}}+{\displaystyle \frac{{\widehat{a}}_{k+1}}{{\widehat{Q}}_k^{(n)}(z){\widehat{Q}}_{k+1}^{(n)}(z)}}(1+{\epsilon }_{k+1}^{(a,-1)})(1+{\epsilon }_{k+1,n}^{(Q,1)})\hfill \\ & ={\displaystyle \frac{{\widehat{a}}_{k+1}}{{\widehat{Q}}_k^{(n)}(z){\widehat{Q}}_{k+1}^{(n)}(z)}}((1+{\epsilon }_{k+1,n}^{(Q,1)}){\epsilon }_{k+1}^{(a,-1)}+{\epsilon }_{k+1,n}^{(Q,1)})\hfill \\ & =G_{k+1}^{(n,-1)}(z)((1+{\epsilon }_{k+1,n}^{(Q,1)}){\epsilon }_{k+1}^{(a,-1)}+{\epsilon }_{k+1,n}^{(Q,1)}).\hfill \end{array}$$

Similarly, for $\sigma =1$, we have

$$\begin{array}{cc}\hfill {\epsilon }_{k,n}^{(Q,1)}& =-{\displaystyle \frac{a_{k+1}}{Q_k^{(n)}(z)Q_{k+1}^{(n)}(z)}}+{\displaystyle \frac{a_{k+1}(1+{\epsilon }_{k+1}^{(a,1)})}{Q_k^{(n)}(z)Q_{k+1}^{(n)}(z)(1+{\epsilon }_{k+1,n}^{(Q,1)})}}\hfill \\ & ={\displaystyle \frac{a_{k+1}}{Q_k^{(n)}(z)Q_{k+1}^{(n)}(z)}}((1+{\epsilon }_{k+1,n}^{(Q,-1)}){\epsilon }_{k+1}^{(a,1)}+{\epsilon }_{k+1,n}^{(Q,-1)})\hfill \\ & =G_{k+1}^{(n,1)}(z)((1+{\epsilon }_{k+1,n}^{(Q,-1)}){\epsilon }_{k+1}^{(a,1)}+{\epsilon }_{k+1,n}^{(Q,-1)}).\hfill \end{array}$$

We prove formula (8) by induction on k, $k=n-1,n-2,\dots ,0$. Since ${\epsilon }_{n,n}^{(Q,\sigma )}=0$, $\sigma \in \{-1,1\}$, for $k=n-1$, we have

$${\epsilon }_{n-1,n}^{(Q,\sigma )}=G_n^{(n,\sigma )}(z){\epsilon }_n^{(a,\sigma )},\sigma \in \{-1,1\}.$$

Assume that formula (8) holds for $k=s+1$

$${\epsilon }_{s+1,n}^{(Q,\sigma )}={\displaystyle \sum _{p=1}^{n-s-1}}(1+{\epsilon }_{s+1+p,n}^{(Q,{(-1)}^p\sigma )}){\epsilon }_{s+1+p}^{(a,{(-1)}^{p+1}\sigma )}{\displaystyle \prod _{m=1}^p}{G}_{s+1+m}^{(n,{(-1)}^{m+1}\sigma )}(z).$$

(10)

Then, substituting (10) into (9), for $k=s$, we obtain

$$\begin{array}{cc}\hfill {\epsilon }_{s,n}^{(Q,\sigma )}& =G_{s+1}^{(n,\sigma )}(z)((1+{\epsilon }_{s+1,n}^{(Q,-\sigma )}){\epsilon }_{s+1}^{(a,\sigma )}+{\epsilon }_{s+1,n}^{(Q,-\sigma )})\hfill \\ & =G_{s+1}^{(n,\sigma )}(z)(1+{\epsilon }_{s+1,n}^{(Q,-\sigma )}){\epsilon }_{s+1}^{(a,\sigma )}+G_{s+1}^{(n,\sigma )}(z){\displaystyle \sum _{p=1}^{n-s-1}}(1+{\epsilon }_{s+1+p,n}^{(Q,{(-1)}^{p+1}\sigma )}){\epsilon }_{s+1+p}^{(a,{(-1)}^{p+2}\sigma )}{\displaystyle \prod _{m=1}^p}{G}_{s+1+m}^{(n,{(-1)}^{m+2}\sigma )}(z)\hfill \\ & =G_{s+1}^{(n,\sigma )}(z)(1+{\epsilon }_{s+1,n}^{(Q,-\sigma )}){\epsilon }_{s+1}^{(a,\sigma )}+{\displaystyle \sum _{p=2}^{n-s}}(1+{\epsilon }_{s+p,n}^{(Q,{(-1)}^p\sigma )}){\epsilon }_{s+p}^{(a,{(-1)}^{p+1}\sigma )}{G}_{s+1}^{(n,\sigma )}(z){\displaystyle \prod _{m=2}^p}{G}_{s+m}^{(n,{(-1)}^{m+1}\sigma )}(z)\hfill \\ & ={\displaystyle \sum _{p=1}^{n-s}}(1+{\epsilon }_{s+p,n}^{(Q,{(-1)}^p\sigma )}){\epsilon }_{s+p}^{(a,{(-1)}^{p+1}\sigma )}{\displaystyle \prod _{m=1}^p}{G}_{s+m}^{(n,{(-1)}^{m+1}\sigma )}(z).\hfill \end{array}$$

□

Lemma 2.

Let $f_n(z)\ne 0$ for all $n\ge 1$ and $z\in \mathrm{\Omega }$. Then, for $n\ge 1,$

$${\epsilon }_n^{(f)}={\epsilon }_0^{(a,1)}+(1+{\epsilon }_0^{(a,1)}){\displaystyle \sum _{p=1}^n}(1+{\epsilon }_{p,n}^{(Q,{(-1)}^{p+1})}){\epsilon }_p^{(a,{(-1)}^p)}{\displaystyle \prod _{m=1}^p}{G}_m^{(n,{(-1)}^m)}(z).$$

(11)

Proof. 

Let n be an arbitrary natural number. Since $f_n=a_0/Q_0^{(n)}$ and ${\widehat{f}}_n={\widehat{a}}_0/{\widehat{Q}}_0^{(n)}$, we have

$$\begin{array}{cc}\hfill {\epsilon }_n^{(f)}& ={\displaystyle \frac{{\widehat{f}}_n-f_n}{f_n}}\hfill \\ & ={\displaystyle \frac{{\widehat{a}}_0/{\widehat{Q}}_0^{(n)}}{a_0/Q_0^{(n)}}}-1\hfill \\ & ={\displaystyle \frac{{\widehat{a}}_0}{a_0}}{\displaystyle \frac{Q_0^{(n)}}{{\widehat{Q}}_0^{(n)}}}-1\hfill \\ & =(1+{\epsilon }_0^{(a,1)})(1+{\epsilon }_{0,n}^{(Q,-1)})-1\hfill \\ & ={\epsilon }_0^{(a,1)}+(1+{\epsilon }_0^{(a,1)}){\epsilon }_{0,n}^{(Q,-1)}.\hfill \end{array}$$

Hence,

$${\epsilon }_n^{(f)}={\epsilon }_0^{(a,1)}+(1+{\epsilon }_0^{(a,1)}){\epsilon }_{0,n}^{(Q,-1)}.$$

(12)

Setting $k=0$, $\sigma =-1$ in formula (8), we obtain

$${\epsilon }_{0,n}^{(Q,-1)}={\displaystyle \sum _{p=1}^n}(1+{\epsilon }_{p,n}^{(Q,{(-1)}^{p+1})}){\epsilon }_p^{(a,{(-1)}^p)}{\displaystyle \prod _{m=1}^p}{G}_m^{(n,{(-1)}^m)}(z).$$

Finally, substituting ${\epsilon }_{0,n}^{(Q,-1)}$ into formula (12), we obtain (11). □

3. Perturbation Stability Sets

The following theorem holds.

Theorem 1.

Let there exist constants A and ${\epsilon }^{(a)}$ such that

$$a_k\le A,{\widehat{a}}_k\le A,k\ge 1,$$

(13)

and the relative errors of the coefficients ${\widehat{a}}_k,$$k\ge 0,$ of the perturbed modified S-fraction (1) satisfy the inequalities

$$|{\epsilon }_k^{(a,1)}| \le {\epsilon }^{(a)}<1,k\ge 0.$$

(14)

Then, the set

$${\mathrm{\Omega }}_C=\{z\in \mathbb{C}:|\mathrm{Re}(z)|\ge C\},C>\sqrt{A}/2,$$

(15)

is a set of uniform stability to perturbations of the modified S-fraction (1), and, in addition, for $n\ge 1,$

$$|{\epsilon }_n^{(f)}| \le {\epsilon }^{(a)}+{\displaystyle \frac{{\epsilon }^{(a)}(1+{\epsilon }^{(a)})}{1-{\epsilon }^{(a)}}}{\displaystyle \frac{A}{C^2}}{\displaystyle \frac{1-{(g_C(A))}^n}{1-g_C(A)}},n\ge 1,$$

(16)

where

$$g_C(A)=\left\{\begin{array}{c}{\displaystyle \frac{\sqrt{A}}{2C}}\mathit{if}{\displaystyle \frac{\sqrt{A}}{2}}<C\le \sqrt{A},\hfill \\ {\displaystyle \frac{A}{C^2+A}}\mathit{if}C>\sqrt{A}.\hfill \end{array}\right.$$

Proof. 

Let us write the set (15) in the form

$${\mathrm{\Omega }}_C={\mathrm{\Omega }}_C^{-}\cup {\mathrm{\Omega }}_C^{+},$$

where

$${\mathrm{\Omega }}_C^{-}=\{z\in \mathbb{C}:\mathrm{Re}(z)\le -C\}$$

and

$${\mathrm{\Omega }}_C^{+}=\{z\in \mathbb{C}:\mathrm{Re}(z)\ge C\}.$$

Let n be an arbitrary natural number. By the induction on $k,$ $0\le k\le n,$ we prove that

$$Q_k^{(n)}(z)\in {\mathrm{\Omega }}_C^{+},0\le k\le n$$

if $z\in {\mathrm{\Omega }}_C^{+}.$

For $k=n$, we have $Q_n^{(n)}(z)=z$. Since $z\in {\mathrm{\Omega }}_C^{+}$, we get $Q_n^{(n)}(z)\in {\mathrm{\Omega }}_C^{+}$. Suppose that for some k, $0\le k\le n-1$, the tail $Q_{k+1}^{(n)}(z)\in {\mathrm{\Omega }}_C^{+}$. Hence, $\mathrm{Re}(Q_{k+1}^{(n)}(z))\ge C>0$. Since $a_{k+1}>0$ and $\mathrm{Re}(Q_{k+1}^{(n)}(z))>0$, we have

$$\mathrm{Re}\left({\displaystyle \frac{a_{k+1}}{Q_{k+1}^{(n)}(z)}}\right)={\displaystyle \frac{a_{k+1}\mathrm{Re}(Q_{k+1}^{(n)}(z))}{|Q_{k+1}^{(n)}{(z)|}^2}}>0.$$

Then

$$\begin{array}{cc}\hfill \mathrm{Re}(Q_k^{(n)}(z))& =\mathrm{Re}\left(z+{\displaystyle \frac{a_{k+1}}{Q_{k+1}^{(n)}(z)}}\right)\hfill \\ & =\mathrm{Re}(z)+\mathrm{Re}\left({\displaystyle \frac{a_{k+1}}{Q_{k+1}^{(n)}(z)}}\right)\hfill \\ & >\mathrm{Re}(z)\ge C.\hfill \end{array}$$

Hence, $Q_k^{(n)}(z)\in {\mathrm{\Omega }}_C^{+}$, $0\le k\le n$.

Since ${\widehat{a}}_k>0$, $k\ge 0$, we similarly prove that ${\widehat{Q}}_k^{(n)}(z)\in {\mathrm{\Omega }}_C^{+}$, $0\le k\le n$, if $z\in {\mathrm{\Omega }}_C^{+}$.

Similarly, consider the case where $z\in {\mathrm{\Omega }}_C^{-}$. By induction on k, we prove that $Q_k^{(n)}(z)\in {\mathrm{\Omega }}_C^{-}$ for all $0\le k\le n$.

For $k=n$, we have $Q_n^{(n)}(z)=z\in {\mathrm{\Omega }}_C^{-}$. Suppose that for some k, $0\le k\le n-1$, the tail $Q_{k+1}^{(n)}(z)\in {\mathrm{\Omega }}_C^{-}$. Then

$$\mathrm{Re}(Q_{k+1}^{(n)}(z))\le -C<0.$$

Since $a_{k+1}>0$ and $\mathrm{Re}(Q_{k+1}^{(n)}(z))<0$, we have

$$\mathrm{Re}\left({\displaystyle \frac{a_{k+1}}{Q_{k+1}^{(n)}(z)}}\right)={\displaystyle \frac{a_{k+1}\mathrm{Re}(Q_{k+1}^{(n)}(z))}{|Q_{k+1}^{(n)}{(z)|}^2}}<0.$$

Then, for the real part of the kth tail, we obtain:

$$\begin{array}{cc}\hfill \mathrm{Re}(Q_k^{(n)}(z))& =\mathrm{Re}(z)+\mathrm{Re}\left({\displaystyle \frac{a_{k+1}}{Q_{k+1}^{(n)}(z)}}\right)\hfill \\ & <\mathrm{Re}(z)\le -C.\hfill \end{array}$$

Hence, $Q_k^{(n)}(z)\in {\mathrm{\Omega }}_C^{-}$. Similarly, ${\widehat{Q}}_k^{(n)}(z)\in {\mathrm{\Omega }}_C^{-}.$

Thus, for any $z\in {\mathrm{\Omega }}_C$, the tails $Q_k^{(n)}(z)$, ${\widehat{Q}}_k^{(n)}(z)$ for all $0\le k\le n$ belong to the same half-plane as the point z.

Let us estimate from above the terms $|G_k^{(n,1)}(z)|$, $1\le k\le n$. From the recurrence relation for the tails of the approximant, we have

$$Q_{k-1}^{(n)}(z)=z+{\displaystyle \frac{a_k}{Q_k^{(n)}(z)}}.$$

Since the variable z and all the tails $Q_k^{(n)}(z)$, $0\le k\le n$, lie in the same half-plane ${\mathrm{\Omega }}_C^{-}$ or ${\mathrm{\Omega }}_C^{+}$, their real parts have the same sign. Then

$$|\mathrm{Re}(Q_{k-1}^{(n)}(z))| =|\mathrm{Re}(z)| + {\displaystyle \frac{a_k|\mathrm{Re}(Q_k^{(n)}(z))|}{|Q_k^{(n)}{(z)|}^2}}.$$

From the last equality, taking into account the estimates $|\mathrm{Re}(z)|\ge C$ and $|\mathrm{Re}(Q_k^{(n)}(z))|\ge C$, we have

$$|\mathrm{Re}(Q_{k-1}^{(n)}(z)) |\ge C + {\displaystyle \frac{a_{k}C}{|Q_k^{(n)}{(z)|}^2}}.$$

Since $|Q_{k-1}^{(n)}(z)|\ge |\mathrm{Re}(Q_{k-1}^{(n)}(z))|$, we get

$$|Q_{k-1}^{(n)}(z)|\ge C+{\displaystyle \frac{a_{k}C}{|Q_k^{(n)}{(z)|}^2}}.$$

Multiplying both sides of the inequality by $|Q_k^{(n)}(z)|>0$ and squaring, we obtain

$$|Q_{k-1}^{(n)}(z)Q_k^{(n)}{(z)|}^2\ge C^2{|Q_k^{(n)}(z)|}^2+2a_k{C}^2+{\displaystyle \frac{a_k^2{C}^2}{|Q_k^{(n)}{(z)|}^2}}.$$

Denote $x=|Q_k^{(n)}{(z)|}^2$. From

$$|Q_k^{(n)}(z)|\ge |\mathrm{Re}(Q_k^{(n)}(z))|\ge C$$

we have $x\ge C^2$. Consider the function of a real variable

$$h(x)=C^{2}x+{\displaystyle \frac{a_k^2{C}^2}{x}}+2a_k{C}^2,x\in [C^2,+\infty ).$$

If $a_k\ge C^2$, then

$$\underset{x\ge C^2}{\min }h(x)=h(a_k)=4a_k{C}^2.$$

If $a_k<C^2$, then

$$\underset{x\ge C^2}{\min }h(x)=h(C^2)={(C^2+a_k)}^2.$$

Hence,

$$|G_k^{(n,1)}(z)|\le g_C(a_k),1\le k\le n,$$

where

$$g_C(x)=\left\{\begin{array}{c}{\displaystyle \frac{\sqrt{x}}{2C}}\mathrm{if}x\ge C^2,\hfill \\ {\displaystyle \frac{x}{C^2+x}}\mathrm{if}0<x<C^2.\hfill \end{array}\right.$$

Since the function $g_C(x)$ is continuous and strictly increasing on $(0,+\infty )$, on every bounded interval $(0,A]$, we have

$$\underset{x\in (0,A]}{\max }{g}_C(x)=g_C(A).$$

Hence,

$$|G_k^{(n,1)}(z)|\le g_C(A),1\le k\le n.$$

(17)

Since the perturbed tails ${\widehat{Q}}_k^{(n)}(z)$, $0\le k\le n$, have the same properties as the exact tails $Q_k^{(n)}(z)$, $0\le k\le n$, and $0<{\widehat{a}}_k\le A$, $k\ge 0$, we have

$$|G_k^{(n,-1)}(z)|\le g_C(A),1\le k\le n.$$

(18)

Moreover, from the estimates $|Q_k^{(n)}(z)|\ge C$, $|{\widehat{Q}}_k^{(n)}(z)|\ge C$, $0\le k\le n$, and the inequalities (13), we obtain

$${\displaystyle \frac{a_k}{|Q_{k-1}^{(n)}(z){\widehat{Q}}_k^{(n)}(z)|}}\le {\displaystyle \frac{A}{C^2}},{\displaystyle \frac{{\widehat{a}}_k}{|{\widehat{Q}}_{k-1}^{(n)}(z)Q_k^{(n)}(z)|}}\le {\displaystyle \frac{A}{C^2}},1\le k\le n.$$

(19)

We write the formula for the relative error of the nth approximant (5) in the form

$${\epsilon }_n^{(f)}={\epsilon }_0^{(a,1)}+(1+{\epsilon }_0^{(a,1)}){\displaystyle \sum _{p=1}^n}{G}_p^{(n,{(-1)}^p)}(z)(1+{\epsilon }_{p,n}^{(Q,{(-1)}^{p+1})}){\epsilon }_p^{(a,{(-1)}^p)}{\displaystyle \prod _{m=1}^{p-1}}{G}_m^{(n,{(-1)}^m)}.$$

(20)

Taking into account that

$$G_p^{(n,{(-1)}^p)}(z)(1+{\epsilon }_{p,n}^{(Q,{(-1)}^{p+1})})=\left\{\begin{array}{c}{\displaystyle \frac{a_p}{Q_{p-1}^{(n)}(z){\widehat{Q}}_p^{(n)}(z)}}\mathrm{if}p\mathrm{is}\mathrm{even},\hfill \\ {\displaystyle \frac{{\widehat{a}}_p}{{\widehat{Q}}_{p-1}^{(n)}(z)Q_p^{(n)}(z)}}\mathrm{if}p\mathrm{is}\mathrm{odd},\hfill \end{array}\right.$$

and the estimates (17)–(19), as well as the fact that from condition (14), for the relative errors ${\epsilon }_k^{(a,-1)}$, the estimate

$$|{\epsilon }_k^{(a,-1)}|={\displaystyle \frac{|{\epsilon }_k^{(a,1)}|}{|1+{\epsilon }_k^{(a,1)}|}}\le {\displaystyle \frac{{\epsilon }^{(a)}}{1-{\epsilon }^{(a)}}},k\ge 0,$$

follows, from formula (20), we obtain

$$\begin{array}{cc}\hfill |{\epsilon }_n^{(f)}|& \le |{\epsilon }_0^{(a,1)}|+(1+|{\epsilon }_0^{(a,1)}|){\displaystyle \sum _{p=1}^n}|G_p^{(n,{(-1)}^p)}(z)(1+{\epsilon }_{p,n}^{(Q,{(-1)}^{p+1})})||{\epsilon }_p^{(a,{(-1)}^p)}|{\displaystyle \prod _{m=1}^{p-1}}|G_m^{(n,{(-1)}^m)}(z)|\hfill \\ & \le {\epsilon }^{(a)}+{\displaystyle \frac{{\epsilon }^{(a)}(1+{\epsilon }^{(a)})}{1-{\epsilon }^{(a)}}}{\displaystyle \frac{A}{C^2}}{\displaystyle \sum _{p=1}^n}{(g_C(A))}^{p-1}\hfill \\ & ={\epsilon }^{(a)}+{\displaystyle \frac{{\epsilon }^{(a)}(1+{\epsilon }^{(a)})}{1-{\epsilon }^{(a)}}}{\displaystyle \frac{A}{C^2}}{\displaystyle \frac{1-{(g_C(A))}^n}{1-g_C(A)}}.\hfill \end{array}$$

Since, by the hypothesis of the theorem, $C>\sqrt{A}/2$, we have $g_C(A)<1$. This allows us to relax the estimate for $|{\epsilon }_n^{(f)}|$, making it independent of n

$$|{\epsilon }_n^{(f)}| <{\epsilon }^{(a)}+{\displaystyle \frac{{\epsilon }^{(a)}(1+{\epsilon }^{(a)})}{1-{\epsilon }^{(a)}}}{\displaystyle \frac{A}{C^2(1-g_C(A))}}.$$

(21)

Denote the right-hand side of inequality (21) by the function $\phi ({\epsilon }^{(a)})$

$$\phi ({\epsilon }^{(a)})={\epsilon }^{(a)}+{\displaystyle \frac{{\epsilon }^{(a)}(1+{\epsilon }^{(a)})}{1-{\epsilon }^{(a)}}}{\displaystyle \frac{A}{C^2(1-g_C(A))}}.$$

The function $\phi ({\epsilon }^{(a)})$ is continuous and strictly increasing on $[0,1)$, and

$$\underset{{\epsilon }^{(a)}\to 0^{+}}{\lim }\phi ({\epsilon }^{(a)})=0.$$

By Definition 1, fix an arbitrary $\epsilon >0$. Using the properties of the function $\phi ({\epsilon }^{(a)})$, there exists ${\delta }_{\epsilon }>0$ such that for all perturbations of the coefficients satisfying the condition $|{\epsilon }_k^{(a,1)}| \le {\epsilon }^{(a)}<{\delta }_{\epsilon }$, the inequality $\phi ({\epsilon }^{(a)})<\epsilon$ holds. It follows that $|{\epsilon }_n^{(f)}| <\epsilon$ for all $n\ge 1$. Hence, the modified S-fraction (1) is stable to perturbations at every point $z\in {\mathrm{\Omega }}_C$. Since ${\delta }_{\epsilon }$ depends only on the constants A and C and does not depend on the choice of the point $z\in {\mathrm{\Omega }}_C$, by Definition 3, the modified S-fraction (1) is uniformly stable on the set ${\mathrm{\Omega }}_C$. □

Corollary 1.

Let there exist constants A and ${\epsilon }^{(a)}$ such that for the coefficients $a_k$, ${\widehat{a}}_k,$$k\ge 0,$ and the relative errors ${\epsilon }_k^{(a,1)},$ $k\ge 0,$ the inequalities (13) and (14) hold, respectively. Then, the set

$${\mathrm{\Omega }}_A=\{z\in \mathbb{C}:|\mathrm{Re}(z)|>\sqrt{A}/2\}$$

(22)

is a set of pointwise stability to perturbations of the modified S-fraction (1), and, in addition, for all $n\ge 1$ and $z_0\in {\mathrm{\Omega }}_A,$

$$|{\epsilon }_n^{(f)}| \le {\epsilon }^{(a)}+{\displaystyle \frac{{\epsilon }^{(a)}(1+{\epsilon }^{(a)})}{1-{\epsilon }^{(a)}}}{\displaystyle \frac{A}{|\mathrm{Re}(z_0){|}^2}}{\displaystyle \frac{1-{(g_{z_0}(A))}^n}{1-g_{z_0}(A)}},$$

(23)

where

$$g_{z_0}(A)=\left\{\begin{array}{c}{\displaystyle \frac{\sqrt{A}}{2|\mathrm{Re}(z_0)|}}if{\displaystyle \frac{\sqrt{A}}{2}}<|\mathrm{Re}(z_0)|\le \sqrt{A},\hfill \\ {\displaystyle \frac{A}{|\mathrm{Re}(z_0){|}^2+A}}if|\mathrm{Re}(z_0)|>\sqrt{A}.\hfill \end{array}\right.$$

Proof. 

Let $z_0$ be an arbitrary fixed point in ${\mathrm{\Omega }}_A$. Then

$$|\mathrm{Re}(z_0)| >{\displaystyle \frac{\sqrt{A}}{2}}.$$

(24)

We set $C=|\mathrm{Re}(z_0)|$. Then, $z_0\in {\mathrm{\Omega }}_C$, where the set ${\mathrm{\Omega }}_C$ is defined by (15). Since the inequality (24) holds, by Theorem 1, the set ${\mathrm{\Omega }}_C$ is a set of uniform stability to perturbations of the modified S-fraction (1). By Theorem 1, for all $z\in {\mathrm{\Omega }}_C$ and all $n\ge 1$, we have the estimate (16). Setting $z=z_0$ and taking into account that $C=|\mathrm{Re}(z_0)|$, we obtain the estimate (23).

By the estimate (23), for an arbitrary $\epsilon >0$, there exists ${\delta }_{\epsilon }(z_0)>0$ such that the condition $|{\epsilon }_k^{(a,1)}| \le {\epsilon }^{(a)}<{\delta }_{\epsilon }(z_0)$, $k\ge 0$, implies the inequality $|{\epsilon }_n^{(f)}| <\epsilon$ for all $n\ge 1$. Hence, the conditions of Definition 1 are satisfied, i.e., the modified S-fraction (1) is stable to perturbations at the point $z_0$. Finally, since the point $z_0\in {\mathrm{\Omega }}_A$ is arbitrary, the set ${\mathrm{\Omega }}_A$ is a set of pointwise stability to perturbations of the modified S-fraction (1). □

Next, we prove the following result.

Theorem 2.

Let there exist constants A and ${\epsilon }^{(a)}$ such that for the coefficients $a_k$, ${\widehat{a}}_k,$ $k\ge 0,$ and the relative errors ${\epsilon }_k^{(a,1)},$ $k\ge 0,$ the inequalities (13) and (14) hold, respectively. Then, the set

$${\mathrm{\Omega }}_{\alpha ,C}=\{z\in \mathbb{C}:|\mathrm{Im}(z)|\le |\mathrm{Re}(z)|\tan \alpha ,|\mathrm{Re}(z)|\ge C\},$$

(25)

is a set of uniform stability to perturbations of the modified S-fraction (1), and, in addition, for $n\ge 1,$

$$|{\epsilon }_n^{(f)}| \le {\epsilon }^{(a)}+{\displaystyle \frac{{\epsilon }^{(a)}(1+{\epsilon }^{(a)})}{1-{\epsilon }^{(a)}}}{\displaystyle \frac{A}{C^2}}{\displaystyle \frac{1-{(g_{\alpha ,C}(A))}^n}{1-g_{\alpha ,C}(A)}}.$$

(26)

where $0\le \alpha <\pi /2,$ $C>\sqrt{A(1-\cos \alpha )},$ and

$$g_{\alpha ,C}(A)={\displaystyle \frac{A}{C^2+A\cos \alpha }}.$$

Proof. 

We split the set ${\mathrm{\Omega }}_{\alpha ,C}$ according to the sign of the real part

$${\mathrm{\Omega }}_{\alpha ,C}={\mathrm{\Omega }}_{\alpha ,C}^{+}\cup {\mathrm{\Omega }}_{\alpha ,C}^{-},$$

where

$${\mathrm{\Omega }}_{\alpha ,C}^{+}=\{z\in \mathbb{C}:\mathrm{Re}(z)\ge C,|\mathrm{Im}(z)|\le \mathrm{Re}(z)\tan \alpha \}$$

and

$${\mathrm{\Omega }}_{\alpha ,C}^{-}=\{z\in \mathbb{C}:\mathrm{Re}(z)\le -C,|\mathrm{Im}(z)|\le -\mathrm{Re}(z)\tan \alpha \}.$$

Let us prove that if $z\in {\mathrm{\Omega }}_{\alpha ,C}^{+}$, then $Q_k^{(n)}(z)\in {\mathrm{\Omega }}_{\alpha ,C}^{+}$ for all $0\le k\le n$.

For $k=n$, we have $Q_n^{(n)}(z)=z\in {\mathrm{\Omega }}_{\alpha ,C}^{+}$. Suppose that $Q_{k+1}^{(n)}(z)\in {\mathrm{\Omega }}_{\alpha ,C}^{+}$ for some $0\le k\le n-1$. Let us prove $Q_k^{(n)}(z)\in {\mathrm{\Omega }}_{\alpha ,C}^{+}$. From the recurrence relation $Q_k^{(n)}(z)=z+a_{k+1}/Q_{k+1}^{(n)}(z)$, we have

$$\mathrm{Re}(Q_k^{(n)}(z))=\mathrm{Re}(z)+{\displaystyle \frac{a_{k+1}\mathrm{Re}(Q_{k+1}^{(n)}(z))}{|Q_{k+1}^{(n)}{(z)|}^2}}$$

and

$$\mathrm{Im}(Q_k^{(n)}(z))=\mathrm{Im}(z)-{\displaystyle \frac{a_{k+1}\mathrm{Im}(Q_{k+1}^{(n)}(z))}{|Q_{k+1}^{(n)}{(z)|}^2}}.$$

Since $a_{k+1}>0$ and $\mathrm{Re}(Q_{k+1}^{(n)}(z))\ge C>0$, we have

$$\begin{array}{cc}\hfill \mathrm{Re}(Q_k^{(n)}(z))& =\mathrm{Re}(z)+a_{k+1}{\displaystyle \frac{\mathrm{Re}(Q_{k+1}^{(n)}(z))}{|Q_{k+1}^{(n)}{(z)|}^2}}\hfill \\ & >\mathrm{Re}(z)\ge C.\hfill \end{array}$$

Applying the triangle inequality to the formula for $\mathrm{Im}(Q_k^{(n)}(z))$ and using the inequalities $|\mathrm{Im}(z)| \le \mathrm{Re}(z)\tan \alpha$ and $|\mathrm{Im}(Q_{k+1}^{(n)}(z))|\le \mathrm{Re}(Q_{k+1}^{(n)}(z))\tan \alpha$, we have

$$\begin{array}{cc}\hfill |\mathrm{Im}(Q_k^{(n)}(z))|& \le |\mathrm{Im}(z)|+a_{k+1}{\displaystyle \frac{|\mathrm{Im}(Q_{k+1}^{(n)}(z))|}{|Q_{k+1}^{(n)}{(z)|}^2}}\hfill \\ & \le \mathrm{Re}(z)\tan \alpha +a_{k+1}{\displaystyle \frac{\mathrm{Re}(Q_{k+1}^{(n)}(z))\tan \alpha }{|Q_{k+1}^{(n)}{(z)|}^2}}\hfill \\ & =\tan \alpha \left(\mathrm{Re}(z)+a_{k+1}{\displaystyle \frac{\mathrm{Re}(Q_{k+1}^{(n)}(z))}{|Q_{k+1}^{(n)}{(z)|}^2}}\right)\hfill \\ & =\mathrm{Re}(Q_k^{(n)}(z))\tan \alpha .\hfill \end{array}$$

It follows that $Q_k^{(n)}(z)\in {\mathrm{\Omega }}_{\alpha ,C}^{+}$.

In the case $z\in {\mathrm{\Omega }}_{\alpha ,C}^{-}$, the reasoning is analogous. Since ${\widehat{a}}_{k+1}>0$, we have ${\widehat{Q}}_k^{(n)}(z)\in {\mathrm{\Omega }}_{\alpha ,C}^{\pm }$, $0\le k\le n$, for all $z\in {\mathrm{\Omega }}_{\alpha ,C}^{\pm }$.

Hence, for all $0\le k\le n$ and $z\in {\mathrm{\Omega }}_{\alpha ,C},$

$$|Q_k^{(n)}(z)|\ge |\mathrm{Re}(Q_k^{(n)}(z))|\ge C,|{\widehat{Q}}_k^{(n)}(z)|\ge |\mathrm{Re}({\widehat{Q}}_k^{(n)}(z))|\ge C.$$

(27)

Moreover, if $z\in {\mathrm{\Omega }}_{\alpha ,C}$, then from the condition $|\mathrm{Im}(z)|\le |\mathrm{Re}(z)|\tan \alpha$, it follows that

$$\begin{array}{cc}\hfill {|z|}^2& ={\mathrm{Re}}^2(z)+{\mathrm{Im}}^2(z)\hfill \\ & \le {\mathrm{Re}}^2(z)+{\mathrm{Re}}^2(z){\tan }^2\alpha \hfill \\ & ={\mathrm{Re}}^2(z)(1+{\tan }^2\alpha )={\displaystyle \frac{{\mathrm{Re}}^2(z)}{{\cos }^2\alpha }}.\hfill \end{array}$$

Hence,

$$|\mathrm{Re}(z)|\ge |z|\cos \alpha .$$

Now, let us estimate the terms $|G_k^{(n,1)}(z)|$, $1\le k\le n$. Since the real parts $\mathrm{Re}(z)$, $\mathrm{Re}(Q_{k-1}^{(n)}(z))$, $\mathrm{Re}(Q_k^{(n)}(z))$ have the same sign, we have

$$|\mathrm{Re}(Q_{k-1}^{(n)}(z))| =|\mathrm{Re}(z)| + a_k{\displaystyle \frac{|\mathrm{Re}(Q_k^{(n)}(z))|}{|Q_k^{(n)}{(z)|}^2}}.$$

Since $Q_k^{(n)}(z)\in {\mathrm{\Omega }}_{\alpha ,C}$, we have $|\mathrm{Re}(Q_k^{(n)})| \ge |Q_k^{(n)}|\cos \alpha$, and

$${\displaystyle \frac{|\mathrm{Re}(Q_k^{(n)}(z))|}{|Q_k^{(n)}{(z)|}^2}}\ge {\displaystyle \frac{|Q_k^{(n)}(z)|\cos \alpha }{|Q_k^{(n)}{(z)|}^2}}={\displaystyle \frac{\cos \alpha }{|Q_k^{(n)}(z)|}}.$$

Taking into account that $|\mathrm{Re}(z)|\ge C$, we get

$$|\mathrm{Re}(Q_{k-1}^{(n)}(z))|\ge C+{\displaystyle \frac{a_k\cos \alpha }{|Q_k^{(n)}(z)|}}.$$

Since $|Q_{k-1}^{(n)}(z)|\ge |\mathrm{Re}(Q_{k-1}^{(n)}(z))|$, multiplying the last inequality by $|Q_k^{(n)}(z)|>0$, we obtain

$$\begin{array}{cc}\hfill |Q_{k-1}^{(n)}(z)||Q_k^{(n)}(z)|& \ge \left(C+{\displaystyle \frac{a_k\cos \alpha }{|Q_k^{(n)}(z)|}}\right)|Q_k^{(n)}(z)|\hfill \\ & =C|Q_k^{(n)}(z)|+a_k\cos \alpha \hfill \\ & \ge C^2+a_k\cos \alpha .\hfill \end{array}$$

Next, since $a_k>0$ and $C^2+a_k\cos \alpha >0$, we have

$$\begin{array}{cc}\hfill |G_k^{(n,1)}(z)|& ={\displaystyle \frac{a_k}{|Q_{k-1}^{(n)}(z)Q_k^{(n)}(z)|}}\hfill \\ & \le {\displaystyle \frac{a_k}{C^2+a_k\cos \alpha }}=g_{\alpha ,C}(a_k),\hfill \end{array}$$

where

$$g_{\alpha ,C}(x)={\displaystyle \frac{x}{C^2+x\cos \alpha }}.$$

We have the same estimate for the perturbed terms $|G_k^{(n,-1)}|(z),$ namely,

$$|G_k^{(n,-1)}(z)|={\displaystyle \frac{{\widehat{a}}_k}{|{\widehat{Q}}_{k-1}^{(n)}(z){\widehat{Q}}_k^{(n)}(z)|}}\le g_{\alpha ,C}({\widehat{a}}_k).$$

Since the function $g_{\alpha ,C}(x))$ is strictly increasing on $(0,+\infty )$ and the coefficients $a_k$, ${\widehat{a}}_k$, $k\ge 0$, satisfy condition (13), we have

$$|G_k^{(n,\sigma )}(z)|\le g_{\alpha ,C}(A),1\le k\le n,\sigma \in \{-1,1\},$$

(28)

where the condition $C>\sqrt{A(1-\cos \alpha )}$ ensures that the inequality $g_{\alpha ,C}(A)<1$ holds. Moreover, taking into account condition (13) and the inequalities (27), we obtain the estimates (19).

Taking into account the estimates (28), (19), from formula (11), we get

$$\begin{array}{cc}\hfill |{\epsilon }_n^{(f)}| & \le |{\epsilon }_0^{(a,1)}|+(1+|{\epsilon }_0^{(a,1)}|){\displaystyle \sum _{p=1}^n}|G_p^{(n,{(-1)}^p)}(z)(1+{\epsilon }_{p,n}^{(Q,{(-1)}^{p+1})})||{\epsilon }_p^{(a,{(-1)}^p)}|{\displaystyle \prod _{m=1}^{p-1}}|G_m^{(n,{(-1)}^m)}(z)|\hfill \\ & \le {\epsilon }^{(a)}+{\displaystyle \frac{{\epsilon }^{(a)}(1+{\epsilon }^{(a)})}{1-{\epsilon }^{(a)}}}{\displaystyle \frac{A}{C^2}}{\displaystyle \sum _{p=1}^n}{(g_{\alpha ,C}(A))}^{p-1}\hfill \\ & ={\epsilon }^{(a)}+{\displaystyle \frac{{\epsilon }^{(a)}(1+{\epsilon }^{(a)})}{1-{\epsilon }^{(a)}}}{\displaystyle \frac{A}{C^2}}{\displaystyle \frac{1-{(g_{\alpha ,C}(A))}^n}{1-g_{\alpha ,C}(A)}}.\hfill \end{array}$$

Since $g_{\alpha ,C}(A)<1$, we have

$$|{\epsilon }_n^{(f)}|<\chi ({\epsilon }^{(a)})={\epsilon }^{(a)}+{\displaystyle \frac{{\epsilon }^{(a)}(1+{\epsilon }^{(a)})}{1-{\epsilon }^{(a)}}}{\displaystyle \frac{A}{C^2(1-g_{\alpha ,C}(A))}}.$$

(29)

The function $\chi :[0,1)\to [0,+\infty )$ is continuous, strictly increasing, and

$$\underset{{\epsilon }^{(a)}\to 0^{+}}{\lim }\chi ({\epsilon }^{(a)})=0.$$

For a fixed $\epsilon >0$, there exists ${\delta }_{\epsilon }>0$, depending only on the parameters $\alpha$, A, C, such that for $|{\epsilon }_k^{(a,1)}| \le {\epsilon }^{(a)}<{\delta }_{\epsilon }$, we have $\chi ({\epsilon }^{(a)})<\epsilon$. From (29), we obtain

$$|{\epsilon }_n^{(f)}|<\epsilon ,n\ge 1.$$

Hence, the modified S-fraction (1) is stable at every point of ${\mathrm{\Omega }}_{\alpha ,C}$. □

Remark 1.

The set ${\mathrm{\Omega }}_{\alpha ,C}^{+}$ is the intersection of the half-plane

$${\mathcal{H}}_C=\{z\in \mathbb{C}:\mathrm{Re}(z)\ge C\}$$

with the angular set

$${\mathcal{H}}_{\alpha }=\{z\in \mathbb{C}:|\arg (z)|\le \alpha \}.$$

The set ${\mathrm{\Omega }}_{\alpha ,C}^{-}$ is the intersection of the half-plane

$${\mathcal{V}}_C=\{z\in \mathbb{C}:\mathrm{Re}(z)\le -C\}$$

with the angular set

$${\mathcal{V}}_{\alpha }=\{z\in \mathbb{C}:|\arg (z)-\pi |\le \alpha \}$$

(see Figure 3). The smaller the angle α, the weaker the condition on the constant C that bounds the set ${\mathrm{\Omega }}_{\alpha ,C}={\mathrm{\Omega }}_{\alpha ,C}^{+}\cup {\mathrm{\Omega }}_{\alpha ,C}^{-}$. For $\alpha =0$, the set ${\mathrm{\Omega }}_{\alpha ,C}$ degenerates into two rays

$${\mathrm{\Omega }}_{0,C}=\{x\in \mathbb{R}:|x|\ge C\},C>0,$$

on the real axis, i.e., the modified S-fraction is uniformly stable to perturbations on the set ${\mathrm{\Omega }}_{0,C}$ for arbitrary bounded coefficients.

Figure 3. Sets of uniform stability to perturbations of modified S-fraction (1).

Figure 3. Sets of uniform stability to perturbations of modified S-fraction (1).

Remark 2.

Comparing Theorem 1 and Theorem 2, we note that ${\mathrm{\Omega }}_{\alpha ,C}\subset {\mathrm{\Omega }}_C$ for any $\alpha \in [0,\pi /2)$ and $C>0$, and as $\alpha \to$$\pi /2$, we have ${\mathrm{\Omega }}_{\alpha ,C}\to {\mathrm{\Omega }}_C$. Since

$$\sqrt{A(1-\cos \alpha )}\le \sqrt{A}/2,\alpha \le \arccos (3/4),$$

the lower bound for C in Theorem 2 is smaller than in Theorem 1, and as $\alpha \to 0^{+}$, it tends to zero ($\sqrt{A(1-\cos \alpha )}\to 0$). For $\alpha \in (\arccos (3/4),\pi /2)$, the lower bound in Theorem 2 is greater than in Theorem 1.

Corollary 2.

Let there exist constants A and ${\epsilon }^{(a)}$ such that for the coefficients $a_k$, ${\widehat{a}}_k,$ $k\ge 0,$ and the relative errors ${\epsilon }_k^{(a,1)},$ $k\ge 0,$ the inequalities (13) and (14) hold, respectively. Then, the set

$${\mathrm{\Omega }}_A^{*}=\{z\in \mathbb{C}:A(1-|\mathrm{Re}(z)|/|z|)<{\mathrm{Re}}^2(z)\}$$

(30)

is a set of pointwise stability to perturbations of the modified S-fraction (1), and, in addition, for each $z_0\in {\mathrm{\Omega }}_A^{*},$

$$|{\epsilon }_n^{(f)}| \le {\epsilon }^{(a)}+{\displaystyle \frac{{\epsilon }^{(a)}(1+{\epsilon }^{(a)})}{1-{\epsilon }^{(a)}}}{\displaystyle \frac{A}{{\mathrm{Re}}^2(z_0)}}{\displaystyle \frac{1-{(g_{z_0}^{*}(A))}^n}{1-g_{z_0}^{*}(A)}},n\ge 1,$$

(31)

where

$$g_{z_0}^{*}(A)={\displaystyle \frac{A|z_0|}{|\mathrm{Re}(z_0)|(|z_0||\mathrm{Re}(z_0)|+A)}}.$$

Proof. 

Let $z_0$ be an arbitrary fixed point in ${\mathrm{\Omega }}_A^{*}$. We set

$$C=|\mathrm{Re}(z_0)|,\alpha =\arctan {\displaystyle \frac{|\mathrm{Im}(z_0)|}{|\mathrm{Re}(z_0)|}}.$$

(32)

Since $|\mathrm{Im}(z_0)|=|\mathrm{Re}(z_0)|\tan \alpha$, we have $z_0\in \partial {\mathrm{\Omega }}_{\alpha ,C}$, where the set ${\mathrm{\Omega }}_{\alpha ,C}$ is defined by (25). By the definition of the set (30)

$$A(1-|\mathrm{Re}(z_0)|/|z_0|)<{(\mathrm{Re}(z_0))}^2.$$

(33)

Since condition (33) coincides with the condition $C>\sqrt{A(1-\cos \alpha )}$ of Theorem 2, the set ${\mathrm{\Omega }}_{\alpha ,C}$ is a set of uniform stability to perturbations of the modified S-fraction (1), and for all $n\ge 1$ and $z\in {\mathrm{\Omega }}_{\alpha ,C}$, the estimate (26) holds for the relative error of the approximant. Taking into account (32), the estimate (26) takes the form (31).

From the estimate (31), it follows that for an arbitrary $\epsilon >0$, there exists ${\delta }_{\epsilon }(z_0)>0$ such that the condition

$$|{\epsilon }_k^{(a,1)}| \le {\epsilon }^{(a)}<{\delta }_{\epsilon }(z_0),k\ge 0,$$

implies the inequality $|{\epsilon }_n^{(f)}(z_0)| <\epsilon$ for all $n\ge 1$. Hence, the modified S-fraction (1) is stable to perturbations at the point $z_0$, and the set ${\mathrm{\Omega }}_A^{*}$ is a set of pointwise stability to perturbations. □

Remark 3.

The boundary

$$\partial {\mathrm{\Omega }}_A^{★}=\{z\in \mathbb{C}:A(1-|\mathrm{Re}(z)|/|z|)={\mathrm{Re}}^2(z)\}$$

of the domain ${\mathrm{\Omega }}_A^{★}$ is a curve symmetric with respect to the coordinate axes, with two vertical asymptotes $\mathrm{Re}(z)=\pm \sqrt{A}$, whereas the boundary of the domain ${\mathrm{\Omega }}_A$ consists of two straight lines $\mathrm{Re}(z)=\pm \sqrt{A}/2.$ On the real axis, ${\mathrm{\Omega }}_A\subset {\mathrm{\Omega }}_A^{★}$. At the same time, as $|\mathrm{Im}(z)|\to +\infty$, the boundary $\partial {\mathrm{\Omega }}_A^{★}$ asymptotically approaches the lines $\mathrm{Re}(z)=\pm \sqrt{A}$, i.e., the domain ${\mathrm{\Omega }}_A^{★}$ narrows to the exterior of the domain

$${\mathcal{O}}_A=\{z\in \mathbb{C}:|\mathrm{Re}(z)|<\sqrt{A}\}.$$

Hence, ${\mathrm{\Omega }}_A^{★}$ is neither a subset nor a superset of ${\mathrm{\Omega }}_A$ (see Figure 4).

Figure 4. Set of stability to perturbations of modified S-fraction (1).

The relation between the set of pointwise stability ${\mathrm{\Omega }}_A^{*}$ and the sets of uniform stability ${\mathrm{\Omega }}_{\alpha ,C}$ is described by

$${\mathrm{\Omega }}_A^{*}=\bigcup _{\begin{array}{c}\alpha \in [0,\pi /2),C>\sqrt{A(1-\cos \alpha )}\end{array}}{\mathrm{\Omega }}_{\alpha ,C},$$

i.e., ${\mathrm{\Omega }}_A^{*}$ is the union of all sets of uniform stability of the form (25) satisfying the condition of Theorem 2. Moreover, every point $z_0\in {\mathrm{\Omega }}_A^{*}$ belongs to the boundary $\partial {\mathrm{\Omega }}_{\alpha ,C}$ of the angular set of uniform stability constructed using the parameters (32).

For $\mathrm{Im}(z)=0$, condition (33) holds for any $A>0$. Hence, we obtain the following corollary.

Corollary 3.

The modified S-fraction (1) is stable to perturbations on the set $\mathbb{R}\setminus \left\{0\right\}$ for arbitrary bounded coefficients $a_k$, ${\widehat{a}}_k,$ $k\ge 0.$

Theorem 3.

Let there exist constants A and ${\epsilon }^{(a)}$ such that for the coefficients $a_k$, ${\widehat{a}}_k,$$k\ge 0,$ and the relative errors ${\epsilon }_k^{(a,1)},$$k\ge 0,$ the inequalities (13) and (14) hold, respectively. Then, the set

$${\mathrm{\Omega }}_R=\{z\in \mathbb{C}:|z|\ge R\},R>2\sqrt{A},$$

(34)

is a set of uniform stability to perturbations of the modified S-fraction (1), and, in addition, for $n\ge 1,$

$$|{\epsilon }_n^{(f)}|\le {\epsilon }^{(a)}+{\displaystyle \frac{{\epsilon }^{(a)}(1+{\epsilon }^{(a)})}{1-{\epsilon }^{(a)}}}{\displaystyle \frac{A}{r^2}}{\displaystyle \frac{1-{(g_R(A))}^n}{1-g_R(A)}},$$

(35)

where $g_R(A)=A/r^2$ and

$$r={\displaystyle \frac{R+\sqrt{R^2-4A}}{2}}.$$

(36)

Proof. 

Let

$$V_r=\{z\in \mathbb{C}:|z|\ge r\},$$

where r is defined according to (36). Note that r is the larger root of the equation $t^2-Rt+A=0$, so $r^2=Rr-A$ whence $R=r+A/r$.

Let n be an arbitrary natural number. By the induction on k, we prove that $Q_k^{(n)}(z)\in V_r$ for all $0\le k\le n$ and $z\in {\mathrm{\Omega }}_R$.

For $k=n$, we have

$$|Q_n^{(n)}(z)|=|z|\ge R=r+A/r>r,$$

so $Q_n^{(n)}(z)\in V_r$. Suppose $|Q_k^{(n)}(z)|\ge r$ for some $0\le k=s+1\le n-1$. Then, for $k=s,$

$$\begin{array}{cc}\hfill |Q_s^{(n)}(z)|& =\left|z+{\displaystyle \frac{a_{s+1}}{Q_{s+1}^{(n)}(z)}}\right|\hfill \\ & \ge |z|-{\displaystyle \frac{a_{s+1}}{|Q_{s+1}^{(n)}(z)|}}\hfill \\ & \ge R-{\displaystyle \frac{A}{r}}=r,\hfill \end{array}$$

i.e., $Q_s^{(n)}(z)\in V_r$. Since ${\widehat{a}}_{k+1}>0$, we similarly prove that $|{\widehat{Q}}_k^{(n)}(z)|\ge r$ for all $0\le k\le n$.

Now, taking into account the estimates $|Q_k^{(n)}(z)|\ge r$, $|{\widehat{Q}}_k^{(n)}(z)|\ge r$, $0\le k\le n$, and the inequalities (13), we will estimate the terms $G_k^{(n,\sigma )}(z)$, $1\le k\le n$, $\sigma \in \{-1,1\}$. For an arbitrary $1\le k\le n$, we have

$$\begin{array}{cc}\hfill |G_k^{(n,1)}(z)|& ={\displaystyle \frac{a_k}{|Q_{k-1}^{(n)}(z)||Q_k^{(n)}(z)|}}\hfill \\ & ={\displaystyle \frac{a_k/|Q_k^{(n)}|}{|z+a_k/Q_k^{(n)}|}}\hfill \\ \hfill & \le {\displaystyle \frac{a_k/|Q_k^{(n)}|}{|z|-a_k/|Q_k^{(n)}|}}\hfill \\ & \le {\displaystyle \frac{A/r}{R-A/r}}={\displaystyle \frac{A}{r^2}}\hfill \end{array}$$

if $\sigma =1,$ and

$$\begin{array}{cc}\hfill |G_k^{(n,-1)}(z)|& ={\displaystyle \frac{{\widehat{a}}_k}{|{\widehat{Q}}_{k-1}^{(n)}(z)||{\widehat{Q}}_k^{(n)}(z)|}}\hfill \\ & ={\displaystyle \frac{{\widehat{a}}_k/|{\widehat{Q}}_k^{(n)}(z)|}{|z+{\widehat{a}}_k/{\widehat{Q}}_k^{(n)}(z)|}}\hfill \\ \hfill & \le {\displaystyle \frac{{\widehat{a}}_k/|{\widehat{Q}}_k^{(n)}(z)|}{|z|-{\widehat{a}}_k/|{\widehat{Q}}_k^{(n)}(z)|}}\hfill \\ & \le {\displaystyle \frac{A/r}{R-A/r}}={\displaystyle \frac{A}{r^2}}\hfill \end{array}$$

if $\sigma =-1.$ Hence,

$$|G_k^{(n,\sigma )}(z)|\le {\displaystyle \frac{A}{r^2}}=g_R(A),1\le k\le n,\sigma \in \{-1,1\}.$$

(37)

Moreover, from $|Q_k^{(n)}(z)|\ge r$ and $|{\widehat{Q}}_k^{(n)}(z)|\ge r$ and the inequalities (13), we have

$${\displaystyle \frac{a_k}{|Q_{k-1}^{(n)}(z){\widehat{Q}}_k^{(n)}(z)|}}\le {\displaystyle \frac{A}{r^2}},{\displaystyle \frac{{\widehat{a}}_k}{|{\widehat{Q}}_{k-1}^{(n)}(z)Q_k^{(n)}(z)|}}\le {\displaystyle \frac{A}{r^2}},1\le k\le n.$$

(38)

Taking into account the estimates (37), (38), and the estimate $|{\epsilon }_k^{(a,-1)}| \le {\epsilon }^{(a)}/(1-{\epsilon }^{(a)})$, from formula (20), we obtain (35).

Since $R>2\sqrt{A}$, we have $r=(R+\sqrt{R^2-4A})/2>\sqrt{A}$. Hence, $g_R(A)=A/r^2<1$. Then

$$|{\epsilon }_n^{(f)}| <{\epsilon }^{(a)}+{\displaystyle \frac{{\epsilon }^{(a)}(1+{\epsilon }^{(a)})}{1-{\epsilon }^{(a)}}}{\displaystyle \frac{A}{r^2(1-g_R(A))}}=\psi ({\epsilon }^{(a)}).$$

(39)

The function $\psi :[0,1)\to [0,+\infty )$ is continuous, strictly increasing, and

$$\underset{{\epsilon }^{(a)}\to 0^{+}}{\lim }\psi ({\epsilon }^{(a)})=0.$$

Since $\psi$ does not depend on the point $z\in {\mathrm{\Omega }}_R$, for an arbitrary $\epsilon >0$, there exists ${\delta }_{\epsilon }>0$ such that for $|{\epsilon }_k^{(a,1)}| \le {\epsilon }^{(a)}<{\delta }_{\epsilon }$, we have $|{\epsilon }_n^{(f)}|<\epsilon$ for all $n\ge 1$ and $z\in {\mathrm{\Omega }}_R$. Hence, by Definition 3, the set ${\mathrm{\Omega }}_R$ is a set of uniform stability to perturbations of the S-fraction (1). □

Remark 4.

It is important to emphasize that the geometric domains ${\mathrm{\Omega }}_C$, ${\mathrm{\Omega }}_{\alpha ,C}$, and ${\mathrm{\Omega }}_R$ established in Theorems 1–3 constitute sufficient conditions for uniform perturbation stability, rather than exact necessary-and-sufficient boundaries. Because these regions are derived using the uniform bounding parameter ${\delta }_{\epsilon }$ to guarantee algorithmic robustness for any arbitrary sequence of bounded coefficients, they represent a conservative inner envelope of stability. Consequently, while individual modified S-fractions with highly regular or rapidly decaying coefficient sequences may exhibit empirical stability beyond these thresholds, the boundaries defined herein provide an absolute, structurally guaranteed safety zone against unbounded rounding-error amplification under worst-case parameter configurations.

Remark 5.

Theorem 3 establishes uniform stability on the exterior of the disk ${\mathrm{\Omega }}_R$, which contains points of the imaginary axis and its neighborhoods, under the condition $|z|\ge R>2\sqrt{A}$. This distinguishes it from Theorems 1 and 2, where the sets of uniform stability are isolated from the imaginary axis by the conditions $\mathrm{Re}z\ge C>\sqrt{A}/2$ and $\mathrm{Re}(z)\ge C>\sqrt{A(1-\cos \alpha )}$, respectively. However, the condition $R>2\sqrt{A}$ of Theorem 3 is stronger than the condition $C>\sqrt{A}/2$ of Theorem 1 for the same value of the parameter ($R=C$)—the minimal admissible radius $R_{\min }=2\sqrt{A}$ in Theorem 3 is four times larger than the minimal value $C_{\min }=\sqrt{A}/2$ in Theorem 1. Thus, Theorem 3 requires greater distance from the origin but ensures stability to perturbations on the imaginary axis.

Corollary 4.

Under the conditions of Theorem 3, the set

$${\mathrm{\Omega }}_A^{**}=\{z\in \mathbb{C}:|z|>2\sqrt{A}\}$$

(40)

is a set of pointwise stability to perturbations of the modified S-fraction (1), and for each $z_0\in {\mathrm{\Omega }}_A^{**}$, the estimate (35) holds with $r=|z_0|$.

Proof. 

Let $z_0$ an arbitrary fixed point in ${\mathrm{\Omega }}_A^{**}$. Then, $|z_0| >2\sqrt{A}$. We set $R= |z_0|$. Then, $z_0\in \partial {\mathrm{\Omega }}_R$, and since $R>2\sqrt{A}$, by Theorem 3, the set ${\mathrm{\Omega }}_R$ is a set of uniform stability to perturbations. From the estimate (35) with $R=|z_0|$, the conditions of Definition 1 are satisfied. By the arbitrariness of $z_0$, the set ${\mathrm{\Omega }}_A^{**}$ is a set of pointwise stability to perturbations of the modified S-fraction (1). □

4. Numerical Experiments

To bridge the theoretical perturbation bounds established in Section 3 with practical computing environments, we consider the constraints imposed by machine precision. When evaluating a modified S-fraction on a digital processor, the continuous coefficients $a_k$ are mapped to a finite-precision representation ${\tilde{a}}_k$. Under the standard IEEE 754 double-precision floating-point format, this representation introduces a deterministic roundoff error governed by the unit roundoff (machine epsilon) $u=2^{-53}\approx 1.11\times {10}^{-16}$, such that the relative coefficient error satisfies

$$|{\epsilon }_k^{(a)}|=\left|{\displaystyle \frac{{\tilde{a}}_k-a_k}{a_k}}\right|\le u.$$

Our framework proves that uniform algorithmic stability is preserved provided that the coefficient perturbations do not exceed the structural threshold ${\delta }_{\epsilon }$. Consequently, within the specified stability sets (${\mathrm{\Omega }}_C$, ${\mathrm{\Omega }}_{\alpha ,C}$, and ${\mathrm{\Omega }}_R$) where the geometric criteria dictate that ${\delta }_{\epsilon }\ge u$, standard double-precision arithmetic is mathematically sufficient to prevent exponential error amplification. In these sets, the intrinsic rounding noise of the machine is safely bounded within the stability sets, guaranteeing that the backward recurrence algorithm remains robust against catastrophic precision loss.

4.1. Empirical Stability Analysis of the Modified S-Fraction for $\arctan (1/z)$

To verify the empirical stability of the modified S-fraction satisfying the conditions of the theoretical results, let us consider the expansion of the function $\arctan (1/z)$ into a continued fraction (see, for example, [2])

$$\arctan {\displaystyle \frac{1}{z}}={\displaystyle \frac{1}{z+\frac{1/3}{z + \frac{4/15}{z + \frac{9/35}{z + \genfrac{}{}{0pt}{}{}{\ddots }}}}}},z\ne 0.$$

(41)

Unlike the expansion of the square of the ratio of Euler gamma functions considered in Section 1.2, where the partial numerators tend to infinity, in this expansion, the coefficients are given by the formula

$$a_n={\displaystyle \frac{n^2}{(2n-1)(2n+1)}},n\ge 1,$$

and are bounded, converging to the limit $1/4$ as $n\to +\infty$.

To study the behavior of the errors of the approximants ${\widehat{f}}_n(z)$, $1\le n\le 2000$, standard double-precision arithmetic compliant with the IEEE 754 standard (where the unit roundoff is $u\approx 1.11\times {10}^{-16}$) was used. The obtained values were compared against high-precision reference values $f_n(z),$$1\le n\le 2000$, computed using 128-bit extended precision arithmetic. The relative error $|({\widehat{f}}_n(z)-f_n(z))/f_n(z)|$ was investigated on a grid of points in the complex plane bounded by $\mathrm{Re}(z)\in [-7.5,7.5]$, $\mathrm{Im}(z)\in [-7.5,7.5]$.

Pointwise analysis of the relative error behavior depending on the approximant order n demonstrates stability of the expansion (see, Figure 5a–h). In most points of the complex plane, the relative error remains at the machine epsilon level (≈${10}^{-16}$), demonstrating resistance to the accumulation of rounding errors even as the approximant order increases up to 2000. At the same time, a loss of accuracy of two orders of magnitude is observed near the imaginary axis and the origin; however, it is significantly smaller compared to the expansion with unbounded partial numerators.

Analyzing Figure 6, we see that high computational precision (at the machine epsilon level ≈${10}^{-16}$) is preserved not only in the domain

$$\mathrm{\Omega }=\left\{z\in \mathbb{C}:|z|>{\displaystyle \frac{2\sqrt{3}}{3}}\right\}$$

but also within it, with the exception of a small neighborhood of the origin along the imaginary axis. This illustrates that the theoretical conditions obtained in this paper are sufficient, but, in practice, the stability domain is wider. The region of accuracy loss (where the error increases up to ${10}^{-14}$) is symmetrically concentrated along the imaginary axis and localized near the origin. Unlike continued fractions with unbounded numerators, where the effects of significant error accumulation spread more widely, the boundedness of the coefficients ensures the damping of errors over a broader region. The fact that practical precision is preserved even in those areas of the complex plane where theoretical results do not yield stability guarantees confirms the high reliability of the modified S-fraction, as well as the current lack of methods for expanding stability domains and the prospects for further research.

Figure 5. Relative errors for continued fraction (41) at selected points z.

Figure 5. Relative errors for continued fraction (41) at selected points z.

Figure 6. Distribution of relative error for continued fraction (41).

Figure 6. Distribution of relative error for continued fraction (41).

4.2. Experimental Studies and Forecasting of Patent Activity Dynamics

To verify the effectiveness and predictive properties of the developed mathematical models, the time series of annual patent application volumes in Ukraine for the period of 2016–2025 was chosen as the baseline experimental dataset [27].

To simplify computational procedures and eliminate the scaling effect, the independent variable x (calendar years) was normalized as $x=\mathrm{Year}-2015$, then $x\in [1,10]$ corresponds to the period of 2016–2025. The values of the function y represent the actual number of registered patents. The experimental data are presented in Table 1.

Table 1. Baseline time series of patent activity.

Year	x	y
2016	1	489,030
2017	2	518,704
2018	3	547,972
2019	4	578,562
2020	5	608,817
2021	6	639,687
2022	7	660,012
2023	8	679,194
2024	9	699,289
2025	10	725,615

Within the experiment, four classes of approximation models were constructed and compared:

• Linear model (1st order):

  $$y=a_{1}x+a_0;$$
• Quadratic model (2nd order):

  $$y=a_2{x}^2+a_{1}x+a_0;$$
• Cubic model (3rd order):

  $$y=a_3{x}^3+a_2{x}^2+a_{1}x+a_0;$$
• Continued fraction model (4th order):

  $$y=b_0+{\displaystyle \frac{a_0}{x+\frac{a_1}{x + \frac{a_2}{x + \frac{a_3}{x}}}}}.$$

The obtained parameter estimation and regression models’ accuracies are as follows:

• Linear model:

  $$a_1\approx 26,210.24,a_0\approx 470,531.87,R^2\approx 0.992640;$$
• Quadratic model:

  $$a_2\approx -807.00,a_1\approx 35,087.24,a_0\approx 452,777.87,R^2\approx 0.998663;$$
• Cubic model:

  $$a_3\approx -15.50,a_2\approx -551.29,a_1\approx 33,907.90,a_0\approx 454,107.53,R^2\approx 0.998676;$$
• Continued fraction model:

  $$b_0\approx 455,566.26,a_0\approx 18,107,084.70,a_1\approx 602.90,a_2\approx 9.84,a_3\approx 138.36,R^2\approx 0.998746.$$

Table 2 presents the residual values (residuals), which are defined as the difference between the actual and calculated value $\Delta y=y_{\mathrm{act}}-y_{\mathrm{pred}}$.

Table 2. Residual values table.

x	y	Linear	Quadratic	Cubic	Continued Fraction
1	489,030	−7712.11	+1971.89	+1581.36	+1366.85
2	518,704	−4248.35	−1020.35	−890.17	−628.34
3	547,972	−1190.59	−2804.59	−2479.15	−2203.89
4	578,562	+3189.16	−1652.84	−1364.59	−1286.95
5	608,817	+7233.92	+777.92	+889.50	+743.95
6	639,687	+11,893.68	+5437.68	+5326.10	+5047.79
7	660,012	+6008.44	+1166.44	+878.19	+608.44
8	679,194	−1019.81	−2633.81	−2959.25	−3084.83
9	699,289	−7135.05	−3907.05	−4037.23	−3935.79
10	725,615	−7019.29	+2664.71	+3055.24	+3375.73

The main criterion for selecting the best model is not only the accuracy of describing past data ($R^2$) but also the predictive efficiency during extrapolation over long time periods. Table 3 shows the results of forecast calculations for the period up to 2035.

Table 3. Comparative table of patent activity forecasts (2026–2035).

Year	x	Continued Fraction	Quadratic Model	Cubic Model	Linear Model
2026	11	739,350	741,091	739,761	758,845
2027	12	754,610	757,617	754,837	785,055
2028	13	768,089	772,529	767,694	811,265
2029	14	779,872	785,827	778,240	837,475
2030	15	790,054	797,512	786,381	863,686
2031	16	798,737	807,582	792,026	889,896
2032	17	806,026	816,038	795,079	916,106
2033	18	812,029	822,880	795,450	942,316
2034	19	816,852	828,108	793,044	968,526
2035	20	820,599	831,723	787,769	994,737

The Compound Annual Growth Rate according to the continued fraction model for the period of 2016–2035 is 2.76%.

Although the cubic polynomial demonstrates a high value of $R^2\approx 0.998676$, the experiment expanding the forecast interval to 2050 clearly demonstrates the overfitting effect and the manifestation of Runge’s phenomenon. On a short interval up to 2035 (see Figure 7), the cubic curve behaves adequately, but over a long distance ($x>20$, after 2035), it loses adequacy and demonstrates a sharp drop (the forecasted value collapses to 301,102 patents in 2050), which contradicts the economic logic of market development (see Figure 8).

Figure 7. Patent activity forecasting up to 2035.

Figure 8. Patent activity forecasting up to 2050.

The linear model predicts unrealistic infinite growth (up to 1.38 million patents in 2050), contrary to the market saturation effect. The quadratic model behaves more realistically, but in the absence of asymptotic constraints, it also demonstrates a decline in the second half of the forecast period. The nonlinear continued fraction model provides the best balance. On the historical interval, it ensures the highest accuracy ($R^2\approx 0.998746$), while on the long-term horizon (up to 2050), it implements the property of asymptotic saturation. The curve smoothly reaches a plateau (stabilizing at the level of ∼803–826 thousand patents), which correctly reproduces the processes of market saturation and the exhaustion of growth potential.

Thus, the continued fraction model is an optimal analytical tool that combines high-quality approximation of past data with adequate behavior during long-term extrapolation.

To evaluate the reliability and stability of the constructed models, a coefficient perturbation experiment was conducted. The essence of the experiment consists of randomly perturbing all coefficients of both the cubic model and the continued fraction model within $\pm 2.0\%$. For each of the five independent perturbation scenarios, we analyzed the quality of approximation on the historical interval (the $R^2$ statistic) and the stability of the forecasted trends over the medium-term (up to 2035) and long-term (up to 2050) intervals. The results of this experiment are presented in Table 4 and Figure 9.

Table 4. Results of the $\pm 2.0\%$ model parameter perturbation experiment and their impact on the $R^2$ metric and forecasting.

Test Case	Cubic Model ${\mathit{R}}^2$	Cubic Forecast 2035	Cubic Forecast 2050	Continued Fraction Model ${\mathit{R}}^2$	Continued Fraction Forecast 2035	Continued Fraction Forecast 2050
Perturbation 1	0.99686	792,501	305,066	0.99854	819,148	801,055
Perturbation 2	0.99591	787,173	305,210	0.99665	821,320	804,057
Perturbation 3	0.99398	780,963	278,571	0.99770	821,578	802,458
Perturbation 4	0.98741	774,276	279,182	0.99058	826,909	807,836
Perturbation 5	0.99545	782,883	298,613	0.99571	817,817	798,292

Figure 9. Stability analysis to coefficient perturbations $(\pm 2.0\%)$.

An analysis of the obtained data indicates significant differences in the behavior of the models under perturbation. The cubic model demonstrates noticeable sensitivity to its coefficients; in the worst-case scenario (Test 4), the coefficient of determination drops to $0.98741$. In contrast, the continued fraction model maintains high data-fitting quality, keeping $R^2$ above $0.99058$, which confirms its lower sensitivity to perturbations. Under the influence of random coefficient perturbations, the forecasted trends of the cubic model exhibit marked fluctuations, forming a range from 774,276 to 792,501 patents (an oscillation amplitude exceeding 18,000 units). Conversely, under perturbations, the continued fraction model provides a more reliable forecast (817,817–826,909), minimizing strategic planning risks in the medium term (up to 2035). When performing long-term forecasting for 2050, perturbations of the cubic model parameters push the forecast down even further into an unrealistic range of 278,571–305,210 patents. On the other hand, the continued fraction model successfully reproduces the market saturation effect (plateau effect), providing a stable forecast for 2050 within the range of 798,292–807,836 patents.

Thus, the experimental results illustrate the perturbation stability property of the continued fraction model, making it a reliable tool for long-term forecasting.

Calculations were performed using Maple software 2022.2 for Windows.

5. Conclusions

This paper investigates the stability of modified Stieltjes continued fractions against coefficient perturbations, a study directly driven by the rigorous demands of modern applied modeling. As functional continued fractions increasingly displace traditional polynomials in high-stakes fields like economic regression forecasting, establishing mathematical guarantees of their structural stability has become critical. By shifting the analytical focus from abstract convergence to explicit numerical stability, this study bridges a vital gap between theoretical continued fraction analysis and stable algorithmic implementation.

The cornerstone of our approach is the development of recursive error formulas (Lemmas 1 and 2), which explicitly map how localized coefficient perturbations propagate into the relative error of the nth approximant. These formulas allowed us to establish clear, geometrically defined zones of computation in the complex plane:

• The half-plane ${\mathrm{\Omega }}_C$: proved to be a set of uniform stability under the condition $C>\sqrt{A}/2$.
• The angular set ${\mathrm{\Omega }}_{\alpha ,C}$: established as a subset of ${\mathrm{\Omega }}_C$ that guarantees uniform stability under a weaker geometric constraint ($C>\sqrt{A(1-\cos \alpha )}$) when $\alpha <\arccos (3/4)$.
• The exterior disk ${\mathrm{\Omega }}_R$: extended uniform stability directly onto the imaginary axis and its immediate neighborhoods, provided that the distance from the origin satisfies $R>2\sqrt{A}$.

Crucially, our pointwise analysis successfully expands the verified domain of computational safety, culminating in the proof that modified S-fractions maintain absolute stability across the entire real axis excluding the origin ($\mathbb{R}\setminus \left\{0\right\}$) for any bounded coefficients. For practical software engineering and applied data science, these results provide a concrete blueprint for error management. The explicit a priori error bounds derived for each stability set eliminate algorithmic guesswork. They definitively indicate where standard double-precision arithmetic (IEEE 754) is entirely sufficient to maintain structural integrity and precisely isolate the volatile zones near the origin where extended precision arithmetic is mandatory.

While this study establishes an authoritative baseline for positive real coefficients, modern empirical data often introduce broader algebraic complexities. Consequently, generalizing this recursive error framework to complex coefficients and directly mapping these stability boundaries onto the explicit expansions of active economic forecasting functions represent the priority directions for our further research.