Behind Jarratt’s Steps: Is Jarratt’s Scheme the Best Version of Itself?

. In this paper, we analyze the stability of the family of iterative methods designed by Jarratt using complex dynamics tools. Tis allows us to conclude whether the scheme known as Jarratt’s method is the most stable among all the elements of the family. We deduce that classical Jarratt’s scheme is not the only stable element of the family. We also obtain information about the members of the class with chaotical behavior. Some numerical results are presented for confrming the convergence and stability results.


Introduction
Te solution of nonlinear equations and systems of equations is among the most important problems, both from a theoretical and a practical point of view, in applied mathematics and other sciences, see for example [1]. Due to the lack of analytical methods for solving such problems, iterative methods are becoming increasingly necessary for approximating the solutions of these equations.
In addition to the classical methods such as Newton, Chebyshev, and Halley among the so-called one-point methods, and multipoint algorithms such as Traub, Jarratt, and Ostrowski, numerous papers have been published in recent years trying to overcome the convergence order of these schemes as well as their stability. In all of them, the authors construct iterative procedures for approximating simple roots α of a nonlinear equation f(x) � 0, where f: I ⊆ R ⟶ R is a real function defned on an open interval I. In the books [2,3], we can found good overviews of this area of numerical analysis.
Te dynamical analysis of an iterative method or a family of schemes is a valuable tool for classifying the diferent iterative formulas, not only in terms of their order of convergence but also in terms of their behavior in terms of the chosen of initial guesses. Tis study also provides useful information on the stability and reliability of the iterative methods. See, for example, [4][5][6].
In this paper, we present a dynamical study of the parametric Jarratt family, a set of fourth-order iterative methods for approximating simple roots α of a nonlinear equation f(x) � 0. In [7], Jarratt designed a fourth-order formula for solving nonlinear equations which require three functional evaluations per iteration, one of f and two of f ′ . Its expression is as follows: where ϕ 1 x k � a 1 w 1 (x) + a 2 w 2 (x), being and a 1 , a 2 , b 1 , b 2 , and c real or complex parameters. Using Taylor's series expansion around a simple zero α of f(x) � 0, Jarratt obtained values of some of the previous parameters to reach fourth-order convergence. Taking θ � b 2 /b 1 + b 2 and c � − 2/3 and expressing Jarratt's class in two steps, we obtain , k � 0, 1, 2, . . ., b 2 � 8θ 2 /3(θ − 1), and θ is an arbitrary parameter that can take real or complex values. θ ≠ 0, 1, otherwise the method is not defned. Tis parametric family includes the so-called Jarratt's method, for θ � 3/2, whose iterative expression is as follows: 1.1. Dynamical Concepts. We are going to analyze the stability of members of family (4). For it, we apply complex dynamics tools to the rational operator obtained when this class is applied on an arbitrary second degree polynomial p(x). We recall some concepts of complex dynamics that we use in this work. For a more general understanding of these concepts, see, for example, [8,9]. Given a rational operator R: C ⟶ C defned on the Riemann sphere, C, the orbit of a point x 0 is the sequence of points.
A fxed point x 0 of operator R is a point such that R(x 0 ) � x 0 . If a fxed point is not a root of polynomial p(x), then it is called a strange fxed point. Fixed points can be classifed according to the behavior of the derivative operator on them. Terefore, a fxed point x 0 is an attracting A critical point of operator R is a point x 0 where the derivative of R cancels out, that is, R ′ (x 0 ) � 0. Critical points that do not coincide with the roots of the polynomial are called free critical points.
Te basin of attraction of an attractor α is defned as the set of preimages of any order that satisfy the following equation: Te rest of the paper is organized as follows. In Section 2, the convergence order of the parametric family (4) is analyzed. Te dynamical behavior of this family as a function of parameter θ is studied in Section 3. First, we determine the rational operator associated with the family and analyze the stability of the corresponding fxed points and critical points of that operator. Te parameter planes of the free critical points are drawning, which allows visualizing the parameter values that make the method stable or unstable. Finally, the dynamical planes are generated, in which the basins of attraction of fxed or periodic points of the method can be visualized for some particular value of the parameter. In Section 4, some numerical tests are presented to compare the family of methods studied with other schemes. Te paper ends with some conclusions, which are presented in Section 5 along with the references used.

Convergence of Jarratt Parametric Family
In this section, the convergence analysis of the Jarratt parametric family is studied. We present an alternative proof to that given by Jarratt. From the error equation, we can observe that all the members of uniparametric family (4) have fourth-order convergence, with independence of parameter θ.
Theorem 1. Let f: I ⊂ R ⟶ R be a sufciently diferentiable function at each point of the open interval I such that α ∈ I is a simple root of f(x) � 0. If we choose an initial estimate x 0 sufciently close to α, sequence x k k ≥ 0 obtained using iterative expression (4), converges to α, with order of convergence p � 4, being the error equation.
Proof. Using Taylor's series expansion of the function f(x k ) and f ′ (x k ) around α, we have Calculating the quotient From the equations (3)-(5) and the frst step of the iterative scheme (4), we have Te Taylor's series expansion of f(y k ) is as follows: and the derivative of function f(y k ) is as follows: Calculating the quotient Substituting equations (11) and (15) in the second step of the iterative expression (4) yields error equation for Jarratt parametric family as follows: Tis completes the proof.
According to Kung and Traub's conjecture (see [10]), the family shown in (4) is optimal.

Complex Dynamical Behavior
In this section, we present a dynamical study of Jarratt's family (4). We begin by calculating the rational operator Discrete Dynamics in Nature and Society associated with the class when it is applied on a quadratic polynomial and then analyze the stability of the fxed and critical points of this operator. From the independent critical points, we generate the parameter spaces, which are graphs that allow us to visually determine the values of the parameter for which a member of the family has stable or unstable behavior.

Rational Operator. We analyze family (4) on a generic quadratic polynomial
Te result is a rational operator, called K, which depends on a, b, and parameter θ: .
Applying the iterative scheme given in equation (4) on p(x) we obtain the rational function K p,θ (x), which depends on roots a, b, and parameter θ ∈ C. Using Möbius transformation on which only depends on the arbitrary parameter θ ∈ C. By factoring the numerator and denominator in (18), it can be proved that for θ ∈ − 3/2, 3/2, 27/10 { } and the operator R θ (x) is simplifed as seen in equations (19)-(21), which completes the proof.

Proposition 3.
Te fxed points of the rational function R θ are x � 0 and x � ∞, which correspond with the roots of p(x), and the following are strange fxed points: 4 Discrete Dynamics in Nature and Society Te total number of fxed points of operator R θ (x) varies as a function of parameter θ, that is, (i) If θ ∈ C and θ ≠ 27/10, then R θ (x) has seven fxed points. (ii) If θ � 27/10, then ex 1 � 1 is not a fxed point and R θ (x) has six fxed points.
Pairs of strange fxed points conjugate to each other satisfy exi − 1/exj � 0 for i ≠ j; these are ex 2 and ex 3 , ex 4 , and ex 5 .
According to Proposition 3, there are at most seven and at least six fxed points for the rational operator R θ (x). In addition, we show the existence of two pairs of strange fxed points conjugate to each other, each pair has the same stability characteristics, and thus, the stability analysis is reduced by half.

Stability of Fixed
Points. In order to analyze the stability of the fxed points, we calculate the frst derivative of operator R θ (x).
It is known that 0 and ∞ are superattracting fxed points, since the methods have order of convergence four, regardless of the value of the parameter θ; however, the stability of the strange fxed points depends on the value of θ. Te stability of strange fxed points ex 1 to ex 5 is established in the following theorems.
Te following results can be demonstrated numerically using the stability functions associated with fxed points. □ Theorem 5 (Stability of ex 2 and ex 3 ).
Te stability of strange fxed points ex 2 and ex 3 , for real values of θ, can be summarized as follows: where D is the region of the cone below the cardioid in Figure 1, then ex 2 and ex 3 are attractors; if θ ≈ − 0.994035, they are superattractors. Te stability surface of strange fxed point ex 1 � 1 in the complex plane can be seen in Figure 3. In it, the zones of attraction (blue surface) and repulsion (gray surface) are shown. Visually, if θ is inside the circumference of the cone, then it is attracting; if θ is on the circumference, it is parabolic and if θ is outside the circumference, it is repulsor.
Te stability surface of strange fxed points ex 2 and ex 3 is shown in Figure 1. Te stability surface of strange fxed points ex 4 and ex 5 is shown in Figure 2. In these fgures, the prevalence of repulsion zones over attraction zones is observed.
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Analysis of Critical Points.
We will calculate the critical points of the rational operator R θ (x) given in equation (18).
Fatou and Julia [11,12] stated that these points are of special interest, since each basin of attraction has at least one critical point, so the free critical points could be in a basin of attraction of some of the solutions of the equation, or be in the basin of some strange fxed point or attracting periodic orbit.
(i) Te free critical points cl 1 (θ) and cl 2 (θ) coincide for θ � 0, (ii) Te free critical points cl 1 (θ) and cl 3 (θ) coincide for the following values of parameter θ: (iv) Te free critical points cl 2 (θ) and cl 4 (θ) coincide for the following values of parameter θ: (v) Te free critical points cl 3 (θ) and cl 4 (θ) coincide for the following values of parameter θ: Proposition 7 states that there is a maximum of six critical points and a minimum of two critical points. Tere are two pairs of free critical points conjugate to each other, each with the same characteristics in terms of stability, simplifying the dynamical analysis.
Parameter values that reduce the number of free critical points are interesting for drawing dynamical planes.

Parameter Spaces.
Te dynamical behavior of operator R θ (x) depends on the values of parameter θ. Parameter spaces are graphs of the independent-free critical values for the method, which allow to visualize parameter values that make the method stable or unstable [13].
We generate the parameter spaces, taking a free critical point cl(θ) as initial estimation for operator R θ (x) and applying the iterative scheme (4) for all values of the parameter θ, defned on a mesh of the complex plane with 800 points on each axis. Tese plots have been generated using Discrete Dynamics in Nature and Society 7 MATLAB R2020b. At a point corresponding to a specifc value of θ, if a method converges to one of the roots of the polynomial in less than 200 iterations and with an error estimate of less than 10 − 3 , then that point is colored red; otherwise, the point is colored black. Te Jarratt parametric family has at most four free critical points, of which there are two pairs conjugate to each other (see Proposition 7); that means there are only two independent free critical points. We then obtain two different parameter spaces: P 1 for x � cl 1 (θ), cl 2 (θ) and P 2 for x � cl 3 (θ), cl 4 (θ), shown in Figure 4.
In P 1 parameter space (Figure 4(a)), the all-red surface means that for any method of the family, in that range of θ values, the critical point cl 1 (θ) is only able to converge to one of the two roots of the polynomial. Tis critical point does not create its own basin; there is no attracting strange free point and no attracting periodic orbit, the only attractors are the roots of the polynomial themselves. Te critical point cl 2 (θ) has the same behavior as cl 1 (θ), being both conjugate to each other.
In the parameter plane P 2 corresponding to the conjugate critical points cl 3 (θ) and cl 4 (θ), the region marked in red corresponds to points where the method has stable behavior, while the regions in black correspond to points where the method has unstable behavior. Regions where strange fxed points are attractors and are also unstable and appear in this parameter space (see Figure 4(b)).

Dynamical Planes.
To study the stability of some methods for Jarratt parametric family, we use dynamical planes. Tese plots allow us to extend the information obtained in the parameter planes; in them, we can visualize the basins of attraction for fxed or periodic points of the method, given some particular value of parameter, θ [13].
For the dynamical analysis we select methods of family (4) corresponding to parameter values located in the stability zone and in the instability zone of the parameter space, and from these we will generate the corresponding dynamical planes, using MATLAB R2020b. In these fgures, a mesh with 800 points on each axis has been drawn, where each point represents a diferent initial estimate that is introduced in the iterative process (see [13]). When a method converges to a solution, in at most 200 iterations and with a tolerance of less than 10 − 3 , then, it is assigned a certain color: orange if it converges to x � 0 and blue if it converges to x � ∞. In case the initial estimate does not converge to any of the roots of the polynomial within the maximum number of iterations, it is assigned the color black; other basins of attraction are colored green and red. Figure 5 shows dynamical planes for values of θ in the stability zone, in which only two basins of attraction corresponding to the roots are observed. Specifcally, some methods appear with global convergence, which is a key fact in some applications, such as the fnding of matrix sign functions by using these iterative methods (see, for example, [14,15]). Figure 6 shows dynamical planes for values of θ outside the stability zone, which can be visually verifed by the existence of black areas of nonconvergence to the solution (in the case of Figures 6(a) and 6(b)) and by the presence of two basins of attraction that do not correspond to roots, but to conjugate strange fxed points (in the case of Figures 6(c) and 6(d)).

Numerical Results
Te numerical tests in this section have been performed using variable precision arithmetic, with 2000 digits of mantissa and a tolerance of 10 − 100 in MATLAB R2020b. Te stopping criterion used is |x k+1 − x k | < 10 − 100 or |f(x k+1 )| < 10 − 100 .
Tables 1-3 summarize the results obtained by applying four diferent methods of the family, some of them stable (FJ2(θ � 1.5) and FJ3(θ � 2.75)) and others unstable (FJ1(θ � − 0.4) and FJ4(θ � 3)), as well as the methods of Chun [16] and Ostrowski [17], which have order of convergence four. Te test functions used are the following: In order to evaluate the stringency of each implemented method with respect to the initial estimate to fnd a solution, we have started the iterations with diferent initial estimates, named according to their proximity to the solution x 0 : close (x 0 ≈ α), far (x 0 ≈ 10α), and very far (x 0 ≈ 100α), respectively.
For each function, the following items have been calculated: approximate root value, error estimates at the last iteration: |x k+1 − x k | and |f(x k+1 )|, the number of iterations required to converge to the solution, the approximate computational convergence order (ACOC), and the elapsed time (e-time), calculated as the arithmetic mean of 10 runs for each method. Table 1 shows that when the initial estimate is close to the root, the presented methods converge, for a minimum of 5 iterations and a maximum of 6 iterations, even in cases of those corresponding to parameter values for family (4) located in regions of instability.
It can also be observed that the lowest error corresponds to Chun's method, followed by the stable FJ3 method. Te number of iterations is in general the same and the order of computational convergence obtained for all methods of the family confrms the theoretical convergence order determined in Section 2 From Tables 2 and 3, we can observe that the presented methods do not always converge to the solution, supporting the results found in the dynamical analysis of Section 3. Te convergence depends on the initial estimation and the nonlinear function used. 8 Discrete Dynamics in Nature and Society   When the initial estimate is far or very far from the root, in general, the FJ1(θ � − 0.4) and FJ4(θ � 3.0) methods diverge, as expected, since these methods correspond to parameter values located in the instability zone.

Conclusions
In this paper, the dynamical study of a family of fourth-order iterative methods has been carried out in order to identify those members of the family that have a better behavior in terms of stability.
Te dynamical behavior of the Jarratt parametric family is generally stable. Tis is shown in the parameter spaces, where the prevalence of the stability regions is observed, and it is confrmed by numerical tests, which yield favorable results on the convergence of the studied methods. Te theoretical order of convergence has been confrmed by ACOC, which is approximately equal to 4.
For initial estimates close to the solution, all methods converge. Divergence cases are verifed for initial estimates far or very far from the solution, especially for methods of the considered family located in the instability zone.
Terefore, we conclude after the analytic, dynamic, and numerical studies performed in this manuscript, that classical Jarratt's scheme is the best one among all the general class of iterative methods proposed originally by Jarratt. In future work, we will extend this scheme to the estimation of matrix sign functions and other nonlinear matrix equations.

Data Availability
No underlying data were collected or produced in this study.

Conflicts of Interest
Te authors declare that they have no conficts of interest.