Research article

Soft strong θ-continuity and soft almost strong θ-continuity

  • We continued the study of "soft strong θ-continuity" and defined and investigated "soft almost strong θ-continuity" which is a generalization of soft strong θ-continuity. We gave characterizations and examined soft composition concerning these two concepts. Furthermore, we derived several soft mapping theorems. We provided several links between these two ideas and their related concepts through examples. Lastly, we looked at the symmetry between them and their topological counterparts.

    Citation: Dina Abuzaid, Samer Al-Ghour. Soft strong θ-continuity and soft almost strong θ-continuity[J]. AIMS Mathematics, 2024, 9(6): 16687-16703. doi: 10.3934/math.2024809

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  • We continued the study of "soft strong θ-continuity" and defined and investigated "soft almost strong θ-continuity" which is a generalization of soft strong θ-continuity. We gave characterizations and examined soft composition concerning these two concepts. Furthermore, we derived several soft mapping theorems. We provided several links between these two ideas and their related concepts through examples. Lastly, we looked at the symmetry between them and their topological counterparts.



    Many theoretical and applied problems in different fields of study, such as physical sciences, biomedical engineering, chemistry, biological and chemical sciences, computer sciences, finance, statistics, economics and applied mathematical sciences contain symmetry, which may result a nonlinear polynomial equation

    ϑ(ϰ)=0, (1.1)

    with m distinct roots 1,...,m. There are numerous numerical iterative methods for computing the roots of polynomial equations. These methods can be classified into two types: numerical iterative methods that approximate a single root at a time and iterative numerical methods that approximate all of the roots of a nonlinear polynomial equation simultaneously.

    The Newton-Raphson method given as

    ϰ(i+1)=ϰ(i)ϑ(ϰ(i))ϑ(ϰ(i)), (1.2)

    is one the most commonly used numerical methods to approximate a single root of Eq (1.1) at a time, thanks to its local quadratic convergence. Iterative methods, such as Eq (1.2), have a significant drawback in that they require derivative evaluation at each iteration, which incurs a significant computational cost, and diverge if ϑ(ϰ(i))=0 or the initial guess is far from the root. See, for example, the work of Akram et al.[1], Cordero et al.[2], Agarwal et al. [3], Chicharro et al. [4], Behl et al.[5], Chun et al. [6], Zafar et al.[7] and many others). Simultaneous methods having global convergence property, are stable and applicable for parallel execution as well. Convergence analysis, computational efficiency and parallel implementation of simultaneous iterative techniques have been studied by McNamee [8], Nedzhibov [9], Proinov et al. [10], Mir et al. [11], Cholakov [12], Proinov and Vasileva [13], Proinov and Petkova [14], Sanchez et al.[15], Ivanov [16], Kanno et al.[17] as well as the references cited therein.

    Weierstrass method (WDM)[18], which approximates all the roots of Eq (1.1) simultaneously, given by

    ϰ(i+1)t=ϰ(i)tϑ(ϰ(i)t)mΠrtr=1(ϰ(i)tϰ(i)r), (1.3)

    is one of the most famous iterative techniques used in comparison to Newton Raphson. The term ϑ(ϰ(i)t)mΠrtr=1(ϰ(i)tϰ(i)r) is called Weierstrass correction. The Weierstrass method, which simultaneously approximates all roots of Eq (1.1), has local quadratic convergence. Weierstrass introduced the method in 1891, and it was rediscovered by Kerner [19], Durand [20], Dochev [21], and Presic [22]. Dochev demonstrated the first local quadratic convergence of the Weierstrass iterative technique in 1962.

    Later on, in 2015, Proinov [23] presented the general convergence theorem for the iterative process and discussed the use of the Weierstrass root approximating technique. In 2016, Nedzibove [24] developed a modified version of Eq (1.3), and Marcheva et al.[25] presented the local convergence theorem of the modified Weierstrass method in 2020. Song et al.[26] created and verified three new derivative-free simultaneous iterative Weierstrass computational methods for calculating all polynomial problem roots.

    The goal of this paper is to develop and analyze the derivative-free family of Weierstrass-type methods with accelerated convergence in order to find the roots of the above problem, which should have better convergence behavior than the Weierstrass method. We analyze the convergence behavior of the proposed method by computing CPU-time, log of residual errors, computational errors, and drawing dynamical planes while solving numerical test problems. Several graphs are plotted to demonstrate that the newly constructed method outperforms existing methods in the literature in efficiency and accuracy.

    First, we present the third order family of the single root finding method (MS), and then we generalize this scheme to simultaneously find all the roots of Eq (1.1):

    ϰ(i+1)=ϰ(i)((3α1)ϑ(ϰ(i))+(1α)ϑ(y(i))(2α1)ϑ(ϰ(i))+ϑ(y(i)))ϑ(ϰ(i))ϑ(ϰ(i)),i=0,1,2,..., (2.1)

    where y(i)=ϰ(i)ϑ(ϰ(i))ϑ(ϰ(i)) and α.

    The convergence theorem that follows validates our assumption.

    Theorem 1. Let ¯T be an open interval and ¯T indicate the precise root of a function that is sufficiently differentiable for ϑ:¯T in an open interval ¯T. The following error equation is satisfied if the convergence order of the family of iterative numerical procedures is three and ϰ(0) is close to :

    ς(i+1)=(22α+222+123)(ς(i))3+O((ς(i))4), (2.2)

    where m=ϑm()m!ϑ(),m2.

    Proof. Let be a single exact-root of ϑ and ϰ(i)=+ς(i), where e(i) is the error at ith iterative step. The following is the result of expanding ϑ(ϰ(i)) around ϰ=, using Taylor's series with ϑ()=0:

    ϑ(ϰ(i))=ϑ()(ς(i)+2(ς(i))2+3(ς(i))3+...), (2.3)

    and

    ϑ(ϰ(i))=ϑ()(1+22ς(i)+33(ς(i))2+...). (2.4)

    Dividing (2.3) by (2.4), we have:

    ϑ(ϰ(i))ϑ(ϰ(i))=ς(i)2(ς(r))2+(22+23)(ς(i))3+..., (2.5)

    and

    y(i)=+2(ς(i))2+(222+23)(ς(i))3+...,
    ϑ(y(i))=ϑ()(1+222(ς(i))2+2(222+23)(ς(i))3+...).

    Using ϑ(y(i)) and ϑ(u(i)) in the 2nd-step of (2.1), we have:

    ((3α1)ϑ(ϰ(i))+(1α)ϑ(y(i))(2α1)ϑ(ϰ(i))+ϑ(y(i)))ϑ(ϰ(i))ϑ(ϰ(i))=e(i)+(22α123222)(ς(i))3+... (2.6)

    Thus,

    ϰ(i+1)=ξ+(22α+123+222)(ς(i))3+O((ς(i))4),

    and it becomes

    ς(i+1)=(22α+222+123)(ς(r))3+O((ς(i))4). (2.7)

    Hence the theorem is proved.

    Now, we convert iterative method (2.1) to simultaneous method by using Weierstrass correction [8] given by

    Δ(ϰ(i)t)=ϑ(ϰ(i)t)mΠrtr=1(ϰ(i)tϰ(i)r),(t,r=1,2,3,...,m)

    as follows:

    ϰ(i+1)t=ϰ(i)tϑ(ϰ(i)t)mΠrtr=1(ϰ(i)tϰ(i)r)((3α1)mΠrtr=1(ϰ(i)tϰ(i)r)+(1α)mΠrtr=1(y(i)ty(i)r)mΠrtr=1(y(i)ty(i)r)+(2α1)mΠrtr=1(ϰ(i)tϰ(i)r)), (2.8)

    where y(i)t=ϰ(i)tϑ(ϰ(i)t)mΠrtr=1(ϰ(i)tˆϰ(i)r),(t,r=1,2,3,...,n) and ˆϰ(i)r is a modified two step Steffensen-type [27] fourth-order convergent method, i.e., ˆϰ(i)r=y(i)rϑ(y(i)r)ϑ[ϰ(i)r,y(i)r]+ϑ[y(i)r,t(i)r]ϑ[ϰ(i)r,t(i)r]+α(y(i)rϰ(i)r)(y(i)rt(i)r), y(i)r=ϰ(i)rϑ(ϰ(i)r)ϑ[ϰ(i)r,t(i)r],t(i)r=ϰ(i)r+ϑ(ϰ(i)r), α. In this case, ϑ[.,.] is the divided difference of order one and two, respectively.

    Remark 1. The family of derivative-free methods (2.8) determines all distinct roots of (1.1) simultaneously and is abbreviated as MM.

    This section discusses MM convergence analysis.

    Theorem 2. Let 1,2,...,m be m simple roots of (1.1). If ϰ(0)1, ϰ(0)2,...,ϰ(0)m are, respectively, the initial estimates of the roots and close to exact roots, then MM has fifth-order convergence.

    Proof. Let ϵt=ϰtt, ϵt=ytt and ϵt=ztt be the residual errors in ϰt,yt and zt estimations, respectively. To simplify things, we leave out iteration index i. From the 1st-step of (2.8), we have:

    ytt=utiΔ(ϰt),ϵt=ϵtϵiΔ(ϰt)ϵt=ϵi(1Gt), (2.9)

    where Δ(ϰt)=ϑ(ϰt)mΠjtj=1(ϰiˆuj) and

    Gt=Δ(ϰt)ϵt=ϑ(ϰt)ϵtmΠjtj=1(ϰiˆuj)=mΠj=1(ϰtξj)ϵtmΠjtj=1(ϰtˆuj)=(ϰtt)mΠjtj=1(ϰtj)ϵtmΠjtj=1(ϰtˆϰj)=mjtj=1(utj)(ϰtˆϰj), (2.10)

    with

    ϰtjϰtˆϰj=ϰtˆϰj+ˆϰjjϰtˆϰj=1+ˆϰjjϰtˆϰj=1+(ϵ4),

    and ˆϰjj=O (ϵ4) (see [27]). Now, for a small enough ϵ: if t is a simple root, then |ϰiϰj| is bounded away from zero, resulting in the following expression:

    mjtj=1(ϰij)(ϰtϰj)=(1+(ϵ4i))m1=1+(m1) O (ϵ4t)=1+(ϵ4t).

    Then it implies that:

    Gi=1+(ϵi4),Gi1=(ϵ4i).

    From (2.9), we have

    ϵt=(ϵ5t). (2.11)

    Now by taking the 2nd-step of Eq (2.8), we have:

    ϰt+1t=ϰttΔ(ϰt)((3α1)mΠjtj=1(ϰtϰj)+(1α)mΠjij=1(ytyj)mΠjtj=1(ytyj)+(2α1)mΠjtj=1(ϰtϰj)),ϵt=ϵtϵiΔ(ϰt)ϵt((3α1)+(1α)mΠjtj=1(yiyj)mΠjtj=1(ϰtϰj)mΠjtj=1(ytyj)mΠjij=1(ϰtϰj)+(2α1)),ϵt=ϵtϵtΔ(ϰt)ϵt((3α1)+(1α)mjtj=1(ytyjϰtϰj)mjij=1(ytyjϰtϰj)+(2α1)), (2.12)

    where

    mjij=1(yiyjϰtϰj)=mjtj=1((ytt+tyj+jj)(ϰit+tϰj+jj))=mjtj=1((ϵtϵj+tj)(ϵtϵj+ij)). (2.13)

    Assume ϵt=ϵj are same order, say O(ϵt), then it follows that

    =mjtj=1(1)=1. (2.14)

    Using Eq (2.14) in Eq (2.12), we have:

    ϵt=ϵtϵt(Δ(xt)ϵt)((3α1)+(1α)1+(2α1))=(ϵ5i). (2.15)

    Hence, the theorem is proved.

    We employ the real and imaginary portions of the initial approximations across a grid of 250×250 of square [2.5×2.5]2C in the complex plane to construct the basins of attraction of the iterative procedures used by MS, MM and the Zhang et al.[28] method (ZPH). Due to the global convergence behavior of the simultaneous technique, the maximum number of iterations was set at 15 and the stopping criterion was |ϑ(ϰ(i)t)|<105. We employ various colors to show the root at which the iterative technique converges, and black in all other cases. Basin colors sharpness signifies fewer iterations. We consider the following two non-linear functions for the generation of basins: ϑ1(ϰ)=ϰ31 with exact roots -1, -0.5+0.866i, -0.5-0.866i, and ϑ2(ϰ)=ϰ61iϰ3+1 with exact roots -1.0167+0.587i, -1.1740i, 1.0167+0.587i, -0.7377-0.4259i, 0.8518i, 0.7377-0.4259i. Figures 1(a, c, e, g) and 2(a, c, e, g), present the basins of attraction of numerical iterative scheme MS for various parameter values of α and Figures 1(b, d, f, h) and 2(b, d, f, h) show basins of attraction of simultaneous methods MM (for various parameter values of α) and ZPH respectively for non-linear equation ϑ1(ϰ)ϑ2(ϰ). Table 1 and color brightness in Figure 1(b, d, f) shows the ascendancy in efficiency of MM as compared to MS and ZPH.

    Figure 1.  (a, c, e, g) depicts the basins of attraction of iterative-scheme MS (for different value of α) and (b, d, f, h) shows the basin of attraction of the MM simultaneous methods (for different value of α), ZPH, respectively, for non-linear equation ϑ1(ϰ).
    Figure 2.  (a, c, e, g) presents the basins of attraction of iterative-scheme MS (for different value of α) and (b, d, f, h) shows basin of attraction of simultaneous methods MM (for different value of α), ZPH, respectively, for nonlinear equation ϑ2(ϰ).
    Table 1.  Elapsed time and number of iterations for generation of basins of attraction.
    Method MS MM ZPH
    Elapsed time Iterations Elapsed time Iterations Elapsed time Iterations
    ϑ1(ϰ) 1.583536 25 0.456321 4 1.432156 7
    ϑ2(ϰ) 2.732514 23 0.632145 5 2.132564 7

     | Show Table
    DownLoad: CSV

    Table 1 clearly demonstrates that the simultaneous method MM surpasses MS and ZPH in terms of actual time and number of cycles required for convergence to the exact root of the equation ϑ1(ϰ) and ϑ2(ϰ) for any initial guessed value used to generate basins of attraction.

    Here, we compare the computation % efficiency of Zhang et al.'s [28] convergence order five technique (abbreviated as ZPH), i.e.,

    ϰ(i+1)t=ϰ(i)t2Δt(ϰ(i)t)1+nrtr=1Δr(ϰ(i)r)ϰ(i)tϰ(i)r+(1+nrtr=1Δr(ϰ(i)r)ϰ(i)tϰ(i)r)2+4Δt(ϰ(i)t)nrtr=1Δr(ϰ(i)t)(ϰ(i)tϰ(i)r)(ϰ(i)tΔt(ϰ(i)t)ϰ(i)r), (4.1)

    with our newly proposed method based on the efficiency index E defined in [29]

    E=logτD, (4.2)

    where D is the cost of numerical computer calculation

    D=D(m)=δasASm+δmMm+δdDm (4.3)

    and τ is the convergence order of iterative numerical schemes. Thus, (4.2) becomes:

    E=logτδasASm+δmMm+δdDm. (4.4)

    By using (4.4) and the information in Table 2, the percentage efficiency ratio of simultaneous numerical iterative methods can be determined employing the following formula forρ((MM),(ZPH)) given by:

    ρ(MM),(ZPH))=(E(MM)E(ZPH)1)×100 (in percent),  (4.5)

    where O(m)=B. The dominance efficiency of MM over ZPH is demonstrated clearly in Figure 3(a, b).

    Table 2.  The number of fundamental mathematical operations.
    Methods ASm Mm Dm
    ZPH 8.0m2+B 6.0m2+B 2.0m2+B
    MM 6.0m2+B 3.0m2+B 2.0m2+B

     | Show Table
    DownLoad: CSV
    Figure 3.  (a, b), compares the computational effectiveness of the simultaneous algorithms MM and ZPH.

    The ZPH and MM family of techniques were utilized to demonstrate the use and effectiveness of several nonlinear problems from biological engineering [12] and applied sciences [29,30,31,32,33,34,35,36] are considered in this section. All calculations utilize CAS-Maple 18 64-digit floating point arithmetic, and the process of iteration is terminated when

    ¯ς(i)t=ϑ(ϰ(i)t)2<∈,

    where ¯ς(i)t represents the absolute error. For simultaneous calculations of all roots of Eq (1.1), we set

    ∈=1030

    as the fixed stopping criteria. In Tables 312, the quantity denoted CPU represents the time required to compute the root using the iterative schemes ZPH and MM, respectively. We use α=13 and β=12 in all numerical calculations. In Tables 4, 6, 8, 10, 12 for random initial vectors ((0)ϑ1-(0)ϑ3) taken from Appendix Table 13, MM and ZPH are represented by MMϑ1MM3ϑ3 and ZPHϑ1ZPHϑ3, respectively.

    Table 3.  Residual errors comparison.
    Method CPU ¯ς(5)1 ¯ς(5)2 ¯ς(5)3 ¯ς(5)4
    ZPH 0.241 1.29e-17 1.62e-17 6.15e-32 4.16e-15
    MM 0.016 3.41e-51 1.15e-51 5.92e-53 5.82e-53

     | Show Table
    DownLoad: CSV
    Table 4.  Residual errors comparison on random initial vector.
    Method CPU ¯ς(17)1 ¯ς(17)2 ¯ς(17)3 ¯ς(17)4
    ZPHϑ1 2.1456 3.34e-07 2.43e-05 4.30e-07 8.35e-04
    ZPHϑ2 3.1255 1.21e-03 0.32e-03 0.30e-08 1.13e-05
    ZPHϑ3 2.9914 4.34e-05 2.31e-04 0.13e-03 4.35e-04
    MMϑ1 1.1201 9.41e-34 4.3e-29 4.13e-32 7.81e-35
    MMϑ2 1.32451 3.10e-32 2.3e-26 1.13e-18 8.03e-21
    MMϑ3 0.93114 1.01e-33 2.3e-21 1.31e-19 8.73e-30

     | Show Table
    DownLoad: CSV
    Table 5.  Residual computational errors comparison.
    Method CPU ¯ς(17)1 ¯ς(17)2 ¯ς(17)3
    ZPH 0.1325 0.0200 0.0200 0.0030
    MM 0.0154 0.14e-09 9.15e-09 4.41e-09

     | Show Table
    DownLoad: CSV
    Table 6.  Residual errors comparison on random initial vector.
    Method CPU ¯ς(37)1 ¯ς(37)2 ¯ς(37)3
    ZPHϑ1 5.4371 1.074e-20 3.398e-10 7.213e-18
    ZPHϑ2 5.1351 2.087e-29 6.385e-11 0.321e-19
    ZPHϑ3 3.1351 5.450e-27 7.378e-13 0.345e-17
    MMϑ1 2.1221 0.494e-37 8.375e-30 3.345e-35
    MMϑ2 1.8321 4.074e-35 0.378e-26 6.345e-35
    MMϑ3 2.9321 6.0548e-31 1.378e-26 1378e-28

     | Show Table
    DownLoad: CSV
    Table 7.  Residual computational errors comparison.
    Method CPU ¯ς(6)1 ¯ς(6)2 ¯ς(6)3 ¯ς(6)4
    ZPH 0.331 2.3e-23 2.3e-24 7.4e-33 3.1e-29
    MM 0.111 2.6e-30 2.6e-35 2.1e-35 2.3e-36

     | Show Table
    DownLoad: CSV
    Table 8.  Residual errors comparison on random initial vector.
    Method CPU ¯ς(11)1 ¯ς(11)2 ¯ς(11)3 ¯ς(11)4
    ZPHϑ1 2.1231 0.07e-30 2.23e-16 0.31e-18 8.73e-31
    ZPHϑ2 2.4131 2.02e-30 1.32e-16 0.53e-18 1.53e-31
    ZPHϑ3 3.1131 1.02e-30 5.35e-16 1.39e-18 8.03e-31
    MMϑ1 1.0212 9.24e-41 4.53e-30 4.32e-32 7.18e-35
    MMϑ2 1.8312 8.02e-47 2.30e-36 2.23e-28 8.23e-41
    MMϑ3 1.7315 7.07e-48 2.37e-36 2.13e-28 1.32e-35

     | Show Table
    DownLoad: CSV
    Table 9.  Residual computational errors comparison of ZPH & MM.
    Method CPU ¯ς(7)1 ¯ς(7)2 ¯ς(7)3 ¯ς(7)4
    ZPH 0.13121 3.02e-30 2.32e-16 2.32e-18 0.32e-31
    MM 0.02154 9.42e-36 4.13e-30 4.32e-32 7.81e-35

     | Show Table
    DownLoad: CSV
    Table 10.  Residual errors comparison on random initial vector.
    Method CPU ¯ς(10)1 ¯ς(10)2 ¯ς(10)3 ¯ς(10)4
    ZPHϑ1 2.031 3.50e-38 2.3e-18 4.38e-18 4.34e-33
    ZPHϑ2 3.534 0.01e-39 2.3e-26 5.87e-18 3.53e-36
    ZPHϑ3 4.1381 4.05e-35 4.31e-26 2.53e-17 2.34e-35
    MMϑ1 3.0914 9.46e-46 0.30e-30 4.3e-32 7.85e-37
    MMϑ2 1.0315 4.04e-45 2.35e-26 3.38e-28 8.13e-35
    MMϑ3 2.1931 1.40e-41 4.35e-26 9.32e-29 1.32e-35

     | Show Table
    DownLoad: CSV
    Table 11.  Residual computational error of simultaneous methods for the (5.9) root.
    Method CPU ¯ς(6)1 ¯ς(6)2 ¯ς(6)3 ¯ς(6)4 ¯ς(6)5 ¯ς(6)6 ¯ς(6)7 ¯ς(6)8 ¯ς(6)9
    ZPH 0.431 3.0e-30 2.3e-25 0.0 0.0 0.0 2.3e-29 2.3e-35 0.0 0.0
    MM 0.221 9.4e-36 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

     | Show Table
    DownLoad: CSV
    Table 12.  Residual errors comparison on random initial vector.
    Method CPU ¯ς(10)1 ¯ς(10)2 ¯ς(10)3 ¯ς(10)4 ¯ς(10)5 ¯ς(10)6 ¯ς(10)7 ¯ς(10)8 ¯ς(10)9
    ZPHϑ1 2.031 4.34e-23 3.50e-28 2.36e-18 4.38e-18 0.34e-23 0.51e-38 0.53e-28 1.31e-18 4.34e-33
    ZPHϑ2 3.534 0.11e-29 2.31e-26 2.31e-26 5.87e-15 3.13e-26 0.01e-39 0.31e-36 1.87e-18
    ZPHϑ3 4.1381 2.34e-35 4.15e-25 4.61e-26 2.53e-17 4.05e-35 1.34e-35 4.05e-35 0.311e-36 1.34e-25
    MMϑ1 3.0914 0.0 0.30e-30 1.30e-30 0.0 9.46e-46 0.0 0.0 5.3e-32 0.85e-47
    MMϑ2 1.0315 0.0 2.35e-46 3.38e-48 8.13e-45 4.04e-45 0.0 0.0 0.0 8.13e-35
    MMϑ3 2.1931 1.40e-47 4.35e-46 0.0 0.0 1.40e-41 4.35e-26 9.32e-29 1.32e-35 1.32e-35

     | Show Table
    DownLoad: CSV

    Example 1. An engineering application.

    A non-linear expression described in [30]

    ϑ3(ϰ)=ϰ47.79075ϰ3+14.7445ϰ2+2.511ϰ1.674, (5.1)

    depicts the minute modification of nitrogen hydrogen (NH) supply at 250 atm and 227 K. The exact roots of Eq (5.1) are 1=3.9485+0.3161i,2=3.94850.31610i,3=0.38410,4=0.2778. The initial guessed values for calculating ϑ3(ϰ) are taken as equal to (0)ϰ1=3.50+0.30i, (0)ϰ2=3.500.30i, (0)ϰ3=0.30+0.010i,(0)ϰ4=1.80+0.010i.

    On the basis of efficiency achieved after five iterations, Table 3 unequivocally shows that the MM technique outperforms the ZPH method. Table 1 displays the global convergence behavior of the proposed numerical schemes on various randomly generated initial guesses (see Appendix Table 13).

    In Table 4, ZPHϑ1ZPHϑ3 and MMϑ1MMϑ3 the numerical scheme ZPH and MM outputs for random initial vectors generated by the MATLAB computing tool. Numerical results of Table 4 clearly indicate numerical schemes are far better than ZPH for solving (5.1) and more complex engineering problems. The number of iterations and CPU time significantly increase when compared to the numerical results obtained in Table 3. Table 4 shows that for any initial vector choice, the method MM converges to exact roots with greater efficiency than ZPH.

    Example 2. Predator-prey model (PPM).

    Consider PPM in which predication rate [7] is denoted by

    ¯P(ϰ)=¯kϰ3¯a3+ϰ3,¯a,¯k>0, (5.2)

    where ϰ is the amount of aphids as victims [31] and female microbes as a predator. Malthusian Model holds G(ϰ)=rϰ, as the growth rate of aphids. For solution of the PPM we take ¯P(ϰ)=¯G(ϰ). This implies

    rϰ3¯kϰ2+i¯a3=0. (5.3)

    Taking ¯k=30 (the way pests are devoured), ¯a=20 (amount of bugs) and i=213 (hourly amount) in (5.3), we get:

    ϑ4(ϰ)=0.7937005260ϰ330ϰ2+6349.604208. (5.4)

    The exact roots of Eq (5.4) are 1=25.198,2=25.198,3=12.84. The initial guessed estimates for ϑ4(ϰ) are selected as:

    (0)ϰ1=1.8+8.7i,(0)ϰ2=1.88.7i,(0)ϰ3=0.1+0.1i.

    After seventeen iterations, Table 5 clearly shows that the MM methodology outperforms the ZPH method in terms of efficiency. Table 6 displays the global convergence behavior of the proposed numerical schemes on various randomly generated initial guesses (see Appendix Table 13).

    Table 6 shows the numerical schemes ZPHϑ1ZPHϑ3 and MMϑ1MMϑ3 outputs for random initial vectors generated by the MATLAB computing tool. The numerical results of the table clearly show that the proposed numerical schemes are superior to ZPH for solving (5.4) and more difficult engineering problems. The number of iterations and CPU time significantly increase when compared to the numerical results obtained in Table 5. Table 6 shows that for any initial vector choice, the method MM converges to exact roots with greater efficiency than ZPH on iteration 37.

    Example 3. Beam positioning problem.

    In this example, we look at a beam positioning problem [32], which yields a nonlinear function

    ϑ5(ϰ)=ϰ4+4ϰ324ϰ2+16ϰ+16. (5.5)

    The exact roots of Eq (5.5) are

    1=2,2=423,3=2,4=4+23.

    The initial estimates for ϑ3(ϰ) are

    (0)ϰ1,2=1.17,(0)ϰ3=7.4641,(0)ϰ4=0.5359.

    On the basis of efficiency achieved after six iterations, Table 7 unequivocally shows that the MM technique outperforms the ZPH method. Table 8 displays the global convergence behavior of the proposed numerical schemes on various randomly generated initial guesses (see Appendix Table 13).

    Table 8 shows the numerical schemes ZPHϑ1ZPHϑ3 and MMϑ1MMϑ3 outputs for random initial vectors generated by the MATLAB computing tool. The numerical results of the Table 8 clearly show that the proposed numerical schemes are superior to ZPH for solving (5.5) and more difficult engineering problems. The number of iterations and CPU time significantly increase when compared to the numerical results obtained in Table 7. Table 8 shows that for any initial vector choice, the method MM converges to exact roots with greater efficiency than ZPH on iteration 11.

    Example 4. [33] Nuclear boiler.

    Consider a nuclear boiler, with the substances ˘A˘A and ˇRˇR, breastfed to the apparatus at rates of ˆQ and q-ˆQ, respectively. The following are the results of the multifaceted reaction system produced in the container:

    ˘A˘A+ˇRˇRßß,ßß+ˇRˇRˇCˇC,ˇCˇC+ˇRˇRˇDˇD,ˇCˇC+ˇRˇR˘E˘E.

    Douglas et al. [34] tested this issue in order to create straightforward feedback control systems as:

    ˆH2.98(ϰ+2.25)(ϰ+1.45)(ϰ+2.85)2(ϰ+4.35)=1, (5.6)

    or

    ˆH2.98(ϰ+2.25)=(ϰ+1.45)(ϰ+2.85)2(ϰ+4.35)1, (5.7)

    where ˆH denotes the proportional controller's gain. This control system is stable for all values of ˆH that yield transfer function roots with negative real parts. Taking ˆH=0 in (5.7), we have:

    ϑ6(ϰ)=ϰ4+11.50ϰ3+47.49ϰ2+83.06325ϰ+51.23266875=0, (5.8)

    ϑ6(ϰ) has the four negative real roots listed below:

    1=1.45,2=2.85,3=2.85,4=4.45.

    We take the root as =4.45 and

    (0)ϰ1=1.0,(0)ϰ2=2.1,(0)ϰ3=1.8,(0)ϰ3=3.9,

    are chosen as initial estimates for finding all roots of (5.8) simultaneously.

    On the basis of efficiency achieved after seventh iterations, Table 9 unequivocally shows that the MM technique outperforms the ZPH method.

    Table 10 displays the global convergence behavior of the proposed numerical schemes on various randomly generated initial guesses (see appendix Table 13).

    Table 10 shows the numerical schemes ZPHϑ1ZPHϑ3 and MMϑ1MMϑ3 outputs for random initial vectors generated by the MATLAB computing tool. The numerical results of Table 10 clearly show that the proposed numerical schemes are superior to ZPH for solving (5.8) and more difficult engineering problems. The number of iterations and CPU time significantly increase when compared to the numerical results obtained in Table 9. Table 10 shows that for any initial vector choice, the method MM converges to exact roots with greater efficiency than ZPH on iteration 10.

    Example 5. [27,35,36] Polynomial of higher degree.

    Consider

    ϑ7(ϰ)=(ϰ+1)(ϰ+3)(ϰ22ϰ+2)(ϰ1)(ϰ24ϰ+5)(ϰ2+4ϰ+5), (5.9)

    with exact roots:

    1=1.0,2=3.0,3,4=1.0±i,5=1.0,6,7=2.0±i,8,9=2.0±1.0i.

    The initial approximations of (5.9) have been taken as:

    (0)ϰ1=1.30+0.20i,(0)ϰ2=2.800.20i,(0)ϰ3=1.2+1.3i,(0)ϰ4=0.81.20i,(0)ϰ5=0.800.30i,(0)ϰ6,7=1.8±1.2i,(0)ϰ8,9=1.80±0.8i.

    On the basis of efficiency achieved after sixth iterations, Table 11 unequivocally shows that the MM technique outperforms the ZPH method. Table 12 displays the global convergence behavior of the proposed numerical schemes on various randomly generated initial guesses (see Appendix Table 13).

    Table 12 shows the numerical schemes ZPHϑ1ZPHϑ3 and MMϑ1MMϑ3 outputs for random initial vectors generated by the MATLAB computing tool. The numerical results of the Table 12 clearly show that the proposed numerical schemes are superior to ZPH for solving (5.9) and more difficult engineering problems. The number of iterations and CPU time significantly increase when compared to the numerical results obtained in Table 11. Table 12 shows that for any initial vector choice, the method MM converges to exact roots with greater efficiency than ZPH on iteration 35.

    ● In comparison to the well-known Weierstrass-type method ZPH, our newly developed methods is more efficient and stable.

    ● The results of the numerical test problems from Tables 3-12, computational efficiency from Figure 3, residual error graphs from Figure 4(a-e), and dynamical planes from Figures 1(a-h) and 2(a-h) show that the proposed family of iterative techniques MM outperforms ZPH in terms of efficiency and convergence behavior.

    Figure 4.  (a–e) show residual graphs of simultaneous methods MM and ZPH for nonlinear polynomials ϑ3(ϰ)ϑ7(ϰ), respectively.

    ● Furthermore, the elapsed time of iterative technique MM from Table 1 is faster than ZPH.

    ● This paper proposes a family of Weierstrass-type derivative-free simultaneous techniques of order five. Tables 112 and Figures 14 show that our MM techniques outperform the Weierstrass-type method ZPH in terms of efficiency, CPU time, basins of attraction, and residual errors. Color brightness inspection in the basins reveals that MM requires fewer iteration steps than the simultaneous method ZPH. The results of the numerical test examples in terms of CPU-time and residual errors demonstrate the effectiveness and rapid convergence of our proposed derivative-free family of iterative algorithms, MM, in comparison to ZPH.

    In summary, the comprehensive empirical results establish that these novel MM techniques outperform ZPH in terms of efficiency, CPU time, basins of attraction, and residual errors. Examination of the color brightness in the basins reveals MM requires fewer iteration steps than ZPH. Overall, the test examples clearly demonstrate the effectiveness, rapid convergence, and computational advantages of the proposed MM algorithms over ZPH.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The work is supported by Provincia autonoma di Bolzano/Alto Adigeâ euro Ripartizione Innovazione, Ricerca, Universitá e Musei (contract nr. 19/34). Bruno Carpentieri is a member of the Gruppo Nazionale per it Calcolo Scientifico (GNCS) of the Istituto Nazionale di Alta Matematia (INdAM) and this work was partially supported by INdAM-GNCS under Progetti di Ricerca 2022.

    All authors declare no conflicts of interest in this paper.

    Table 13.  Random initial vectors generated using built-in commands in MATLAB.
    Inital Vector (0)ϰ1 (0)ϰ2 (0)ϰ3 (0)ϰ4 (0)ϰ5 (0)ϰ6 (0)ϰ7 (0)ϰ8 (0)ϰ9
    ϑ(0)1 0.1234 0.2541 0.0415 0.04125 0.25410 025412 0.5412 0.0354 0..35214
    ϑ(0)2 0.2134 0.5412 0.03251 0.05412 0.10351 0.0541 0.09854 0.0214 0.874515
    ϑ(0)3 0.3251 0.6541 0.210 0.14521 0.2321 0.0012 0.5214 0.2145 0.022145

     | Show Table
    DownLoad: CSV


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