Four-Point Optimal Sixteenth-Order Iterative Method for Solving Nonlinear Equations

Abstract

We present an iterative method for solving nonlinear equations. The proposed iterative method has optimal order of convergence sixteen in the sense of Kung-Traub conjecture (Kung and Traub, 1974); it means that the iterative scheme uses five functional evaluations to achieve 16(=2⁵⁻¹) order of convergence. The proposed iterative method utilizes one derivative and four function evaluations. Numerical experiments are made to demonstrate the convergence and validation of the iterative method.

1. Introduction

According to Kung and Traub conjecture, a multipoint iterative method without memory could achieve optimal convergence order 2^𝑛−1 by performing 𝑛 evaluations of function or its derivatives [1]. In order to construct an optimal sixteenth-order convergent iterative method for solving nonlinear equations, we require four and eight optimal-order iterative schemes. Many authors have been developed the optimal eighth-order iterative methods, namely, Bi et al. [2], Bi et al. [3], Geum and Kim [4], Liu and Wang [5], Wang and Liu [6], and Soleymani et al. [7–9]. Some recent applications of nonlinear equation solvers in matrix inversion for regular or rectangular matrices have been introduced in [10–12].

For the proposed iterative method, we developed new optimal fourth- and eighth-orders iterative methods to construct optimal sixteenth-order iterative scheme. On the other hand, it is known that rational weight functions give a better convergence radius. By keeping this fact in mind, we introduced rational terms in weight functions to achieve optimal sixteenth order.

For the sake of completeness, we list some existing optimal sixteenth-order convergent methods. Babajee and Thukral [13] suggested 4-point sixteenth-order king family of iterative methods for solving nonlinear equations (BT): 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛,𝑧_𝑛=𝑦_𝑛−1+𝛽𝑡₁1+𝛽−2𝑡₁𝑓𝑦_𝑛𝑓^′𝑥_𝑛,𝑤_𝑛=𝑧_𝑛−𝜃₀+𝜃₁+𝜃₂+𝜃₃𝑓𝑦_𝑛𝑓^′𝑥_𝑛,𝑥_𝑛+1=𝑤_𝑛−𝜃₀+𝜃₁+𝜃₂+𝜃₃+𝜃₄+𝜃₅+𝜃₆+𝜃₇𝑓𝑤_𝑛𝑓^′𝑥_𝑛, (1) where 𝑡₁=𝑓𝑦_𝑛𝑓𝑥_𝑛,𝑡₂=𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑡₃=𝑓𝑧_𝑛𝑓𝑦_𝑛,𝑡₄=𝑓𝑤_𝑛𝑓𝑥_𝑛,𝑡₅=𝑓𝑤_𝑛𝑓𝑧_𝑛,𝑡₆=𝑓𝑤_𝑛𝑓𝑦_𝑛,𝜃₀=1,𝜃₁=1+𝛽𝑡₁+3/2𝛽𝑡²₁1+𝛽−2𝑡₁+3/2𝛽−1𝑡²₁𝜃−1,₂=𝑡₃,𝜃₃=4𝑡₂,𝜃₄=𝑡₅+𝑡₁𝑡₂,𝜃₅=2𝑡₁𝑡₅+41−𝛽𝑡³₁𝑡₃+2𝑡₂𝑡₃,𝜃₆=2𝑡₆+7𝛽²−472𝛽+14𝑡₃𝑡⁴₁+2𝛽−3𝑡²₂+5−2𝛽𝑡₅𝑡²₁−𝑡³₃,𝜃₇=8𝑡₄+−12𝛽+2𝛽²+12𝑡₅𝑡³₁−4𝑡³₃𝑡₁+−2𝛽²+12𝛽−22𝑡²₃𝑡³₁+−10𝛽³+1272𝛽²−105𝛽+46𝑡₂𝑡⁴₁. (2) In 2011, Geum and Kim [14] proposed a family of optimal sixteenth-order multipoint methods (GK2): 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛,𝑧_𝑛=−𝐾_𝑓𝑢_𝑛𝑓𝑦_𝑛𝑓^′𝑥_𝑛,𝑠_𝑛=𝑧_𝑛−𝐻_𝑓𝑢_𝑛,𝑣_𝑛,𝑤_𝑛𝑓𝑧_𝑛𝑓^′𝑥_𝑛,𝑥_𝑛+1=𝑠_𝑛−𝑊_𝑓𝑢_𝑛,𝑣_𝑛,𝑤_𝑛,𝑡_𝑛𝑓𝑠_𝑛𝑓^′𝑥_𝑛, (3) where 𝑢_𝑛=𝑓𝑦_𝑛𝑓𝑥_𝑛,𝑣_𝑛=𝑓𝑧_𝑛𝑓𝑦_𝑛,𝑤_𝑛=𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑡_𝑛=𝑓𝑠_𝑛𝑓𝑧_𝑛,𝐾_𝑓𝑢_𝑛=1+𝛽𝑢_𝑛+−9+5/2𝛽𝑢²_𝑛1+𝛽−2𝑢_𝑛+−4+𝛽/2𝑢²_𝑛,𝐻_𝑓=1+2𝑢_𝑛+(2+𝜎)𝑤_𝑛1−𝑣_𝑛+𝜎𝑤_𝑛,𝑊_𝑓=1+2𝑢_𝑛1−𝑣_𝑛−2𝑤_𝑛−𝑡_𝑛+𝐺𝑢_𝑛,𝑣_𝑛,𝑤_𝑛, (4) one of the choices for 𝐺(𝑢_𝑛,𝑣_𝑛,𝑤_𝑛) along with 𝛽=24/11 and 𝜎=−2: 𝐺𝑢_𝑛,𝑣_𝑛,𝑤_𝑛=−6𝑢³_𝑛𝑣_𝑛−244𝑢11^4𝑛𝑤_𝑛+6𝑤²_𝑛+𝑢_𝑛2𝑣²_𝑛+4𝑣³_𝑛+𝑤_𝑛−2𝑤²_𝑛. (5) In the same year, Geum and Kim [15] presented a biparametric family of optimally convergent sixteenth-order multipoint methods (GK1): 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛,𝑧_𝑛=−𝐾_𝑓𝑢_𝑛𝑓𝑦_𝑛𝑓^′𝑥_𝑛,𝑠_𝑛=𝑧_𝑛−𝐻_𝑓𝑢_𝑛,𝑣_𝑛,𝑤_𝑛𝑓𝑧_𝑛𝑓^′𝑥_𝑛,𝑥_𝑛+1=𝑠_𝑛−𝑊_𝑓𝑢_𝑛,𝑣_𝑛,𝑤_𝑛,𝑡_𝑛𝑓𝑠_𝑛𝑓^′𝑥_𝑛, (6) where 𝑢_𝑛=𝑓𝑦_𝑛𝑓𝑥_𝑛,𝑣_𝑛=𝑓𝑧_𝑛𝑓𝑦_𝑛,𝑤_𝑛=𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑡_𝑛=𝑓𝑠_𝑛𝑓𝑧_𝑛,𝐾_𝑓𝑢_𝑛=1+𝛽𝑢_𝑛+−9+5/2𝛽𝑢²_𝑛1+𝛽−2𝑢_𝑛+−4+𝛽/2𝑢²_𝑛,𝐻_𝑓=1+2𝑢_𝑛+(2+𝜎)𝑤_𝑛1−𝑣_𝑛+𝜎𝑤_𝑛,𝑊_𝑓=1+2𝑢_𝑛+(2+𝜎)𝑣_𝑛𝑤_𝑛1−𝑣_𝑛−2𝑤_𝑛−𝑡_𝑛+2(1+𝜎)𝑣_𝑛𝑤_𝑛+𝐺𝑢_𝑛,𝑤_𝑛, (7) one of the choices for 𝐺(𝑢_𝑛,𝑤_𝑛) along with 𝛽=2 and 𝜎=−2: 𝐺𝑢_𝑛,𝑤_𝑛1=−2𝑢_𝑛𝑤_𝑛6+12𝑢_𝑛+24−11𝛽𝑢²_𝑛+𝑢³_𝑛𝜙₁+4𝜎+𝜙₂𝑤²_𝑛,𝜙₁=11𝛽²,𝜙−66𝛽+136₂=2𝑢_𝑛𝜎²−2𝜎−9−4𝜎−6. (8)

2. A New Method and Convergence Analysis

The proposed sixteenth-order iterative method is described as follows (MA): 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛,𝑧_𝑛=𝑦_𝑛−1+2𝑡₁−𝑡²₁1−6𝑡²₁𝑓𝑦_𝑛𝑓^′𝑥_𝑛,𝑤_𝑛=𝑧_𝑛−1−𝑡₁+𝑡₃1−3𝑡₁+2𝑡₃−𝑡₂𝑓𝑧_𝑛𝑓^′𝑥_𝑛,𝑥_𝑛+1=𝑤_𝑛−𝑞₁+𝑞₂+𝑞₃+𝑞₄+𝑞₅+𝑞₆+𝑞₇𝑓𝑤_𝑛𝑓^′𝑥_𝑛, (9) where 𝑡₁=𝑓𝑦_𝑛𝑓𝑥_𝑛,𝑡₂=𝑓𝑧_𝑛𝑓𝑦_𝑛,𝑡₃=𝑓𝑧_𝑛𝑓𝑥_𝑛,𝑡₄=𝑓𝑤_𝑛𝑓𝑥_𝑛,𝑡₅=𝑓𝑤_𝑛𝑓𝑦_𝑛,𝑡₆=𝑓𝑤_𝑛𝑓𝑧_𝑛,𝑞₁=11−2𝑡₁+𝑡²₁+𝑡³₁+𝑡⁴₁+𝑡⁵₁+𝑡⁶₁+𝑡⁷₁,𝑞₂=4𝑡₃1−31/4𝑡₃,𝑞₃=𝑡₂1−𝑡₂−20𝑡³₂,𝑞₄=8𝑡₄1−𝑡₄+2𝑡₅1−𝑡₅+𝑡₆1−𝑡₆,𝑞₅=15𝑡₁𝑡₃1−131/15𝑡₃,𝑞₆=54𝑡²₁𝑡₃1−𝑡²₁𝑡₃,𝑞₇=7𝑡₂𝑡₃+2𝑡₁𝑡₆+6𝑡₆𝑡²₁+188𝑡₃𝑡³₁+18𝑡₆𝑡³₁+9𝑡²₂𝑡₃+648𝑡⁴₁𝑡₃. (10)

Theorem 1. Let 𝑓∶𝐷⊆ℝ→ℝ be a sufficiently differentiable function, and 𝛼∈𝐷 is a simple root of 𝑓(𝑥)=0, for an open interval 𝐷. If 𝑥₀ is chosen sufficiently close to 𝛼, then the iterative scheme (9) converges to 𝛼 and shows an order of convergence at least equal to sixteen.

Proof. Let error at step 𝑛 be denoted by 𝑒_𝑛=𝑥_𝑛−𝛼 and 𝑐₁=𝑓^′(𝛼) and 𝑐_𝑘=(1/𝑘!)(𝑓^(𝑘)(𝛼)/𝑓^′(𝛼)), 𝑘=2,3,…. We provided Maple based computer assisted proof in Algorithm 1 and got the following error equation: 𝑒_𝑛+1=−𝑐₄𝑐₃𝑐²₂𝑐₅𝑐₃𝑐²₂+2𝑐₄𝑐₂𝑐²₃−20𝑐⁴₃−51𝑐³₃𝑐²₂+522𝑐²₃𝑐⁴₂−2199𝑐₃𝑐⁶₂+2𝑐⁸₂−30𝑐₄𝑐₃𝑐³₂+54𝑐₄𝑐⁵₂𝑒_𝑛¹⁶+𝑂𝑒_𝑛¹⁷. (11)

fxn:= c1*(e+c2*e∧2+c3*e∧3+c4*e∧4+c5*e∧5+c6*e∧6+c7*e∧7+c8*e∧8+c9*e∧9+c10*e∧10+     c11*e∧11+c13*e∧12+c13*e∧13+c14*e∧14+c15*e∧15+c16*e∧16+c17*e∧17); dfxn:= diff (fxn, e); ye:= simplify (taylor (e−fxn/dfxn, e = 0, 17)); fyn:= c1*(ye+c2*ye∧2+c3*ye∧3+c4*ye∧4+c5*ye∧5+c6*ye∧6+c7*ye∧7+c8*ye∧8+c9*ye∧9+     c10*ye∧10+c11*ye∧11+c12*ye∧12+c13*ye∧13+c14*ye∧14+c15*ye∧15+c16*ye∧16+c17*ye∧17); fyn:= simplify (taylor (fyn, e = 0, 17)); t1:= simplify (taylor (fyn/fxn, e = 0, 17)); fydfx:= simplify (taylor (fyn/dfxn, e = 0, 17)); ze:= factor (simplify (taylor (ye−(1+2*t1−t1∧2)*fydfx/(1−6*t1∧2), e = 0, 17))); factor (simplify (taylor (ye−(1+2*t1−t1∧2)*fydfx/(1−6*t1∧2), e = 0, 7)));     −c2*c3*e∧4+(−2*c3∧2+2*c3*c2∧2+2*c2∧4−2*c2*c4)*e∧5+(−3*c2*c5−7*c3*c4+6*c2*c3∧2+       12*c3*c2∧3+3*c4*c2∧2−14*c2∧5)*e∧6+O(e∧7)   fzn:= c1*(ze+c2*ze∧2+c3*ze∧3+c4*ze∧4+c5*ze∧5+c6*ze∧6+c7*ze∧7+c8*ze∧8+c9*ze∧9+     c10*ze∧10+c11*ze∧11+c12*ze∧12+c13*ze∧13+c14*ze∧14+c15*ze∧15+c16*ze∧16+c17*ze∧17); fzn:= simplify (taylor (fzn, e = 0, 17)); t2:= simplify (taylor (fzn/fyn, e = 0, 17)); t3:= simplify (taylor (fzn/fxn, e = 0, 17)); fzdfx:= simplify (taylor (fzn/dfxn, e = 0, 17)); we:= simplify (taylor (ze−(1−t1+t3)*fzdfx/(1−3*t1+2*t3−t2), e = 0, 17)); simplify (taylor (ze−(1−t1+t3)*fzdfx/(1−3*t1+2*t3−t2), e = 0, 10));       −c4*c3*c2∧2*e∧8+(−2*c5*c3*c2∧2−4*c2*c4*c3∧2+4*c3*c4*c2∧3−2*c4∧2*c2∧2+2*c4*c2∧5     −3*c3∧3*c2∧2+4*c3∧2*c2∧4+4*c3*c2∧6)*e∧9+O(e∧10)   fwn:= c1*(we+c2*we∧2+c3*we∧3+c4*we∧4+c5*we∧5+c6*we∧6+c7*we∧7+c8*we∧8+c9*we∧9+c10*we∧10+      c12*we∧12+c13*we∧13+c14*we∧14+c15*we∧15+c16*we∧16+c17*we∧17); fwn:= simplify (taylor (fwn, e = 0, 17)); fwdfx:= simplify (taylor (fwn/dfxn, e = 0, 17)); t4:= simplify (taylor (fwn/fxn, e = 0, 17)); t5:= simplify (taylor (fwn/fyn, e = 0, 17)); t6:= simplify (taylor (fwn/fzn, e = 0, 17)); q1:= simplify (taylor (1/(1−2*(t1+t1∧2+t1∧3+t1∧4+t1∧5+t1∧6+t1∧7)), e = 0, 17)); q2:= simplify (taylor (4*t3/(1−(31/4)*t3), e = 0, 17)); q3:= simplify (taylor (t2/(1−t2−20*t2∧3), e = 0, 17)); q4:= simplify (taylor (8*t4/(1−t4)+2*t5/(1−t5)+t6/(1−t6), e = 0, 17)); q5:= simplify (taylor (15*t1*t3/(1−(131/15)*t3), e = 0, 17)); q6:= simplify (taylor (54*t1∧2*t3/(1−t1∧2*t3), e = 0, 17)); q7:= simplify (taylor (7*t2*t3+2*t1*t6+6*t6*t1∧2+188*t3*t1∧3+18*t6*t1∧3+               9*t2∧2*t3+648*t1∧4*t3, e = 0, 17)); x[n+1]:= simplify (taylor (we−fwdfx*(q1+q2+q3+q4+q5+q6+q7), e = 0, 17)); for i to 16 do p:= factor (simplify (coeff (x[n+1], e, i))) end do;               0               0               0               0               0               0               0               0               0               0               0               0               0               0               0  −c4*c3*c2∧2*(c5*c3*c2∧2+2*c2*c4*c3∧2−20*c3∧4−51*c3∧3*c2∧2+522*c3∧2*c2∧4−2199*c3*c2∧6  +2*c2∧8−30*c3*c4*c2∧3+54*c4*c2∧5)

Algorithm 1 

The Maple code for finding the error equation.

3. Numerical Results

If the convergence order 𝜂 is defined as lim_𝑥→∞𝑒_𝑛+1𝑒_𝑛^𝜂=𝑐≠0, (12) then the following expression approximates the computational order of convergence (COC) [16] as follows: 𝑥𝜌≈ln_𝑛+1−𝛼/𝑥_𝑛−𝛼ln𝑥_𝑛−𝛼/𝑥_𝑛−1−𝛼, (13) where 𝛼 is the root of nonlinear equation. A set of five nonlinear equations are used for numerical computations in Table 1. Three iterations are performed to calculate the absolute error (|𝑥_𝑛−𝛼|) and computational order of convergence. Table 2 shows absolute error and computational order of convergence, respectively.

Table 1 

Set of six test nonlinear functions.

Table 2 

Numerical comparison of absolute error |𝑥_𝑛−𝛼|, number of iterations = 3.

4. Conclusion

An optimal sixteenth-order iterative scheme has been developed for solving nonlinear equations. A Maple program is provided to calculate error equation, which actually shows the optimal order of convergence in the sense of Kung-Traub conjecture. The computational order of convergence also verifies our claimed order of convergence. The proposed scheme uses four functions and one derivative evaluation per full cycle, which gives 1.741 as the efficiency index. We also have shown the validity of our proposed iterative scheme by comparing it with other existing optimal sixteenth-order iterative methods. The numerical results show that the performance of iterative scheme is competitive as compared to other methods.

Acknowledgments

The first and the second authors were funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. They, therefore, acknowledge with thanks DSR technical and financial support while the third author is supported for this research under the Spanish MEC Grants AYA2010-15685.