Page 135 - IJOCTA-15-3

A novel ninth-order root-finding algorithm for nonlinear equations with implementations in various ...
and the first and second derivatives are Thus, the proposed algorithm has an optimal log-
1
well-behaved. That is, there exists a con- arithmic complexity of O(log( )).
∈
′′
′
stant L such that |f (x)| ≤ L|f (x)|. As discussed previously, we have proven
(ii) The initial guess x 0 must satisfy that the semi-local convergence of the proposed
′
|f(x 0 )f (x 0 )| < 2 , ensuring that the method has a ninth-order rate. We assumed that:
L
3
error does not grow uncontrollably. (i) f ∈ C is on an open interval D ⊂ R, (ii)
′
Sensitivity to initial conditions: although a α ∈ D is a simple root of f : f(α) = 0, f (α) ̸= 0,
′
higher order of convergence reduces sensitivity to and there exists δ > 0 such that: f (x) ̸= 0 for
′′ f (3)

initial conditions, the algorithm may still behave f (x) (x) are
′
′
f (x) , and f (x)
all x ∈ B(α, δ) ⊂ D,
unpredictably for functions with closely spaced bounded in B(α, δ). Now, the goal was to prove
roots. Nevertheless, it tends to be more stable that the method converges α from an initial point
than Newton-type methods. x 0 ∈ B(α, δ), under these assumptions. The key
If the function satisfies smoothness and ideas in the proof consist of analyzing the error
bounded derivative conditions, the method is e n = x n − α and showing that
guaranteed to converge for a broad range of ini-
tial guesses. The exponential correction step ex- |e n+1 | ≤ K|e n | 9
pands the region of attraction, improving robust-
for some constant K > 0, when |e n | is suf-
ness. The summary of the convergence properties ficiently small but not necessarily infinitesimal
is provided in Table 1. (semi-local).
The proposed root-finding algorithm exhib- In the literature, numerous root-finding algo-
ited high stability due to its rapid convergence rithms exist (see, e.g., [39–43]) each with its own
and reduced function evaluations per iteration. limitations.
Stability was analyzed using fixed-point theory
and the basin of attraction, ensuring that small 3. Numerical experiments
perturbations in the initial guess do not lead to
divergence. The method’s high-order accuracy al- In this section, the practical applicability of the
lowed for faster convergence while minimizing nu- proposed scheme is demonstrated using numerical
merical errors. Polynomiography results in Sec- examples.
tion 5 indicate that the method has wider and Example 1. Consider the nonlinear equation
smoother basins of attraction, reducing sensitiv- to illustrate the proposed algorithm:
ity to initial conditions compared to NR and HM. 2 2
In the following section, we present several sin (x) − x + 1 = 0. (13)
numerical examples to illustrate the proposed al- An approximate solution of Equation (13) is
gorithm, and comparisons are made to show its
accuracy and efficiency. x ≈ 1.404491648215341226035086817786868077
As mentioned in Section 1, the efficiency in- (14)
dex E is defined as E = p 1/c , where is the order of Using the proposed algorithm with the initial
convergence of the method and c is the number of guess x 0 = 1, we obtained the following values in
function evaluations per iteration. The proposed the first iteration:
method has ninth-order convergence (p = 9) and
six function evaluations per iteration (c = 6), con- 2f(x 0 )f (x 0 )
′
sisting of three function evaluations (x), two first t 0 = x 0 − ′ 2 ′′ = 1.352266356364,
′
derivative evaluations (f (x)), and one second de- 2f (x 0 ) − f(x 0 )f (x 0 )

′′
rivative evaluation (f (x)). An explicit calcula- s 0 = t 0 exp −f(t 0 ) = 1.40790110417003320,
′
tion gives E = 9 1/6 ≈ 1.4422. t 0 f (t 0 )
The computational complexity of an iterative f(t 0 ) + f(s 0 )
x 1 = t 0 − = 1.4030669959818645244254.
root-finding method is determined by the follow- f (t 0 )
′
ing: (i) the number of function evaluations per The function value and absolute error after
iteration, (ii) the arithmetic operations per iter- the first iteration are:
ation, and (iii) the convergence order and total
iterations required. As mentioned earlier, each it-
|f(x 1 )| = 0.00353271303535116810231715,
eration requires six function evaluations. As each
step involves only a constant number of opera- x 1 − x 0 = 0.2872756590641623253773259

tions, the arithmetic complexity per iteration is x 1
(1), and the number of iterations required to reach respectively. Repeating this process, we have:
1
a precision ϵ is approximately k = O(log ( )). Iteration 2
9 ∈
507