Page 134 - IJOCTA-15-3

Srinivasarao Thota et al. / IJOCTA, Vol.15, No.3, pp.503-516 (2025)
2.1. Local convergence analysis
3
′ 2
′
′′
′
Local convergence analysis examines how the it- f (x j ) f (x j ) = f (α) e j + f (α) f (α) e 2
j
2
erative method behaves when the initial approx-
1 ′ ′′′ ′′ 2 3 4
imation is sufficiently close to the root α. The + f (α) f (α) + f (α) e + φ(e ).
j
j
2
aim was to determine the order of convergence by
Now, the Taylor expansion of the denominator
deriving an error equation.
in the fractional term in the first step of Equation
Let α be a simple root of (x), i.e., f(α) =
′
0, f (α) ̸= 0. Expanding f(x) in a Taylor series (9), multiplied by its numerator, results in:
around α, we obtain: ′
2f (x j ) f (x j ) 1 ′′ 2 3
= e j − 4f (α) f (α) e j
2
′ 2
′′
′
2f (x j ) − f (x j ) f (x j )
′′
′
3
2
f(x) = f (α)(x−α)+ f (α) (x − α) +O((x − α) ) 1
′
′′′
2 − f (α) 5f (α) f (α) − 6f ′′ 2 (α) e 4
′′
j
′ 3
Similarly, the derivatives are: 4f (α)
5
+ φ(e ),
j
′′
′
′
2
f (x) = f (α) + f (α)(x − α) + O((x − α) ), Following the proof of Theorem 1 in Kalantari
. 38
′′
′′
f (x) = f (α) + O(x − α). and Hans Lee, we obtain:
The error at the nth iteration is defined as f (α) 8
′′
9
10
e n = x n − α. The following theorem guarantees e j+1 = c ′ e + φ(e ),
j
j
f (α)
that the nonlinear three-step scheme presented in
where c is a constant. Hence, the proposed
Equation (9) for solving f(x) = 0 has ninth-order
algorithm has ninth-order convergence. □
accuracy.
Theorem 2. Let f : I → R. Suppose α ∈ I
Theorem 1. Let α be the root of a sufficiently is a simple root of f(x) = 0 and δ is a suffi-
differentiable function f : θ → R, where θ ⊆ R ciently small neighborhood of α. Let f (x) ex-
′′
is an open interval. Then, the three-step scheme ists and f (x) ̸= 0 in δ. Then the iterative
′′
presented in Equation (9) has ninth-order conver- Equation (9) produces a sequence of iterations
gence. {x n : n = 0, 1, 2, . . . } with the order of conver-
gence nine
Proof. The proof of this theorem is similar to
that of proof of Theorem 1 in Kalantari and Hans Proof. Let
Lee. 38 For simplicity, we include the main steps
of the proof. R(x) = t − f(t) + f(s) ,
′
Let α be the root of f(x) = 0, x j be the jth f (t)
approximation to the root provided by Equation where
(IX), and
′
−f(t) 2f(x)f (x)
s = t exp and t = x− .
2
′
′′
′
e j = x j − α (10) tf (t) 2f (x) − f(x)f (x)
5
be the error term after the jth iteration. Us- As in Abbasbandy, we observed that:
ing Taylor’s expansion for f(x j ) about α, we ob-
tain: ′ (8) (9)
R(α) = α, R (α) = 0, . . . , R (α) = 0, R (α) ̸= 0.
1 Hence, the proposed algorithm has conver-
2
′′
′
3
f (x j ) = f (α) e j + f (α) e + φ(e ). (11) gence of order nine. □
j
j
2
3
Here, φ(e ) describes the asymptotic behavior
j 2.2. Semi-local convergence analysis
of a function Equation (10) as e n → 0, particu-
larly focusing on the dominant order of smallness. Semi-local analysis examines the behavior of the
It represents an upper bound on the rate at which algorithm when the initial guess is not arbitrarily
the function shrinks near the root. Expanding close to the root. It ensures that the algorithm
f (x j ) using Taylor’s series about α, we also have: converges for a broader class of functions.
Kantorovich-type conditions: To ensure semi-
1 ′′′ 2 3 local convergence, we verified whether the func-
′′
′
′
f (x j ) = f (α) + f (α)e j + f (α)e + φ(e ) tion satisfies the following:
j
j
2
(12) (i) The function (x) is continuously differen-
Multiplying Equations (11) and (12), we get: tiable in a neighborhood around the root,
506