EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 3, 2022, 1301-1306
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A New Secant Type Method with Quadratic-Order
Convergence

Zinah F. Salih1,∗, Basim A. Hassan1, Barah M. Sulaiman1

1 Department of Mathematics, College of Computers Sciences and Mathematics,
University of Mosul, Iraq

Abstract. Using the development of the second approximation, a variation of the standard secant
technique for nonlinear problems has been developed. The iterative formula is created using Taylor
series expansion, which includes an estimate of the second derivative of Θ(µ). It is demonstrated
that the new approaches have quadratic convergence. In comparison with the Newton technique
employing seven test functions in the practical application, it turns out that the performance of
the method is efficient.

2020 Mathematics Subject Classifications: 65K10

Key Words and Phrases: New Secant Type Method, Iteration Methods, Test functions

1. Introduction

Consider the scalar function Θ, which is determined by a single independent variable,
µ. Assume we wish to determine the value of µ where Θ(µ) is the minimum value, as
shown in [11].

Gradient-based minimization approaches look for spots that satisfy the optimality
conditions to find a local minimum. Finding µ∗ that satisfies the first-order optimality
requirements, i.e.:

Θ
′
(µ∗) = 0 (1)

is equivalent to finding the roots of the function to be a minimum of its first derivative.
Because of this, root finding methods are effective in function reduction and may be used
to identify stationary locations. See [2][7] for further information.

In regard to different approximation approaches in terms of convergence, Newton’s
method is the most appealing, as illustrated in [10][1]. It works by locally approximating
the objective function using a quadratic model that decides on it at a given place. The

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4460

Email addresses: zn−f2020@uomosul.edu.iq (Z. F. Salih),
basimah@uomosul.edu.iq (B. A. Hassan), barah−mahmood82@uomosul.edu.iq (B. M. Sulaiman)

https://www.ejpam.com 1301 © 2022 EJPAM All rights reserved.



Z. F. Salih, B. A. Hassan, B. M. Sulaiman / Eur. J. Pure Appl. Math, 15 (3) (2022), 1301-1306 1302

technique can be recurrent indefinitely until the approximation function is optimal. New-
ton’s approach, on the other hand, is widely recognized for requiring assessment of the 1st

and 2nd derivatives in each iteration, as shown by the iteration formula: See [14][13] for
further information.

µi+1 = µi −
Θ

′
(µi)

Θ′′(µi)
(2)

To avoid this, certain Newton-type methods are referred to as secant methods since
they do not require the second derivative. Due to the time-consuming and difficulty
involved in determining the second derivative of a function, the formula for iteration,
often known as:

µi+1 = µi −
Θ

′
(µi)(µi − µi−1)

Θ′(µi)−Θ′(µi−1)
(3)

having been developed, more information may be found in [8]. The calculation of Θ and
its derivatives consumes the majority of the computational time. Numerous studies have
focused on nonlinear issues that do not require the second derivative (see [10][14][4] and
[3], despite the fact that a lot of new modified techniques with a higher convergence order
have been proposed (e.g. [5][6][15][9]).

The derivation of a secant approach based on the second-order Taylor expansion is
presented in this study. In comparison to the Newton method, by employing seven test
functions, the new method’s performance is evaluated using the most common and exten-
sively used criterion: the iterations’ number.

2. A New Secant Type Method

An important category of nonlinear optimization problems is the second-order Taylor
expansion. We have [12] for the second-order Taylor expansion around the point µi:

Θ(µ) ∼= Θ(µi) + Θ
′
(µi)(µ− µi) +

1

2
Θ

′′
(c)(µ− µi)

2 (4)

The minimal position at which the derivative of Θ(µ) equals zero may be found by:

Θ
′
(µ) ∼= Θ

′
(µi) + Θ

′′
(µi)(µ− µi) = 0 (5)

We got the following results from Eqs. (4) and (5):

Θ
′′
(µi)(µ− µi)

2 ∼= Θ(µi)−Θ(µ)− 1

2
Θ

′
(µi)(µ− µi) (6)

This results in:

Θ
′′
(µi) =

Θ(µi)−Θ(µ)− 1
2Θ

′
(µi)(µ− µi)

(µ− µi)2
(7)

The preceding equation may be simplified by substituting µi−1 for µ:

Θ
′′
(µi) =

Θ(µi)−Θ(µi−1)− 1
2Θ

′
(µi)(µi−1 − µi)

(µi−1 − µi)2
(8)



Z. F. Salih, B. A. Hassan, B. M. Sulaiman / Eur. J. Pure Appl. Math, 15 (3) (2022), 1301-1306 1303

When we plug Eq. (8) into Eq. (2), we get:

µi+1 = µi −
Θ

′
(µi)(µi−1 − µi)

2

Θ(µi)−Θ(µi−1)− 1
2Θ

′(µi)(µi−1 − µi)
(9)

which is the new secant type method.
Based on a new secant type technique, the following algorithm is presented:

1. Put ϵ, µ0 (initial point) and the function Θ(µ0), set i = 0.

2. Let i = i+ 1.

3. Set µi = µi−1 − Θ
′
(µi−1)

Θ′′ (µi−1)

4. Compute µi+1 = µi − Θ
′
(µi)(µi−1−µi)

2

Θ(µi)−Θ(µi−1)− 1
2
Θ′ (µi)(µi−1−µi)

.

5. Stop, if the absolute value of |µi+1 − µi| ≤ ϵ.

Convergence rate refers to how quickly an iterative process approaches the numerical
solution.

3. Analysis of Convergence

A more explicit investigation of the convergence of the new secant iteration has been
done since it is critical for the development of optimization algorithms.

Theorem 1. Let a sufficiently differentiable function Θ : I → R, where I is an open
interval and let µ∗ ∈ I be a zero of Θ. If µ0 converges to µ∗, then the new secant type
approach has a quadratic convergence rate.

Proof. The new secant’s type method is as:

µi+1 = µi −
Θ

′
(µi)(µi−1 − µi)

2

Θ(µi)−Θ(µi−1)− 1
2Θ

′(µi)(µi−1 − µi)
(10)

We may get the following result by removing µ∗ from both sides of Eq. (10):

ei+1 = ei −
Θ

′
(µi)(ei−1 − ei)

2

Θ(µi)−Θ(µi−1)− 1
2Θ

′(µi)(ei−1 − ei)
(11)

Where ei = µi+1 − µ∗, as the error after i iterations in the approximate minimum.
Now, using Taylor’s theorem in its mean value form:

Θ(µ∗) = Θ(µi) + Θ
′
(µi)(µ

∗ − µi) +
1

2!
Θ

′′
(µi)(µ

∗ − µi)
2 +

1

3!
Θ

′′′
(c)(µ∗ − µi)

3 (12)



Z. F. Salih, B. A. Hassan, B. M. Sulaiman / Eur. J. Pure Appl. Math, 15 (3) (2022), 1301-1306 1304

For some c between µ∗ and µi, and because the derivative of the given equation is zero,
we get:

−Θ
′
(µi) = −eiΘ

′′
(µi) +

e2i
2
Θ

′′′
(c) (13)

By putting Eq. (13) into Eq. (11), we can get the following conclusion:

ei+1 = e2i
1

2

Θ
′′′
(c)

Θ′′(µi)
(14)

This indicates that the convergence order is quadratic.

4. Application Examples

In this study, the offered methodologies will be utilized to solve seven test functions.
The functions are listed in the tables, along with the minimum points (µ∗), number of
iterations (NI), and Execution time (T).

A comparison with the Newton Procedure (N- Procedure) is shown to assess the new
procedure performance.

By using the Matlab program, the calculations have been performed with double pre-
cision. If |µi+1 − µi| ≤ ϵ where ϵ = 10−10 achieves in a numerical test, the search is
stopped. Lastly, it is evident that the proposed approach is superior to the N-Procedure
for all functions where fewer iterations are required to get the desired outcomes.

EXAMPLE 1. Function Θ(µ) = cos(µ) + (µ− 2)2, with µ0 = 2.

Procedure (NI) µi (T)

N- Procedure 4.0 2.3542 0.5790
New Procedure 4.0 2.4333 0.0780

EXAMPLE 2. Function Θ(µ) = eµ − 3µ2, with µ0 = 0.25.

Procedure (NI) µi (T)

N- Procedure 4.0 0.2045 0.2660
New Procedure 3.0 0.2036 0.0160

EXAMPLE 3. Function Θ(µ) = e−µ + µ2, with µ0 = 1.

Procedure (NI) µi (T)

N- Procedure 5.0 0.3517 0.3130
New Procedure 4.0 0.2025 0.0780



REFERENCES 1305

EXAMPLE 4. Function Θ(µ) = −µe−µ, with µ0 = 0.

Procedure (NI) µi (T)

N- Procedure 7.0 1 0.3900
New Procedure 5.0 0.2535 0.0790

EXAMPLE 5. Function Θ(µ) = 0.65− 0.75/(1 + µ2)− 0.65µ tan−1(1/µ), with
µ0 = 0.1.

Procedure (NI) µi (T)

N- Procedure 6.0 0.4809 0.7970
New Procedure 5.0 0.2901 0.0620

EXAMPLE 6. Function Θ(µ) = 0.5µ2 − sin(µ), with µ0 = 2.

Procedure (NI) µi (T)

N- Procedure 5.0 0.7391 0.4210
New Procedure 3.0 0.7262 0.0630

EXAMPLE 7. Function Θ(µ) = µ4 + 2µ2 − µ− 3, with µ0 = 1.

Procedure (NI) µi (T)

N- Procedure 7.0 0.2367 0.6870
New Procedure 5.0 0.7624 0.0780

5. Conclusion

This research aims to develop a new determined second derivative using a second-order
Taylor expansion and a new secant equation. Therefore, it is acceptable to assume that the
new secant type technique is more efficient than the Newton method. Some strategies may
not always converge to the global minimum, which happens when they fail to converge to
the minimum in some circumstances.

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