EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5639
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Modified Conjugate Gradient Method with Taylor
Approximation: Applications in Electric

Circuits and Image Restoration

Ahmad Alhawarat1,∗, Sultanah Masmali2, Ibtisam Masmali2,
Mehiddin Al-Baali3, Shahrina Ismail4,5

1 Department of Mathematics, Faculty of Arts and Science, Amman Arab University,
Amman 11953, Jordan
2 Department of Mathematics, College of Science, Jazan University, Jazan, Saudi Arabia
3 Department of Mathematics, College of Science, Sultan Qaboos University, Oman
4 Financial Mathematics Program, Faculty of Science and Technology, Universiti Sains Islam
Malaysia, Bandar Baru Nilai, 71800, Nilai, Negeri Sembilan, Malaysia
5 Risk and Analytics Research Group, Faculty of Science and Technology, Universiti Sains
Islam Malaysia, Bandar Baru Nilai, 71800, Nilai, Negeri Sembilan, Malaysia

Abstract. In this century, the application of optimization methods is frequently utilized in nu-
merous fields like image restoration, electrical engineering, medical science, machine learning (ML),
signal processing as well as many others. In this paper, we choose to improve and maintain one
of the most popular, low-memory, and simple algorithms of optimization methods. This method
is known as the conjugate gradient method (CGM). Here, we develop a new 3-term CGM with
several search directions in the third term suitable for any CGM related to the Fletcher-Reeves
method. Apart from that, we propose a new 3-term CGM with mild conditions for any method
in relation to the Polak-Ribière-Polyak method. The proposed methods satisfy the descent and
convergence properties. Moreover, in the numerical findings section, we perform a comparison of
the new method with several renowned methods that have emerged in this century, such as CG-
Descent 6.8 and nonnegative Dai-Liao methods utilizing more than 180CUTEst library functions.
The numerical findings indicate that the novel approach surpasses recent methodologies. These
numerical findings encompass the count of gradient assessments, function assessments, CPU dura-
tion as well as iteration count. Additionally, we discussed the implementation of the CG method
in image restoration and pi-electric circuits.

2020 Mathematics Subject Classifications: 65K05, 90C30

Key Words and Phrases: Global convergence, Conjugate gradient method, Unconstrained
optimization, Pi-electric circuit

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5639

Email addresses: a.alhawarat@aau.edu.jo (A. Alhawarat),
smasmali@jazanu.edu.sa (S. Masmali), ibtisam234@hotmail.com (I. Masmali),
albaali@squ.edu.om (M. Al-Baali), shahrinaismail@usim.edu.my (S. Ismail)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 2 of 27

1. Introduction

The conjugate gradient (CG) method is broadly employed to solve unconstrained optimiza-
tion problems since we can employ the CG method in several fields like image restoration,
electrical engineering, machine learning (ML) [24, 25], medical science, signal processing,
as well as many others. The general form of optimization problems that we want to solve
employing the CG method is expressed as given below:

min f(x), x ∈ Rn, (1)

in which the objective function, denoted byf , is smooth. The primary iterative formula
with regard to the CG method is expressed below:

xk+1 = xk + αkdk, k = 1, 2, ... (2)

Here, αk may be gain from inexact or exact line search.
Subsequently, the Strong Wolfe-Powell (SWP) line [22, 23] is an inexact line search

commonly used and may be expressed as below:

f(xk + αkdk) ≤ f(xk) + δαk∇fT
k dk, (3)

and
|∇f(xk + αkdk)

Tdk| ≤ σ|∇fT
k dk|, (4)

On the other hand, the weak Wolfe-Powell (WWP) line search is represented in Eqn. (3)
and Eqn. (5) as follows:

∇f(xk + αkdk)
Tdk ≥ σ∇fT

k dk, (5)

with 0 < δ < 1
2 , δ < σ < 1.

Here,dk refers to a search direction often expressed by:

dk =

{
−∇fk, if k = 1,

−∇fk + βkdk−1, if k ≥ 2.
(6)

in which βk refers to the CG parameter. Here, the CG parameters are categorized
into2 groups. The first group is an effective one. This comprises the methods proposed
by Hestenes-Stiefel (HS) [13], Polak-Ribière-Polyak (PRP) [20] as well asLiu and Storey
(LS) [17].

βHS
k =

∇fT
k yk−1

dTk−1yk−1
, βPRP

k =
∇fT

k yk−1

∥yk−1∥2
, βLS

k = −
∇fT

k yk−1

dTk−1∇fk
,

in which yk−1 = ∇fk − ∇fk−1. Nonetheless, this group encounters issues related to
convergence, provided that their values become negative [21]. On the other hand, the
second group, while less efficient, demonstrates strong global convergence. It is important
to note that this category encompasses the Fletcher-Reeves (FR) [10], Fletcher (CD) [9]



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 3 of 27

as well as Dai-Yuan (DY) [7] methods. The equations defining these methods are provided
below:

βFR
k =

∥∇fk∥2

∥∇fk−1∥2
, βCD

k = − ∥∇fk∥2

dTk−1∇fk
, βDY

k =
∥∇fk∥2

dTk−1yk−1
.

Dai and Liao [6] recommended the formula of CG and re-expressed it below:

βDL
k =

∇fT
k yk−1

dTk−1yk−1
−

∇fT
k sk−1

dTk−1yk−1
= βHS

k −
∇fT

k sk−1

dTk−1yk−1
,

where sk−1 = xk −xk−1.To avoid the convergence problem mentioned by [21], the authors
in [6] use the following restriction:

βDL+
k = max{βHS

k , 0} − t
∇fT

k sk−1

dTk−1yk−1
,

Meanwhile, Li et al. [16] suggested the CG formula as expressed below:

β0∗
k =

∇fT
k y

∗
k−1

dTk−1y
∗
k−1

− t
∇fT

k sk−1

dTk−1y
∗
k−1

,

where

y∗k−1 = yk−1 +
θk−1

∥sk−1∥2
sk−1,

θk−1 = 2(fk−1 − fk) + (∇fk−1 +∇fk)
T sk−1.

Hager and Zhang [12] stated the CG formula expressed by:

βHZ
k = max{βN

k , ηk},

in which βN
k = 1

dTk yk
(yk − 2dk

∥yk∥2

dTk yk
)T gk, ηk = − 1

∥dk∥ min{η, ∥∇fk∥} , while η > 0 denotes

a constant.
Alhawarat et al. [3] introduced an effective hybrid CG method that incorporates the

SWP line search, as described below:

dPRP ∗
k =


−∇fk, k = 1,

−∇fk +
(
∇fT

k ∇fk−∇fT
k ∇fk−1

∥∇fk−1∥2

)
dk−1, if ∥∇fk∥2 > ∇fT

k ∇fk−1, k ≥ 2,

βNPRP
k , elsewhere.

where βNPRP
k

=
gTk gk−

∥gk∥
∥gk−1∥ |g

T
k gk−1|

∥gk−1∥2
.

To introduce a positive CG method, Alhawarat et al. [4] expressed the positive CG formula
having a novel restart property given below:

βAZPRP
k =

{∥∇fk∥2−µk∇fT
k ∇fk−1

∥∇fk−1∥2
, if ∥∇fk∥2 > µk|∇fT

k ∇fk−1|,
0, otherwise.



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 4 of 27

in which ∥·∥ resembles the Euclidean norm, while µk is expressed as:

µk =
∥xk − xk−1∥

∥yk−1∥
.

Jiang et al. [14] recommended the CG method expressed by:

βJJSL
k =

gTk yk−1

gTk−1yk−1
.

To enhance the efficacy of prior methods, they established a restart criterion, which is
specified below.

dk =


−gk, k = 1,

−gk + βNSL
k dk−1 +

gTk dk−1

dTk−1(gk−gk−1)
gk−1, if 0 ≤ gTk gk−1 ≤ ∥gk∥2, k ≥ 2,

−gk +
gTk gk−1

∥gk−1∥2
gk−1, k ≥ 2, otherwise.

Based on Jiang et al. [14] modification, Alhawarat et al. [2] present a 3-term CG
method expressed below:

dk =


−gk, k = 1,

−gk + βHS
k dk−1 +

gTk dk−1

dTk−1(gk−gk−1)
gk−1, if ∥gk∥2 > gTk gk−1, k ≥ 2,

−µk
gTk sk−1

dTk−1(gk−gk−1)
dk−1, otherwise.

Ma et al.[18] presented the 3-term CG method expressed by:

dk =


−∇fk, k = 1,

−∇fk + βLS
k dk−1 + ξ

∇fT
k qk

∥qk−1∥qk−1, k ≥ 2, βLS
k ∥∇fT

k dk−1∥ < ∥∇fk∥2,

−∇fk + ξ
∇fT

k qk
∥qk−1∥2

qk−1, otherwise.

where 0 ≤ ξ ≤ 1, 0 ≤ µ ≤ 1, while qk−1 is any non-zero vector.
Jiang et al. [15] developed the CG method expressed by:

dk =


−∇fk, k = 1,

−∇fk + βTDL
k dk−1, k ≥ 2, βTDL

k ∇fT
k dk−1 < ξ∥∇fk∥2,

−∇fk + ξ
∇fT

k qk
∥qk−1∥2

qk−1, otherwise.

in which βTIDL
k = min{βIDL

k , κ ∥∇fk∥
∥dk−1∥}, β

IDL
k =

∇fT
k yk−1−tk∇fT

k dk−1

dk−1(∇fk−∇fk−1)
, κ is a positive con-

stant, while 0 < tk < ρ, where ρ is a positive constant.
Moreover, Alhawarat et al. [2] established the 4-term CGM by employing the directions
expressed below:

−∇fk, dk−1, yk−1, and sk−1



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 5 of 27

dFTCGHS
k = −∇fk − tk

∇fT
k sk−1

yTk−1sk−1
dk−1 −

(
∇fT

k dk−1

yTk−1dk−1

)
(yk−1 + sk−1).

The descent condition (DC), called the downhill condition, is written as

∇fT
k dk < 0, ∀k ≥ 1. (7)

is crucial in examining CG methods, which is also a crucial part in proving the global
convergence analysis. For example, Al-Baali [1] altered Eqn.(7) as follows and employed
it in proving the FR method:

∇fT
k dk ≤ −c∥∇fk∥2, ∀k ≥ 1, c ∈ (0, 1). (8)

in which c ∈ (0, 1). Here, Eqn. (10) given below resembles the sufficient DC. Furthermore,
it has better performance than Eqn.(9) because the quantity of may be regulated by
utilising ∥∇fk∥2:

2. Proposed CG method related to βFR
k

To maintain or improve the former useful behavior and rectify any formula related to βFR
k ,

we construct a new 3-term CG method relying on [18] and based on Taylor’s expansion as
given below:
Let f(xk−1) ≈ f(xk)−∇fT

k sk−1 +
1
2s

T
k−1G(xk)sk−1,

where G(xk) is a Hessian matrix.

−(−f(xk−1) + f(xk)) = −∇fT
k sk−1 +

1

2
sTk−1G(xk)sk−1

2(f(xk)− f(xk−1))

∥sk−1∥2
sk−1 = 2gk −G(xk)sk−1

2(f(xk)− f(xk−1))

∥sk−1∥2
sk−1 − 2gk = −G(xk)sk−1.

Let
(f(xk)− f(xk−1))

∥sk−1∥2
sk−1 = εk.

Thus, it is acceptable to add εk to the search direction with regards to the CG method
given below:

dk = −∇fk + βkdk−1 + εk

Utilizing Eqn. (3), βFR
k , and Eqn. (6), the new search direction become given by:

dPPR
k =


−∇fk, k = 1,

−∇fk + βk
∥∇f⊤

k sk−1∥
∥sk−1∥2

sk−1, k ≥ 2, |βk| ≤ βPPR
k ,

−∇fk − δ
∥∇f⊤

k sk−1∥
∥sk−1∥ sk−1, otherwise.

(9)



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 6 of 27

In this section, we employ βFR
k as a CG formula. Eqn. (9) can be used for any CG

formula related to the FR method, i.e. |βk| ≤ βFR
k . Algorithm 1 illustrates the steps

acquired in obtaining the stationary point of the optimization function.

Algorithm 1
Step 1. Establish an initial point x1, initial direction d1 = −g1 as well as setk := 1
Step 2.Provided that the stopping criterion is met, we may stop.
Step 3. Calculate the search direction dk according to Eqn. (2) utilizing Eqn. (9).
Step 4. Calculate these step sizes αkutilizing Eqns. (3) and (4).
Step 5. Update xk relying on Eqn. (2).
Step 6. Set k := k + 1 and move forward to Step 2.

3. Global Convergence Properties for Eqn. 11

The objective function is considered to be subject to the following presumption.
Assumption 1
I. The level set Ψ = {x ∈ Rn : f(x) ≤ f(x1)} is bounded, which implies the existence of a
nonnegative constant, ρ provided that

∥x∥ ≤ ρ,∀x ∈ Ψ.

II. In several neighborhoods W of Ψ, f refers to a differentiable as well as a continuous
function. Moreover, it has a gradient that is Lipchitz continuous. This implies that, for
any x, y ∈ W, a constant L > 0 exists given that

∥∇f(x)−∇f(y)∥ ≤ L ∥x− y∥ .

This assumption suggests the existence of a positive constant η given that
∥∇f(u)∥ ≤ η,∀η ∈ W.

To establish the CG method’s convergence characteristics with multiple line searches,
which include SWP as well as WWP line searches, the following Lemma, introduced by
Zoutendijk [26], is typically utilized.
Lemma 3.1 Let Assumption 1 to be true. We now take into consideration any form with
regard to Eqn. (2),in which αk fulfills the WWP line search with the DC expressed in
Eqn. (8).Therefore, the inequality given below holds.

∞∑
k=1

(
∇fT

k dk
)2

∥dk∥2
< ∞. (10)

The theorem given below demonstrates that the new formula meets the DC (8).Note that
the proof is analogous to the one provided in [2].
Theorem 3.1 Assume the sequences {xk} as well as {dk} be generated by the methods
(2) as well as (9). We take into consideration the line search method derived from Eqns.
(3) as well as (4). Consequently, the sufficient DC specified in Eqn. (8) is met.
Proof.



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 7 of 27

Case 1 |βk| ≤ βFR
k

Upon multiplying Eqn. (9) with gTk gives

∇fT
k dk = −∥∇fk∥2 +

∥∇fk∥2

∥∇fk−1∥2
∇fT

k dk−1 + δ
∇fT

k sk−1

∥sk−1∥2
∇fT

k sk−1.

Utilizing a SWP line search, we now have

(δ − 1) + σ
∇fT

k−1dk−1

||∇fk−1||2
≤

∇fT
k dk

||∇fk||2
≤ (δ − 1)− σ

∇fT
k−1dk−1

||∇fk−1||2

δ −
k−1∑
j=0

(σ)j ≤
∇fT

k dk
||∇fk||2

≤ δ − 2 +

k−1∑
j=0

(σ)j

k−1∑
j=0

(σ)j ≤ 1− (σ)k

1− σ
,

We can now write

δ − 1− (σ)k

1− σ
≤

∇fT
k dk

||∇fk||2
≤ δ − 2 +

1− (σ)k

1− σ
.

When σ ≤ 1
2 − δ, we have 1−(σ)k

1−σ < 2− δ. Suppose c = (2− δ)− 1−(σ)k

1−σ .
Then,

c− (2− δ) ≤
∇fT

k dk
||∇fk||2

≤ −c

∇fT
k dk ≤ −c||∇fk||2.

Case 2 βk > βFR
k

∇fT
k dk = −||∇fk||2 − δ

∇fT
k
pk−1

∥pk−1∥2
∇fT

k pk−1 = −||∇fk||2 − δ

∥∥∇f
k
pk−1

∥∥2
∥pk−1∥2

≤ −c||∇fk||2.

This completes the proof.

Theorem 3.2 Suppose Assumption 1 is met. Additionally, presume that sequences {gk}
as well as {dk} are produced through Algorithm 1, in which αk is gained from Eqns. (3)
and (4) with σ ≤ 1

2 − δ. We now have that lim
k→∞

inf ∥∇fk∥ = 0.

Proof.
Case 1: |βk| ≤ βFR

k .
The theorem is proved via contradiction. Let

∥∇fk∥ ≥ ε for all k ≥ 0. (11)



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 8 of 27

By taking square with respect to both sides of Eqn. (6), we obtain

∥dk∥2 = ∥∇fk∥2 − 2βk∇fT
k dk−1 + β2

k ∥dk−1∥2 .

Applying the formula in Eqn. (9) yields

∥dk∥2 ≤ ∥∇fk∥2 − 2 ∥∇fk∥2

∥∇fk−1∥2
∇fT

k dk−1 − 2δ
∇fT

k pk−1

∥pk−1∥2
∇fT

k dk−1 +
∥∇fk∥4

∥∇fk−1∥4
∥dk−1∥2 + δ2

(∇fT
k sk−1)

2

∥sk−1∥4
∥dk−1∥2

+ 2δ ∥∇fk∥2

∥∇fk−1∥2
|∇fT

k sk−1|
∥sk−1∥2

∥dk−1∥2 ,

We then have the equation given below.

∥dk∥2 ≤ (1 + δ)2 ∥∇fk∥2 +
∥∇fk∥4

∥∇fk−1∥4
∥dk−1∥2 + (2 + 2δ)

∥∇fk∥2

∥∇fk−1∥2
∣∣∇fT

k dk−1

∣∣ .
Dividing both sides of the equation by ∥∇fk∥4 and employing Theorem 3.1 yields

∥dk∥2

∥∇fk∥4
≤ ∥dk−1∥2

∥∇fk−1∥4
+

σ c(2 + 2δ) + (1 + δ)

∥∇fk∥2
2

Repeating similar process and utilising the relation 1
∥∇f0∥ = 1

∥d0∥ gives

∥dk∥2

∥∇fk∥4
≤ (σ c(2 + 2δ) + (1 + δ)2)

k∑
i=0

1

∥∇f i∥2
.

Now, by using Eqn. (3), we obtain

∥∇fk∥4

∥dk∥2
≥ ε2

(σ c(2 + 2t) + (1 + t)2) k
.

Thus,
∞∑
k=0

∥∇fk∥4

∥dk∥2
= ∞,

contradicting Eqn. (10). Therefore, lim
k→∞

inf ∥∇fk∥ = 0, completing the proof.

Case 2: βk > βFR
k

Similar to Case 1, let
∥∇fk∥ ≥ ε for all k ≥ 0.

From Eqn. (3), we have

∥dk∥ =

∥∥∥∥−∇fk − δ
∇fT

k sk−1

∥sk−1∥2
sk−1

∥∥∥∥ ≤ (1 + δ) ∥∇fk∥ ≤ 1 + η = ε1,

where ε1 is some positive constant, yielding



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 9 of 27

∞∑
k=1

∥∇fk∥4

∥dk∥2
≥

∞∑
k=1

ε4

ε21
= ∞,

which contradicts Eqn. (10). Thus, in all cases, we have

lim
k→∞

inf ∥∇fk∥ = 0

which completes the proof.

4. Proposed CG method that is related to βHS
k or βPRP

k

Even thoughβHS
k and βPRP

k are efficient CG methods while βHS
k inherits the conjugacy

condition, both of them can not satisfy the global convergence and the descent proper-
ties. Both of these issues are considered open problems. To address these difficulties, we
construct the search direction given below relying on the approaches from [18] and [2].

d∗
k
=


−∇fk, k = 1,

−∇fk + (βHS
k or βPRP

k )dk−1 + δ
∇fT

k
sk−1

∥sk−1∥2
sk−1, k ≥ 2, if ∥∇fk∥2 >

∣∣∇fT
k
∇fk−1

∣∣
−∇fk − µk

∇fT
k
sk−1

∥sk−1∥2
sk−1. elsewhere

(12)
As a special case, we employ βHS

k and the condition ∥∇fk∥2 >
∣∣∇fT

k
∇fk−1

∣∣ in Eqn. (4)
as follows:

dHS+δ
k =



−∇fk, k = 1,

−∇fk +

(
∇fT

k ∇fk−∇fT
k ∇fk−1

dTk−1yk−1

)
dk−1 + δ

∇fT
k sk−1

∥sk−1∥2
sk−1, k ≥ 2, if ∥∇fk∥2 > ∥∇fT

k ∇fk∥,

−∇fk − µk
∇fT

k sk−1

∥sk−1∥2
sk−1, elsewhere.

(13)

Eqn. (4) possesses the descent property as well as convergence analysis. On the other
hand, the numerical results show that Eqn. (4) outperforms DL+ and CG-Descent 6.8.
Theorem 3.3Let the sequences {xk} as well as {dk} be generated by the Eqns. (2) and
(4).We take into consideration the line search method derived utilizing Eqns. (3) as well
as (4). Consequently, the sufficient DC (8) is met withσ < 1

3and δ ≤ 1−3σ
1−σ .

Proof.
Case 1: ∥∇fk∥2 >

∣∣∇fT
k
∇fk−1

∣∣
Upon multiplying Eqn. (4) with ∇fT

k gives

∇fT
k dk = −∥∇fk∥2 +

∇fT
k
∇fk −∇fT

k
∇fk−1

dTk−1yk−1
∇fT

k dk−1 + δ
∇fT

k sk−1

∥sk−1∥2
∇fT

k sk−1.



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 10 of 27

By using the condition in Eqn. (4), we obtain

∇fT
k dk ≤ (δ − 1) ∥∇fk∥2 +

2 ∥∇fk∥2

dTk−1yk−1
∇fT

k dk−1.

Utilizing the SWP line search, we now have

∇fT
k dk

||∇fk||2
≤ (δ − 1)− 2σ

∇fT
k−1dk−1

(σ − 1)∇fT
k−1dk−1

= (δ − 1) +
2σ

(1− σ)
.

When σ < 1
3and δ ≤ 1−3σ

1−σ , we have

∇fT
k dk < 0.

This completes the proof.
Case 2: ∥∇fk∥2 <

∣∣∇fT
k
∇fk−1

∣∣
We have

dk = −∇f
k
− µk

∇fT
k
sk−1

∥sk−1∥2
sk−1

Multiply the equation by
∇fT

k ,

we obtain

∇fT
k dk = −∥∇fk∥2 − µk

∇fT
k
sk−1

∥sk−1∥2
∇fT

k sk−1 = −∥∇fk∥2 − µk
−
∥∥∇fT

k sk−1

∥∥2
∥sk−1∥2

< 0.

Thus, in both cases, the DC is satisfied.
Gilbert and Nocedal [11] named a property called Property* to perform a specialized
function in research on CG formulas associated with the PRP method given below.
Property*
We take into consideration a method of the form (2) as well as (6) and let

0 < γ ≤ ∥gk∥ ≤ γ̄.

The method has Property* provided that ∃ a constant b > 1 as well as λ > 0. Note
that ∀ k ≥ 1, we have |βk| ≤ b. Moreover, if ∥xk − xk−1∥ ≤ λ, this implies that

|βk| ≤
1

2b
.

Lemma 3.2 Let Assumption 1 is met. We take into consideration any form of Eqns. (2)
as well as (3). Then, βHS

k meets Property* and the proof is similar to the one given in
[11].

Proof. Let b = 2γ̄2

c((1−δ)−σ)γ2 and λ = c((1−δ)−σ)γ2

2(Lλγ̄)b . Following from here, utilizing βHS
k and

SWP line search, we get

∣∣ βHS
k

∣∣ ≤ ∣∣∣∣∣gTk (gk − gk−1)

dTk−1yk−1

∣∣∣∣∣ ≤
∥gk∥2 +

∣∣gTk gk−1

∣∣
c((1− δ)− σ) ∥gk∥2

≤ 2γ̄2

c((1− δ)− σ)γ2
= b.



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 11 of 27

Provided that ∥xk+1 − xk∥ ≤ λ satisfies Assumption 1, we then have∣∣βHS
k

∣∣ ≤ ∣∣∣∣∣
∥∥gTk ∥∥ (∥gk − gk−1∥)

dTk−1yk−1

∣∣∣∣∣ ≤ (Lλγ̄)

c((1− δ)− σ)γ2
≤ 1

2b
.

Lemma 3.3 We now have an assumption that Assumption 1 is satisfied, while sequences
{gk} as well as {dk} are formed by applying Algorithm 1. Following from here, the step
size αk is created utilizing the SWP line search provided that the sufficient DC is satisfied.
Given that βk ≥ 0, a constant γ > 0 exists in which ∥gk∥ > γ∀k ≥ 1. Hence, dk ̸= 0 and

∞∑
k=0

∥uk+1 − uk∥2 < ∞. (14)

in which uk = dk
∥dk∥ .

Proof. First, provided that dk = 0. Thus, following the sufficient DC, we now obtain
gk = 0. Therefore, we assume that dk ̸= 0 with

γ̄ ≥ ∥gk∥ ≥ γ > 0, ∀k ≥ 1. (15)

We now define
uk = wk + δkuk−1,

in which

wk =
−gk + θk
∥dk∥

, θk = δ
gTk sk−1

∥sk−1∥2
sk−1, δk = βHS

k

∥dk−1∥
∥dk∥

.

Provided that uk expresses a unit vector, we now obtain

∥wk∥ = ∥uk − δkuk−1∥ = ∥δkuk − uk−1∥ .

Following the triangular inequality and δk ≥ 0, we now have

∥uk − uk−1∥ ≤ (1 + δk) ∥uk − uk−1∥ = ∥uk − δkuk−1 − (uk−1 − δkuk)∥ . (16)

≤ ∥uk − δkuk−1∥+ ∥uk−1 − δkuk∥ = 2 ∥wk∥ .

Next, we express

ν = −gk + δ
gTk sk−1

∥sk−1∥2
sk−1.

By utilizing the triangular inequality, we gain:

∥ν∥ ≤ (1 + δ) ∥gk∥ = (1 + δ)γ̄ = T. (17)

Then, ∥ν∥ ≤ T . From Eqn. (16), we have ∥uk − uk−1∥ ≤ 2w.



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 12 of 27

By Eqns. (16) and (17), we get

∞∑
k=0

∥uk+1 − uk∥2 ≤ 4

∞∑
k=0

∥w∥2 ≤ 4T 2
∞∑
k=0

1

∥dk∥2
< ∞,

completing the proof.
By Lemmas 4.1 and 4.2 in [11], we obtained the results given below.

Theorem 3.4 Suppose sequences {xk} as well as{dk} be generated by Eqns. (2) as well
as (3) by utilizing the CG method given in Eqn. (4). Moreover, suppose the step size
satisfies (3) as well as (4). By employing Lemmas 3.2, 3.3, as well as Lemmas 4.1 4.2 in
[11], we gain the findings such that lim inf

k→∞
∥gk∥ = 0.

5. Numerical Findings and Discussions

To assess the effectiveness of the new search direction as defined in Eqn. (??), we se-
lected over 180 test functions from the CUTEr [5] library. We conducted a comprehensive
comparison with robust CG coefficients, including the DL+ and FR CG methods. The
comparison criteria included CPU time, the number of function evaluations, the number
of iterations as well as the number of gradient evaluations. By analyzing these metrics,
we aimed to determine how the new search direction performs relative to the benchmark
methods. We used the SWP line search setting δ = 0.01 as well as σ = 0.1 for all methods.
The results for the DL+, PFR, and FR methods were obtained by executing a modified
version of the CG-Descent code. This modified code is available on the Hager webpage(see
here). By adapting the original CG-Descent code, we ensured compatibility with our spe-
cific requirements for testing these methods.

The norm of the gradient was used as the stopping criterion for all algorithms, particularly
when ∥gk∥ ≤ 10−6. The computations were performed on a host computer running Ubuntu
20.04.2.0 LTS OS, equipped with an 11th Gen Intel(R) Core(TM) i5-1155G7 @ 2.50GHz
processor and 8.00 GB RAM. The performance findings are depicted in Figures 1 to 4,
utilizing a performance measure established by Dolan and More [8]. Figure 1 illustrates
that the PFR method outperforms the DL+ and FR methods in terms of the number of
iterations required. As seen in Figure 2, PFR significantly surpasses all other methods in
terms of the number of function evaluations. Meanwhile, Figure 3 demonstrates that PFR
outperforms DL+ and FR in the context of gradient evaluations. Additionally, Figure 4
indicates that the PFR method not only strongly outperforms the FR CG method with
regards to the CPU time but is also competitive with the DL+ CG method.

The numerical findings presented below validate the efficacy of the proposed search di-
rection HS+TA. Detailed results are provided in Table 1. In this study, we utilized over
180 test functions from the CUTEr library to rigorously assess the performance of our



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 13 of 27

Figure 1: Number of iterations graph Figure 2: Number of function evaluations graph

Figure 3: Number of gradient evaluations graph Figure 4: CPU time graph

proposed methods. As an example of using |βk| ≤ βFR
k in (9), we employ βAZPRP

k as
follows:

dTAPRP
k

=

{
−∇fk, k = 1,

−∇fk + βAZPRP
k dk−1 + δ

∇fT
k
sk−1

∥sk−1∥2
sk−1, k ≥ 2.

All computations were performed on a host machine running the Ubuntu 20.04 operat-
ing system, equipped with an AMD A4-7210 CPU and 4 GB of RAM. Here, we compared
the modified search directions dHS+TA

k
, dTAPRP

k , DL+ and CG-Descent 6.8 methods uti-
lizing a SWP line search in obtaining the step length having σ = 0.1 for dHS+TA

k
, dTAPRP

k

and DL+ with δ = 0.01. We executed the code using the default parameters specified
by the authors for CG-Descent 6.8. The results are illustrated in Figures 5-7, utilizing
a performance measure developed by Dolan and More [5]. From Figures 5-7, it can be
observed that the new search directions strongly outperformed DL+, CG-Descent 6.8, and
dTAPRP
k with regards to the function evaluation, number of iterations as well as CPU time.



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 14 of 27

Figure 5: Number of iteration graph Figure 6: Number of function evaluations graph

Figure 7: CPU time graph

In Figure 8, we present the function number that is listed in the Table 2 (see in Appendix
below) with its gradient for dHS+TA

k
. We observe that the number of success functions is

more than 99%.

Figure 8: Function number with the gradient for
HS+TA method.



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 15 of 27

6. Application on Pi-Electric circuit

Figure 9 depicts a Pi-Electric circuit, which is utilized as the output network of automati-
cally tuned transmitters [19]. In this circuit, the tuning capacitor C1 as well as the loading
capacitor C2 are adjusted using semi-independent control loops. Here, the objectivesfor
this electric circuit are given below:
1. To achieve a 180◦ phase shift between I1(jw) and the voltage across C1 at a given
carrier frequency ω.
2. To attain a specific level of power output to an antenna system.
3. To determine the values of C1 as well as C2 that maximize the power delivered to R2.
Suppose the power output across R2 be expressed by P0 given below:

P0 =
|V0(jw)|2

R2

Let values of τ1 = R1C1 and τ2 = R2C2. Then, from the circuit analysis, we have

V0(s) =
−R1R2I1(s)

s3(Lτ1τ1) + s2L(τ1 + τ2) + s(L+R1τ2 +R2τ1) +R1 +R2

Figure 9: Pi-Electric circuit [8].

V0(jw) =
I1R1√

RE2 + IM2

where I1 ≡ |I1(jw)|

RE = 1 +
R1

R2
− w2Lτ1

R2
− w2Lτ2

R2

IM =
wL

R2
+

R1wτ2
R2

+ wτ1 −
w3Lτ1τ2

R2

To find P0, minimize f such that

f = [Re(V0(jw))]
2 + [IM(V0(jw))]

2

having dimensionless parameters stated below:

a =
R1

R2

b =
wL

R2

x1 = wτ1



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 16 of 27

x2 = wτ2

For the case when a = 10 and b = 1, the objective function that we want to minimize is
given by:

f(x1, x2) = (11− x1 − x2)
2 + (1 + 10x2 + x1 − x1x2)

2.

Its graph is shown in Figure 10.

Figure 10: Graph of f(x1, x2) by using MATLAB.

By using Algorithm 1, we found that:
x1 = 7,
x2 = −2,

with function value 40.

7. Application on Image Restoration

Among the important applications of the CG method is restoring damaged images. In this
study, we applied Gaussian noise having an SD of 25% to the original images presented
in Table 3 and then used Algorithm 1 to restore these images. To illustrate the efficacy
of the suggested method (HS+TA), we compared Algorithm 1 with CG-Descent 6.8 and
DL+ with regards to the number of iterations, CPU time, as well as RMSE. Moreover, the
RMSE between the restored image as well as the original true image was used to evaluate
the restoration quality.

RMSE =
∥τ − τk∥2

∥τ∥
.

The restored image is expressed by τk, while the true image byτ . Moreover, the RMSE
is used to assess the restored image’s quality, with lower RMSE values indicating higher
image quality. The data presented in Table 2 demonstrates that the new search direction
surpasses the performance of CG-Descent 6.8 and DL+ in several aspects, including the
number of iterations, CPU time, including the RMSE value. The stopping criteria for the
process is

∥xk+1 − xk∥2
∥xk∥2

< ε.



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 17 of 27

This context yields ε = 10−3. Provided that ε = 10−4 or ε = 10−6, the RMSE remains
constant, which indicates that a fixed RMSE value can correspond to a varying number of
iterations. This implies that while the RMSE does not change, the number of iterations
obligatory to achieve this RMSE may fluctuate.

Figure 11: The number of iterations for image
restoration represented in Table 2.

Figure 12: CPU time for image restorationrepre-
sented in Table 2.

Figure 13: RMSE for image restoration repre-
sented in Table 2.

Figures 11, 12, and 13 demonstrate that the latest modification HS+TA outperforms
DL+ and CG-Descent in the number of iterations, CPU as well as the RMSE value.
On the other hand, we found that CG-Descent outperforms DL+ in all figures. Table 3
below presents the results of restoring corrupted images utilizing HS+TA, highlighting its
effectiveness and efficiency. These outcomes suggest that HS+TA is a robust method for
image restoration, successfully recovering images to a high standard.



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 18 of 27

Table 1: Restoration of corrupted images of Moon, Cameraman, Mandi, Coins, Kids, as well as M.83 using
HS+TA.

Image Original image Image with Gaussian noise Restored image

Moon (128 pixels)

Cameraman (128 pixels)

Mandi (128 pixels)

Mandi (256 pixels)

Kids (512 pixels)

M.83 (1024 pixels)

8. Conclusion

This research develops a modified CG method based on Taylor expansion. The proposed
method satisfies both descent and convergence properties. Moreover, numerical findings
demonstrate that the suggested method is robust and effective, outperforming or matching
the performance of CG-Descent 6.8 and DL+. In future work, we plan to apply these
methods to machine learning applications to explore their potential in this domain.



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 19 of 27

Acknowledgements

The editors and reviewers are gratefully acknowledged by the authors for any comments
and recommendations that might make this work better. We appreciate Dr. William
Hager’s contribution to the CG method’s source code.

Availability of data and material

All data is available in the work itself.

Competing interests

No competing interests are declared by the author.

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Table 2: The test function with the number of iterations, function evaluations, gradient evaluations, function
value, gradient value, and CPU time for HS+TA.

Function
num-
ber

Function DIM Iteration
(HS+TA)

Fun.
Evalu-
ations

Grad.
Eval-
ua-
tions

Final f Final g Time

1 AKIVA 2 8 20 15 6.166042 6.97E-07 0.02

2 ALLINITU 4 9 25 18 5.744385 1.61E-09 0.02

3 ARGLINA 200 1 3 2 200 1.24E-11 0.02

4 ARGLINB 200 1 3 2 99.62547 6.69E-06 0.02

5 ARWHEAD 5000 6 16 12 0 1.84E-07 0.03

6 BARD 3 12 32 22 0.008215 3.28E-07 0.02

7 BDQRTIC 5000 61 162 141 20006.26 5.64E-07 0.02

8 BEALE 2 11 33 26 1.66E-19 2.57E-09 0.02

9 BENNETT5LS3 14 44 34 0.000539 6.73E-08 0.02

10 BIGGS6 6 24 64 44 0.005656 2.03E-07 0.02

11 BOX 10000 7 25 21 -1864.54 7.17E-07 0.02

12 BOX3 3 10 23 14 6.37E-15 9.22E-08 0.02

13 BOXBODLS 2 12 75 69 1168.009 2.64E-08 0.02

14 BOXPOWER 20000 29 72 47 3.3E-12 7.03E-08 0.03

15 BRKMCC 2 5 11 6 0.169043 6.22E-08 0.02

16 BROWNAL 200 5 64 61 1.47E-09 1.18E-06 0.02

17 BROWNBS 2 10 24 18 0 0 0.02

18 BROWNDEN 4 16 38 31 85822.2 1.53E-10 0.02

19 BROYDN7D 5000 57 104 79 3441.346 9.15E-07 0.02

20 BRYBND 5000 45 105 64 0.538446 0.000967 0.02

21 CHAINWOO 4000 12798 26461 14190 1651.345 5.92E-07 3.06

22 CHWIRUT1LS3 15 43 34 2384.477 3.18E-07 0.02

23 CHWIRUT2LS3 15 35 25 513.048 3.58E-07 0.02

24 CLIFF 2 10 46 39 0.199787 1.95E-07 0.02

25 COSINE 10000 10 48 42 -9999 4.37E-07 0.03

26 CRAGGLVY 5000 88 178 149 1688.215 9.4E-07 0.06

27 CUBE 2 17 48 34 1.65E-20 5.82E-09 0.02

28 CURLY10 10000 49278 69394 78474 -1003163 9.98E-07 24.87

29 CURLY20 10000 69850 92869 116727 -1003163 9.99E-07 49.13

30 CURLY30 10000 74607 98792 125185 -1003163 9.87E-07 89.41

31 DANWOODLS2 8 32 28 0.004317 9.93E-08 0.02

32 DECONVU 63 415 834 420 2.24E-08 9.73E-07 0.02

33 DENSCHNA 2 6 16 12 1.32E-14 3.21E-07 0.02

34 DENSCHNB 2 6 18 15 3.2E-19 1.43E-09 0.02

35 DENSCHNC 2 11 36 31 5.27E-15 2.76E-07 0.02

36 DENSCHND 3 14 46 40 5.24E-12 2.34E-07 0.02

37 DENSCHNE 3 12 43 38 1.46E-13 7.56E-07 0.02

38 DENSCHNF 2 9 31 26 3.18E-22 2.52E-10 0.02

39 DIXMAANA 3000 6 15 11 1 3.94E-13 0.02

40 DIXMAANB 3000 6 16 12 1 2.61E-08 0.02

41 DIXMAANC 3000 6 14 9 1 2.06E-08 0.02

42 DIXMAAND 3000 7 18 13 1 1.12E-07 0.02

43 DIXMAANE 3000 218 245 417 1 9E-07 0.05



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 22 of 27

Function
num-
ber

Function DIM Iteration
(HS+TA)

Fun.
Evalu-
ations

Grad.
Eval-
ua-
tions

Final f Final g Time

44 DIXMAANF 3000 111 227 119 1 9.46E-07 0.02

45 DIXMAANG 3000 171 347 179 1 9.88E-07 0.02

46 DIXMAANH 3000 175 356 185 1 9.89E-07 0.02

47 DIXMAANI 3000 3308 3403 6529 1 9.08E-07 0.56

48 DIXON3DQ 10000 10000 10007 19995 3.16E-13 7.17E-08 2.81

49 DJTL 2 75 1163 1148 -8951.55 3.11E-07 0.02

50 DQDRTIC 5000 5 11 6 3.36E-17 1.14E-08 0.02

51 DQRTIC 5000 16 44 36 1.18E-06 4.47E-07 0.05

52 ECKERLE4LS3 2 6 4 0.699696 3.81E-07 0.02

53 EDENSCH 2000 24 56 48 12003.28 3.24E-07 0.02

54 EG2 1000 3 8 5 -998.947 4.88E-07 0.02

55 EIGENALS 2550 8775 15326 11017 2.11E-10 7.02E-07 23.2

56 EIGENBLS 2550 14010 28030 14023 1.92E-06 9.28E-07 31.64

57 EIGENCLS 2652 10008 19716 10333 2.18E-11 8.83E-07 21.02

58 ENGVAL1 5000 20 41 35 5548.668 5.18E-07 0.02

59 ENGVAL2 3 26 73 55 3.27E-23 6.59E-10 0.02

60 ENSOLS 9 22 47 27 788.5398 3.55E-07 0.02

61 EXPFIT 2 9 29 22 0.240511 3.34E-07 0.02

62 EXTROSNB 1000 2063 4728 2849 3.81E-07 7.66E-07 0.7

63 FBRAIN2LS 4 79 259 204 0.318972 2.1E-07 0.48

64 FBRAIN3LS 6 1308 3934 3080 0.242722 9.73E-07 1.77

65 FBRAINLS 2 9 27 21 0.416603 3.37E-07 0.03

66 FLETCHCR 1000 209 410 238 7.01E-15 8.69E-07 0.02

67 FMINSRF2 5625 284 578 299 1.000024 9.19E-07 0.13

68 FMINSURF 5625 332 673 345 1 9.45E-07 0.16

69 FREUROTH 5000 29 64 58 608159.2 8.17E-07 0.02

70 GAUSS1LS 8 49 113 74 1315.822 8.44E-09 0.02

71 GAUSS2LS 8 49 124 87 1247.528 4.65E-09 0.02

72 GBRAINLS 2 8 20 13 28.51586 4.33E-08 0.02

73 GENROSE 500 1100 2230 1170 1 6.15E-07 0.03

74 GROWTHLS 3 109 431 369 1.004041 6.87E-07 0.02

75 GULF 3 33 95 72 1.15E-17 1.15E-17 0.02

76 HAHN1LS 7 5 56 53 8522.662 4.03E-08 0.02

77 HAIRY 2 17 82 68 20 7.89E-09 0.02

78 HATFLDD 3 17 49 37 2.55E-07 1.39E-07 0.02

79 HATFLDE 3 13 37 30 2.73E-06 3.38E-07 0.02

80 HATFLDFL 3 21 68 54 6.39E-05 9.84E-07 0.02

81 HEART6LS 6 375 1137 876 8.8E-17 2.23E-07 0.02

82 HEART8LS 8 253 657 440 2.97E-16 2.3E-07 0.02

83 HELIX 3 23 60 42 2.88E-19 1.3E-08 0.02

84 HIELOW 3 13 30 21 874.1654 5.19E-08 0.02

85 HILBERTA 2 2 5 3 5.75E-32 5.75E-32 0.02

86 HILBERTB 10 4 9 5 9.95E-19 2.27E-09 0.02



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 23 of 27

Function
num-
ber

Function DIM Iteration
(HS+TA)

Fun.
Evalu-
ations

Grad.
Eval-
ua-
tions

Final f Final g Time

87 HIMMELBB 2 4 18 18 2.17E-16 4.97E-09 0.02

88 HIMMELBF 4 23 59 46 318.5717 4.39E-07 0.02

89 HIMMELBG 2 7 22 17 3.04E-22 4.92E-11 0.02

90 HIMMELBH 2 5 13 9 -1 2.77E-07 0.02

91 HUMPS 2 45 223 202 9.15E-16 1.35E-08 0.02

92 INDEF 5000 1 46 147 3.184123e-
314

3.184008e-
314

0.02

93 INTEQNELS 502 6 13 7 1.08E-11 5.53E-07 0.02

94 JENSMP 2 12 47 41 124.3622 1.46E-08 0.02

95 JIMACK 3549 8327 16656 8329 0.866793 9.85E-07 193.23

96 KIRBY2LS 2 54 222 201 3.905074 3.69E-06 0.02

97 KOWOSB 4 16 46 32 0.000308 9.46E-07 0.02

98 LANCZOS1LS6 61 177 135 2.83E-07 9.05E-07 0.02

99 LANCZOS2LS6 60 169 125 2.75E-07 7.32E-07 0.02

100 LANCZOS3LS6 61 164 118 3.41E-07 5.02E-07 0.02

101 LOGHAIRY 2 26 196 179 0.182322 7.9E-07 0.02

102 LSC1LS 3 31 108 89 7.711852 1.73E-07 0.02

103 LSC2LS 3 37 106 86 13.33387 2.98E-08 0.02

104 MANCINO 100 12 30 19 1.95E-21 5.14E-08 0.02

105 MARATOSB 2 589 2885 2585 -1 4.08E-07 0.02

106 MEXHAT 2 14 59 55 -0.04001 1.01E-08 0.02

107 MEYER3 3 19 76 63 9.387247e-
323

6.952756e-
310

0.02

108 MGH09LS 4 25 82 72 0.001019 4.57E-07 0.02

109 MGH10SLS 19 1082 4052 4968 1.29E+09 6.04E-20 0.02

110 MGH17LS 5 84 323 265 0.024518 1.55E-07 0.02

111 MISRA1ALS 2 33 147 145 0.124551 2.38E-09 0.02

112 MISRA1BLS 2 26 113 101 0.075465 5.97E-09 0.02

113 MOREBV 5000 161 168 317 1.09E-10 9.94E-07 0.3

114 MSQRTALS 1024 2795 5599 2806 6.72E-10 9.2E-07 1.09

115 MSQRTBLS 1024 2235 4476 2247 1.7E-10 9.71E-07 0.95

116 NCB20 5010 906 2105 1348 -82.2049 9.39E-07 0.02

117 NELSONLS 3 1101 5415 7690 25.60412 4.25E-29 0.2

118 NONCVXU2 5000 6476 12572 6855 11584.55 9.83E-07 3.02

119 NONDQUAR 5000 2575 5220 2691 3.09E-06 7.14E-07 0.55

120 OSBORNEA 5 82 230 174 5.46E-05 1.94E-07 0.02

121 OSBORNEB 11 57 134 84 0.040138 7.26E-07 0.02

122 OSCIGRAD 100000 117 191 201 77276.13 9.71E-07 6.48

123 OSCIPATH 500 10 20 14 0.999967 9.3E-07 0.02

124 PALMER1C 8 12 27 28 0.097605 1.28E-07 0.02

125 PALMER1D 7 10 24 23 0.652674 8.27E-11 0.02

126 PALMER2C 8 11 21 22 0.024369 2.2E-09 0.02

127 PALMER3C 8 11 21 21 0.029538 7.92E-10 0.02

128 PALMER4C 8 11 21 21 0.050311 3.55E-09 0.02



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 24 of 27

Function
num-
ber

Function DIM Iteration
(HS+TA)

Fun.
Evalu-
ations

Grad.
Eval-
ua-
tions

Final f Final g Time

129 PALMER5C 6 6 13 7 2.128087 3.75E-12 0.02

130 PALMER6C 8 11 24 24 0.026387 1.88E-08 0.02

131 PALMER7C 8 11 20 20 0.601987 1.52E-07 0.02

132 PALMER8C 8 11 19 19 0.159768 2.75E-09 0.02

133 PARKCH 15 699 1501 1190 1623.743 9.84E-07 1.38

134 PENALTY2 200 191 225 363 4.71E+13 8.84E-07 0.02

135 PENALTY3 200 104 323 256 0.000999 3.82E-07 1.94

136 POWELLBSLS2 50 211 234 1.14E-07 3.02E-07 0.02

137 POWER 10000 358 730 379 1.74E-09 9.4E-07 0.11

138 POWERSUM 4 4 10 6 37.43145 8.13E-09 0.02

139 PRICE3 2 10 25 17 3.41E-18 3.19E-08 0.02

140 PRICE4 2 9 30 23 3.42E-24 3.13E-10 0.02

141 QING 100 67 134 85 2.62E-14 7.84E-07 0.02

142 QUARTC 5000 16 44 36 1.18E-06 4.47E-07 0.03

143 RAT42LS 3 18 53 44 8.056523 1.36E-07 0.02

144 RAT43LS 4 44 156 122 8786.405 1.17E-07 0.02

145 RECIPELS 3 16 49 38 5.12E-13 7.12E-07 0.02

146 ROSENBR 2 28 84 65 5.87E-16 7.61E-07 0.02

147 ROSENBRTU2 37 175 153 3.24E-14 1.44E-07 0.02

148 ROSZMAN1LS4 27 85 66 0.039541 1E-08 0.02

149 S308 2 7 21 17 0.773199 3.3E-09 0.02

150 SCHMVETT 5000 41 71 58 -14994 9.53E-07 0.02

151 SENSORS 100 26 67 47 -2079 1.64E-07 0.06

152 SINEVAL 2 46 181 153 1.49E-23 1.55E-10 0.02

153 SINQUAD 5000 15 44 36 -6757014 1.12E-08 0.02

154 SISSER 2 5 19 19 4.6E-10 4.93E-07 0.02

155 SNAIL 2 61 251 211 7E-17 1.59E-08 0.02

156 SPARSINE 5000 25740 26045 51185 1.15E-10 8.98E-07 14.19

157 SPARSQUR 10000 17 70 70 2.75E-10 1.33E-07 0.02

158 SPMSRTLS 49999 204 413 227 3.49E-11 9.41E-07 0.06

159 SROSENBR 5000 9 23 15 1.3E-12 7.94E-07 0.02

160 SSBRYBND 5000 9465 16566 14479 6.13E-15 9.31E-07 4.56

161 SSI 3 307 1162 990 3.94E-06 5.82E-07 0.02

162 STRATEC 10 170 419 283 2212.262 4.3E-08 1

163 TESTQUAD 5000 1573 1580 3141 1.33E-13 9.19E-07 0.27

164 TOINTGOR 50 122 219 159 1373.905 7.64E-07 0.02

165 TOINTGSS 5000 4 9 5 10.002 1.87E-07 0.02

166 TOINTPSP 50 147 323 241 225.5604 8.83E-07 0.02

167 TOINTQOR 50 29 36 53 1175.472 4.46E-07 0.02

168 TQUARTIC 5000 8 23 18 3.61E-12 2.91E-09 0.02

169 TRIDIA 5000 781 788 1557 4.64E-15 9.37E-07 0.16

170 TRIGON1 10 19 41 22 4.09E-15 3.93E-07 0.02

171 TRIGON2 10 22 57 43 2.966531 2.88E-08 0.02



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 25 of 27

Function
num-
ber

Function DIM Iteration
(HS+TA)

Fun.
Evalu-
ations

Grad.
Eval-
ua-
tions

Final f Final g Time

172 VANDANMSLS22 2 5 3 9.623043 2.85E-09 0.02

173 VAREIGVL 5000 34 76 45 1.66E-12 7.68E-07 0.16

174 VESUVIALS 8 1262 1954 3155 991.4145 1.6E-18 1.23

175 VESUVIOLS 8 79 198 173 991.41 1.6E-18 0.17

176 VESUVIOULS8 79 211 173 0.4771138 1.3E-07 0.02

177 VIBRBEAM 8 98 255 174 0.156446 2.83E-08 0.02

178 WAYSEA1 2 11 55 50 6.75E-22 2.99E-09 0.02

179 WAYSEA2 2 9 28 23 1.07E-17 1.37E-09 0.02

180 WATSON 12 60 137 82 1.62E-07 8.55E-07 0.02

181 WOODS 4000 64 184 136 8.34E-15 9.5E-07 0.01

182 YATP1LS 123200 13 34 25 3.88E-16 2.32E-07 0.64

183 YFITU 3 68 208 167 6.67E-13 1.72E-08 0.02

184 ZANGWIL2 2 1 3 2 -18.2 3.79E-15 0.02



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 26 of 27

Table 3: Numerical results for images with Gaussian noise, incorporating a standard deviation of 25%, via the
Dai-Liao CG, AHS, and CG-Descent 6.8 methods.

Image Algorithm Number of
iterations

CPU time
(seconds)

RMSE

Mandi 128 pixels DL+ 127 1.724e+000 0.1003

HS+TA 125 1.563e+000 0.1001

CG-Descent
6.8

134 1.825e-001 0.1004

Cameraman 128
pixels

DL+ 165 1.631e+000 0.1257

HS+TA 160 1.359e+000 0.1146

CG-Descent
6.8

163 1.854e+000 0.1148

Coins 128 pixels DL+ 135 1.542e+000 0.0832

HS+TA 133 1.391e+000 0.0828

CG-Descent
6.8

133 1.491e+000 0.0831

Mandi 256 pixels DL+ 120 1.856e+001 0.0519

HS+TA 118 1.745e+001 0.0517

CG-Descent
6.8

119 1.656e+001 0.0991

Moon 256 pixels DL+ 170 1.678e+001 0.0355

HS+TA 166 1.234e+001 0.0350

CG-Descent
6.8

166 1.45e+001 0.0368

Cameraman 256
pixels

DL+ 166 1.856e+001 0.0894

HS+TA 164 1.797e+001 0.0890

CG-Descent
6.8

162 1.925e+001 0.0892

Coins 256 pixels DL+ 134 1.447e+001 0.0506

HS+TA 129 1.264e+001 0.0505

CG-Descent
6.8

130 1.564e+001 0.0508

Mandi 512 pixels DL+ 114 7.981e+001 0.0371

HS+TA 110 6.955e+001 0.0370

CG-Descent
6.8

116 7.314e+001 0.0472

kids 512 pixels DL+ 57 6.955e+001 0.0377

HS+TA 55 5.425e+001 0.0383

CG-Descent
6.8

55 5.634e+001 0.0395

Coins 512 pixels DL+ 129 7.323e+001 0.0326

HS+TA 128 5.948e+001 0.0326

CG-Descent
6.8

127 6.323e+001 0.0503



A. Alhawarat et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5639 27 of 27

Cameraman 512
pixels

DL+ 148 9.650e+001 0.0533

HS+TA 146 9.438e+001 0.0533

CG-Descent
6.8

143 9.727e+001 0.0532

Moon 1024 pix-
els

DL+ 155 1.238e+002 0.0183

HS+TA 150 1.115e+002 0.0081

CG-Descent
6.8

150 1.203e+002 0.0082

Cameraman
1024 pixels

DL+ 146 3.950e+002 0.0534

HS+TA 125 2.643e+002 0.0289

CG-Descent
6.8

130 2.853e+002 0.0298

Coins 1024 pix-
els

DL+ 128 3.441e+002 0.0326

HS+TA 113 2.049e+002 0.0173

CG-Descent
6.8

124 2.897e+002 0.0289