On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of Trigonometric Function


CAUCHY – Jurnal Matematika Murni dan Aplikasi 
Volume 7(1) (2021), Pages 84-96 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: July 15, 2021 Reviewed: August 19, 2021 Accepted: October 06, 2021 
DOI: https://doi.org/10.18860/ca.v7i1.12934    

On the Modification of Newton-Secant Method in Solving 
Nonlinear Equations for Multiple Zeros of Trigonometric Function 

Juhari  

Department of Mathematics, Faculty of Science and Technology, Maulana Malik Ibrahim 
Islamic State University of Malang 

 

Email: juhari@uin-malang.ac.id   

ABSTRACT 

This study discusses the construction of mathematical model modification of Newton-Secant 
method and solving nonlinear equations for multiple zeros by using a modified Newton-Secant 
method. A nonlinear equations for multiple zeros or multiplicity  𝑚 > 1 is an equation that has 
more than one root. The first step is to construct of mathematical model Newton-Secant method 
and its modification, namely to construct a mathematical model of the Newton-Secant method 
using the concept of the Newton method and the concept of the Secant method. The second step 
is to construct a modified mathematical model of the Newton-Secant method by adding the 
parameter 𝜃. After obtaining the formula for the modification Newton-Secant method, then 
applying the method to solve a nonlinear equations for multiple zeros. In this case, it is applied to 
the nonlinear equation trigonometric function 𝑓(𝑥) = (𝑐𝑜 𝑠2 𝑥 + 𝑥)5 which has a multiplicity of 
𝑚 =  5. The solution is done by selecting four different initial guess, namely −2; −0,8; −0,2 and 
2. Furthermore, to determine the effectivity of this method, the researcher compared the result 
with the Newton-Raphson method, the Secant method, and the Newton-Secant method that has 
not been modified. The obtained results from the construction of mathematical model Newton-
Secant method and its modification is an iteration formula modification of Newton-Secant 
method. And for the result of 𝑓(𝑥) using a modification of Newton-Secant method with four 
different initial guess, the root of 𝑥 is obtained approximately, namely −0.641714371 with fewer 
iterations if compared to using the Newton method, the Secant method, and the Newton-Secant 
method. Based on the problem to find the root of the nonlinear equation 𝑓(𝑥) it can be 
concluded that the modification of Newton-Secant method is more effective than the Newton 
method, the Secant method, and the Newton-Secant method. 

Keywords: modification; Newton-Secant method; nonlinear equation; multiple zeros; 
trigonometric function 

INTRODUCTION 

 In the fields of science, engineering and economics often involve problems 
mathematics. Mathematical problems are often found in the form of nonlinear equations 
[1]. Nonlinear equations in the form of functions 𝑓(𝑥) can be form of algebraic equations 
and transcendent equations. Transcendent equations or non-algebraic equations are 
equations that cannot be expressed in algebraic operations. This equation consists of 
logarithmic functions, exponential functions, hyperbolic functions and trigonometric 

https://doi.org/10.18860/ca.v7i1.12934
mailto:juhari@uin-malang.ac.id


On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of 
Trigonometric Function 

Juhari 85 

functions [2]. In finding a solution to a nonlinear equation means making that equation 
becomes zero, i.e. 𝑓(𝑥) = 0 [3]. In determining the solution of a complex nonlinear 
equation will be difficult if done using analytical methods so the numerical will be 
solution to this problem [4]. The solution obtained from the numerical method an 
approximate solution. The approximate solution is different from the exact solution, so 
there is the difference between exact solution and approximate solution. This difference 
is often referred to as an error [5]. 
 In finding the roots of a nonlinear equation, it is not always singular or simple, 
sometimes nonlinear equations are in the form of multiple nonlinear equations, meaning 
the equation has a multiplicity 𝑚 > 1. The following is the definition of multiplicity:  
 
Definition 1 (Multiplicity) 
The root 𝛼 of 𝑓(𝑥) is said to have multiplicity (𝑚) if 

𝑓(𝑥) = (𝑥 − 𝛼)𝑚ℎ(𝑥) 
for ℎ(𝑥) a continuous function with ℎ(𝑥) ≠ 0, and 𝑚 is a positive integer. If 𝑚 = 1 then 𝛼 
is called a simple root. If 𝑚 ≥ 2 then 𝛼 is called a multiple root [6]. Thus it can be said 
that a function has multiplicity if the multiplicity of a function is more than one [5]. 
 The most famous numerical method for solving nonlinear equations is the 
Newton-Raphson method. In finding solutions to nonlinear equations, this method 
requires one initial guess and function derivative value. This method will fail if the initial 
guess selection gives the derivative value zero [7]. The following is the formula of 
Newton-Raphson method : 

                   𝑥𝑛+1 = 𝑥𝑛 −
𝑓(𝑥𝑛)

𝑓 ′(𝑥𝑛)
                                                             (1) 

with 𝑛 = 0,1,2, . . .. and 𝑓 ′(𝑥𝑛) ≠ 0. 
 The numerical method that is no less famous is the Secant method. This method 
able to overcome the weakness of the Newton-Raphson method. In the Newton-Raphson 
method is required first derivative of the function 𝑓(𝑥). The process of finding the 
derivative function 𝑓(𝑥) does not always easy, sometimes there are some functions that 
are difficult to find the derivative value. To overcome the weakness of Newton's method 
then in the Secant method derivative function is replaced by another equivalent form. So 
the secant method does not require another derivative of the function but requires two 
initial guesses [8]. The following is the formula of Secant method : 

 𝑥𝑛+1 = 𝑥𝑛 −
𝑓(𝑥𝑛) ∙ (𝑥𝑛−1 − 𝑥𝑛 )

𝑓(𝑥𝑛−1) − 𝑓(𝑥𝑛)
 (2) 

with 𝑛 = 1,2,3 . . .. 
 In the calculation process using numerical method such as Newton's method and 
the Secant method are needed initial guess. Determining the initial value will be easier if 
you pay attention the theory related the intermediate value theorem. The intermediate 
value theorem is a theorem that is used to determine the presence or absence of a 
solution at a certain interval limit. 
 
Theorem 1 (Intermediate Value Theorem) 
If 𝑓 ∈ 𝐶[𝑎, 𝑏] and 𝐾 is any number between 𝑓(𝑎) and 𝑓(𝑏), then there exists a number 𝑐 
in (𝑎, 𝑏) for which 𝑓(𝑐) = 𝐾. [9] 
Another theorem that is almost similar with the intermediate value theorem is Bolzano's 
theorem. 
 
  



On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of 
Trigonometric Function 

Juhari 86 

Theorem 2 (Bolzano's Theorem) 
If 𝑓: [𝑎, 𝑏] ⊂ ℝ → ℝ is a continuous function and if 𝑓(𝑎) ∙ 𝑓(𝑏) < 0, then there is at least 
one root 𝑥 ∈ (𝑎, 𝑏) such that 𝑓(𝑥) = 0. [10] 
 Each numerical method has a different order of conver gence. Order of 
convergence is the speed of an iteration method in finding the roots simultaneously 
approximation of the equation of function 𝑓. The following is the definition of 
convergence : 
 
Definition 2 (convergence) 
Let the sequence 𝑥1, 𝑥2, . . . , 𝑥𝑛 convergence to 𝛼 and 𝑒𝑛 = 𝑥𝑛 − 𝛼 where 𝑛 ≥ 0. If the 
order of convergence 𝑝 > 0 and error constant 𝐶 ≠ 0, with 

lim
𝑛→∞

|𝑥𝑛+1 − 𝛼|

|𝑥𝑛 − 𝛼|
𝑝

= lim
𝑛→∞

|𝑒𝑛+1|

|𝑒𝑛|
𝑝

= 𝐶 

then the sequence {𝑥𝑛} converges to 𝛼 with the order of convergence 𝑝 [11]. 
 If 𝑝 = 1 then the iteration method has a linear convergence order. If 𝑝 = 2 then 
the iteration method has a quadratic order of convergence. If 1 < 𝑝 < 2 then iteration 
method has a superlinear order of convergence [12]. If 𝑝 = 3 then the iteration method 
has a cubic order of convergence [13]. Numerical methods such as the Newton-Raphson 
method have the quadratic order of convergence. While the Secant method has a 
superlinear order of convergence [12]. 
 Numerical methods often experience developments. The development aims to 
find methods that are considered more effective in solving existing problems [14]. Based 
on this in 2002 Kasturiarachi combine Newton's method and the Secant method become 
a new method, namely the leap-frogging Newton‘s method or Newton-Secant method 
[13]. This method has a cubic convergence when used to solve simple nonlinear 
equations. Whereas if used to solve multiple zeros of nonlinear equations the 
convergence to be linear. Therefore, in [15] modified Newton-Secant method with the 
addition of parameter 𝜃. The purpose of this modification, namely to maintain the order 
of convergence Newton-Secant method to remain cubic, if used to find the roots of 
nonlinear equations [15]. The following theorem related to the parameter 𝜃 used to 
modify the Newton-Secant method : 
 
Theorem 3  
Let 𝛼 ∈ 𝐷 be multiple root of a sufficiently differentiable function 𝑓 ∶  𝐷 ⊂ 𝑹 →  𝑹 on an 
open interval 𝐷 with multiplicity 𝑚 > 1, which includes 𝑥0 as an initial approximation of 
𝛼. Then, the modification of Newton-Secant method has order three and 𝜃 =

(
−1+𝑚

𝑚
)

−1+𝑚

, 𝑚 ∈ 𝑍+. 

Proof:  
Let 𝛼 is multiple zero of equation 𝑓(𝑥) = 0, then 𝑓(𝛼) = 0 and 𝑓′(𝛼) ≠ 0. Next, suppose  

𝑒𝑛 ≔ 𝑥𝑛 − 𝛼 
𝑒𝑛,�̅� ≔ 𝑥𝑛̅̅ ̅ − 𝛼 

𝑐𝑖 ≔
𝑚!

(𝑚 + 𝑖)!

𝑓 (𝑚+𝑖)(𝛼)

𝑓 (𝑚)(𝛼)
 

Using the Taylor expansion of 𝑓(𝑥𝑛) around 𝑥𝑛 = 𝛼 we get 
𝑓(𝑥𝑛) = 𝑐0𝑒𝑛

𝑚 + 𝑐1𝑒𝑛𝑒𝑛
𝑚 + 𝑐2𝑒𝑛

2𝑒𝑛
𝑚 + 𝑐3𝑒𝑛

3𝑒𝑛
𝑚 + 𝑂(𝑒𝑛

4)                            
simplified to  
 𝑓(𝑥𝑛) = 𝑒𝑛

𝑚(𝑐0 + 𝑐1𝑒𝑛 + 𝑐2𝑒𝑛
2 + 𝑐3𝑒𝑛

3) + 𝑂(𝑒𝑛
4), (3) 

and 



On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of 
Trigonometric Function 

Juhari 87 

 𝑓 ′(𝑥𝑛) = 𝑒𝑛
𝑚−1(𝑚 + (𝑚 + 1)𝑐1𝑒𝑛 + (𝑚 + 2)𝑐2𝑒𝑛

2 + (𝑚 + 3)𝑐3𝑒𝑛
3 + 𝑂(𝑒𝑛

4)). (4) 

If equation (3) is divided by equation (4), then we get 

 
𝑓(𝑥𝑛)

𝑓 ′(𝑥𝑛)
= 𝑒𝑛 (

(𝑐0 + 𝑐1𝑒𝑛 + 𝑐2𝑒𝑛
2 + 𝑐3𝑒𝑛

3) + 𝑂(𝑒𝑛
4)

(𝑚 + (𝑚 + 1)𝑐1𝑒𝑛 + (𝑚 + 2)𝑐2𝑒𝑛
2 + (𝑚 + 3)𝑐3𝑒𝑛

3 + 𝑂(𝑒𝑛
4))

) (5) 

Furthermore, equation (5) can be written as 

 
𝑓(𝑥𝑛)

𝑓 ′(𝑥𝑛)
=

1

𝑚
𝑒𝑛 −

𝑐1
𝑚2𝑐0

𝑒𝑛
2 +

−(1 + 𝑚)𝑐1
2 + 2𝑚𝑐0𝑐2

𝑚3𝑐0
2

𝑒𝑛
3 + 𝑂(𝑒𝑛

4), (6) 

and since 

 

𝑒𝑛,�̅� = 𝑥𝑛̅̅ ̅ − 𝛼

=
−1 + 𝑚

𝑚
𝑒𝑛 −

𝑐1
𝑚2𝑐0

𝑒𝑛
2 +

−(1 + 𝑚)𝑐1
2 + 2𝑚𝑐0𝑐2

𝑚3𝑐0
2

𝑒𝑛
3 + 𝑂(𝑒𝑛

4). 
(7) 

For 𝑓(𝑥𝑛̅̅ ̅) we have 

 𝑓(𝑥𝑛̅̅ ̅) = 𝑒𝑛,�̅�
𝑚 (𝑐0 + 𝑐1𝑒𝑛,�̅� + 𝑐2𝑒𝑛,�̅�

2 + 𝑐3𝑒𝑛,�̅�
3 ) + 𝑂(𝑒𝑛,�̅� 

4 ) (8) 

Substituting (3)-(8) in modification of Newton-Secant method formula, which the 
method will be constructed in the Research Results section. So we get 

𝑒𝑛+1 = 𝐷1𝑒𝑛 + 𝐷2𝑒𝑛
2 + 𝐷3𝑒𝑛

3 + 𝑂(𝑒𝑛
4), 

where 

𝐷1 = 1 +
𝜃

𝑚 (−𝜃 + (
−1+𝑚

𝑚
)

𝑚

)
    , 

and 

𝐷2 =
𝜃𝑚−2+𝑚(−𝑚(−1 + 𝑚)𝑚 + 𝜃𝑚𝑚(−1 + 𝑚))𝑐1

(−1 + 𝑚)((−1 + 𝑚)𝑚 − 𝜃𝑚𝑚)2𝑐0
   , 

and 

𝐷3 =
𝜃𝑚−3+𝑚𝐴

2(−1 + 𝑚)2((−1 + 𝑚)𝑚 − 𝜃𝑚𝑚)3𝑐0
2

    , 

where 
𝐴 = (−1 + 𝑚)2𝑚(−1 + 𝑚 + 2𝑚2)(𝑚𝑐1

2 − 2(−1 + 𝑚)𝑐0𝑐2)

+ 2𝜃2(−1 + 𝑚)2𝑚2𝑚((1 + 𝑚)𝑐1
2 − 2𝑚𝑐0𝑐2)

− 𝜃(−1 + 𝑚)1+𝑚(𝑚(3 + 4𝑚)𝑐1
2 + 2(1 + 𝑚 − 4𝑚2)𝑐0𝑐2).    

Therefore, to provide the three order of convergence, it is need to choose 𝐷𝑖 =
0   (𝑖 = 1, 2), so we have 

𝜃 = (
−1 + 𝑚

𝑚
)

−1+𝑚

   , 

and the error equation becomes 

𝑒𝑛+1 = (
𝑚𝑐1

2 − 2(−1 + 𝑚)𝑐0𝑐2

2𝑚2𝑐0
2

) 𝑒𝑛
3 + 𝑂(𝑒𝑛

4)  , 

and modification of Newton-Secant method has convergence order of three [15].  
 Based on the description above the researcher intends to construct the 
modification of Newton-Secant method in solving nonlinear equations for multiple 
zeros. Then  apply the method to solve the nonlinear equations for multiple zeros. In this 
case, it is applied to a nonlinear trigonometric function. To determine the effectivity of 
modification of Newton-Secant method then the solution will also be compared with 
Newton method, Secant method, and Newton-Secant method. 
 



On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of 
Trigonometric Function 

Juhari 88 

METHODS 

Research Steps 
1. Construction of mathematical model Newton-Secant method and its modification 

with the following steps: 
a) Construction of mathematical model Newton-Secant method. 

 Analyzing the theory related to the origin of the Newton-Secant 
method based on Kasturiarachi’s article 2002 entitled Leap-
Frogging Newton’s Method. 

 Performing analysis to obtain Newton's approximation by using 
the equation of the tangent (𝑥0, 𝑓(𝑥0)) that intersects (𝑥0̅̅ ̅, 0). 

 Create an equation of a secant connecting the points (𝑥0, 𝑓(𝑥0)) and 
(𝑥0̅̅ ̅, 𝑓(𝑥0̅̅ ̅)) using the equation of the line that passes through two 
point and assumes a secant that satisfies the 𝑥-axis on point (𝑥1, 0). 

 Substituting Newton's approximation into the equation of secant 
connecting the points (𝑥0, 𝑓(𝑥0)) and (𝑥0̅̅ ̅, 𝑓(𝑥0̅̅ ̅)). 

 Write the iteration formula based on the process that has been 
done at the stages above. 

b) Construction of mathematical model modification of Newton-Secant 
method with adding parameter 𝜃 to the second term of the Newton-Secant 
method. In this case, theorem 3 is used which is in the Introduction 
section.  

2. Solving nonlinear equation that have a multiplicity 𝑚 > 1 using modification of 
Newton-Secant method. In this case the solution is done by selecting two 
different initial values. After that the researcher compared the result with the 
Newton-Raphson method, Secant method, and Newton-Secant method that has 
not been modified.  This comparison aims to determine the effectivity of 
modification of Newton-Secant method if when viewed from the iterations, 
convergence, and time needed to solve a nonlinear equation having a multiplicity 
of 𝑚 > 1.  
 

RESULTS AND DISCUSSION 

1. Construction of mathematical model Newton-Secant method and its 
modification 

a. Construction of mathematical model Newton-Secant method 

 Newton-Secant method or also known as leap-frogging Newton's method is a 
combination of Newton method and Secant method. Based on this to do construction of 
mathematical model Newton-Secant method used the concept of Newton's method and 
the concept of the secant method. Suppose that the function 𝑓(𝑥) has the zero 𝛼 in the 
interval [𝑎, 𝑏] and 𝑓 ∈ 𝐶2[𝑎, 𝑏]. Let 𝑥0 be the initial guess. If the equation of the tangent 
line at (𝑥0, 𝑓(𝑥0)) intersects (𝑥0̅̅ ̅, 0) then by using the concept of Newton's method 
geometrically as follows : 



On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of 
Trigonometric Function 

Juhari 89 

 
Figure 1. Newton's Approximation Curve 

Then we get Newton's approximation : 

                       𝑓 ′(𝑥𝑛) =
Δ𝑦

Δ𝑥
 

                                      𝑓 ′(𝑥0) =
𝑓(𝑥0) − 0

𝑥0 − 𝑥0̅̅ ̅
 

            𝑓 ′(𝑥0)(𝑥0 − 𝑥0̅̅ ̅) = 𝑓(𝑥0) 

                             𝑥0 − 𝑥0̅̅ ̅ =
𝑓(𝑥0)

𝑓 ′(𝑥0)
 

 
            𝑥0̅̅ ̅ = 𝑥0 −

𝑓(𝑥0)

𝑓 ′(𝑥0)
 

(9) 

In this case used 𝑥0̅̅ ̅ instead of 𝑥1 because this is only used as an intermediate 
approximation. 
 Furthermore find the equation of the secant line connecting the points (𝑥0, 𝑓(𝑥0)) 
and (𝑥0̅̅ ̅, 𝑓(𝑥0̅̅ ̅)). To find these equations is used the concept of the Secant method 
geometrically. Look at the following curve : 

 
Figure 2. The Concept of Secant Method to Find the Equation of the Secant line 

 

Based on Figure 2 above, the following gradient is obtained:  

                       𝑓 ′(𝑥0) =
Δ𝑦

Δ𝑥
 

                                                                                =
[𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)]

𝑥0 − 𝑥0̅̅ ̅
                    

Tangent to the curve at 𝑥0 with 
gradient 𝑓 ′(𝑥0) 



On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of 
Trigonometric Function 

Juhari 90 

The gradient is used to get the equation of the line connecting the points (𝑥0, 𝑓(𝑥0)) and 
(𝑥0̅̅ ̅, 𝑓(𝑥0̅̅ ̅)). Based on the gradient formula to find the equation of the line that through 
two points, the following equation is obtained :   

 𝑦 − 𝑓(𝑥0) =
𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)

𝑥0 − 𝑥0̅̅ ̅
(𝑥 − 𝑥0) (10) 

 Assume the secant line meets the x-axis at the point (𝑥1, 0). So with substituting a 
value of 𝑥1 in 𝑥 and value of 0 in 𝑦 in equation (10) then we get, 
 

                                               0 − 𝑓(𝑥0) =
[𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)]

𝑥0 − 𝑥0̅̅ ̅
(𝑥1 − 𝑥0) 

 

 
                            

0 − 𝑓(𝑥0)

𝑥1 − 𝑥0
=

[𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)]

𝑥0 − 𝑥0̅̅ ̅
 

 

 
                                

−𝑓(𝑥0)

𝑥1 − 𝑥0
=

[𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)]

(𝑥0 − 𝑥0̅̅ ̅)
 

 

 [𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)] ∙ (𝑥1 − 𝑥0) = −𝑓(𝑥0) ∙ (𝑥0 − 𝑥0̅̅ ̅)  
 

                                     𝑥1 − 𝑥0 =
−𝑓(𝑥0) ∙ (𝑥0 − 𝑥0̅̅ ̅)

[𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)]
 

 

 
                                                     𝑥1 = 𝑥0 −

𝑓(𝑥0) ∙ (𝑥0 − 𝑥0̅̅ ̅)

[𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)]
 (11) 

 Then substitute Newton's approximation (9) into equation (11) so that optained, 

𝑥1 = 𝑥0 −

𝑓(𝑥0) ∙ (𝑥0 − (𝑥0 −
𝑓(𝑥0)

𝑓′(𝑥0)
))

[𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)]
 

      = 𝑥0 −
𝑓(𝑥0) ∙ (

𝑓(𝑥0)

𝑓′(𝑥0)
)

[𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)]
                          

     = 𝑥0 −

[𝑓(𝑥0)]
2 

𝑓′(𝑥0)

[𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)]
                         

       = 𝑥0 −
[𝑓(𝑥0)]

2

𝑓 ′(𝑥0)[𝑓(𝑥0) − 𝑓(𝑥0̅̅ ̅)]
         (12) 

Repeating this above process, the iteration formula can be written as following : 

where 

𝑥𝑛+1 = 𝑥𝑛 −
[𝑓(𝑥𝑛)]

2

𝑓 ′(𝑥𝑛)[𝑓(𝑥𝑛) − 𝑓(𝑥𝑛̅̅ ̅)]
   ,     𝑛 = 0, 1, 2,   .  .  .  . 

 

𝑥𝑛̅̅ ̅ = 𝑥𝑛 −
𝑓(𝑥𝑛)

𝑓 ′(𝑥𝑛)
                           

(13) 

Thus, we get equation (13) is the iteration formula for the Newton-secant method. 
 
b. Construction of mathematical model modification of Newton-Secant method 

 In solving simple nonlinear equations the Newton-Secant method has cubic 
convergence. While to solve the nonlinear equations for multiple zeros or multiplicity 
𝑚 > 1 the convergence is not cubic but becomes linear. Therefore, to maintain the 
convergence of the Newton-Secant method to remain cubic it is necessary to modify the 
method. The process modification of Newton-Secant method is done by adding 
parameter 𝜃 to the second term of the Newton-Secant method formula. The addition of 
these parameter resulted in the convergence of the modification of Newton-Secant 
method is cubic.  



On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of 
Trigonometric Function 

Juhari 91 

 The formula for the Newton-Secant method (13) can be written in the following 
form, 

𝑥𝑛+1 = 𝑥𝑛 −
𝑓(𝑥𝑛)

𝑓(𝑥𝑛) − 𝑓(𝑥𝑛̅̅ ̅)
∙

𝑓(𝑥𝑛)

𝑓 ′(𝑥𝑛)
  ,     𝑛 = 0, 1, 2,   .  .  .  .      

Refer to theorem 3 in the Introduction section to the construction of mathematical 
model modification of Newton-Secant method is done by adding parameter 𝜃 to the 
second term Newton-Secant method. So that the iteration formula is obtained as 
following :  

where 

𝑥𝑛+1 = 𝑥𝑛 −
𝜃𝑓(𝑥𝑛)

𝜃𝑓(𝑥𝑛 ) − 𝑓(𝑥𝑛̅̅ ̅)
∙

𝑓(𝑥𝑛)

𝑓 ′(𝑥𝑛)
  ,     𝑛 = 0, 1, 2,   .  .  .  . 

 

𝑥𝑛̅̅ ̅ = 𝑥𝑛 −
𝑓(𝑥𝑛)

𝑓 ′(𝑥𝑛)
         and         𝜃 = (

−1 + 𝑚

𝑚
)

−1+𝑚

 

(14) 

Thus, we get equation (14) is modification of Newton-Secant method formula which can 
be used to find solution to nonlinear equations for multiple zeros. 
2. Solving Nonlinear Equation for Multiple Zeros (Trigonometric Function) Use 

Modification of Newton-Secant Method 

 In the previous explanation, the construction of mathematical model Newton-
Secant method and its modification has been explained. To know effectivity of 
modification of Newton-Secant method then it is necessary to apply a modified Newton-
Secant method to nonlinear equations for multiple zeros. In this case, it is taken an 
example of nonlinear equation of trigonometric function, namely  

𝑓(𝑥) = (cos2 𝑥 + 𝑥)5 
Solving equation 𝑓(𝑥) use modification of Newton-Secant method will be applied with 
four different initial guess, namely 𝑥 = −2; 𝑥 = −0,8; 𝑥 = −0,2 and 𝑥 = 2.  
 The following are the steps to find a solution to the equation  
𝑓(𝑥) = (𝑐𝑜𝑠2 𝑥 + 𝑥)5 using modification of Newton-Secant method: 

1) Determine the initial guess to be used 

The initial guess selection is based on theorem 1 (intermediate value theorem) in 

the Introduction section. So that the initial guess are chosen, namely 𝑥 = −2; 𝑥 =

−0,8; 𝑥 = −0,2 and 𝑥 = 2. 

2) Finding the derivative of 𝑓(𝑥) 

𝑓(𝑥) = (cos2 𝑥 + 𝑥)5  

𝑓 ′(𝑥) = 5(𝑐𝑜𝑠2𝑥 + 𝑥)4(−2 cos 𝑥 sin 𝑥 + 1).  

3) Determine the multiplicity of 𝑓(𝑥) 

Based on the definition of multiplicity in the Introduction section, it can be said 

that the multiplicity of 𝑓(𝑥) is 𝑚 = 5. 

4) Set the error to be used 

In this case the author sets the error used, namely 10−10. 

5) Calculating the value of 𝜃 



On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of 
Trigonometric Function 

Juhari 92 

𝜃 = (
−1 + 𝑚

𝑚
)

−1+𝑚

= (
−1 + 5

5
)

−1+5

= (
4

5
)

4

= 0,4096 

6) Perform iteration using the modification of Newton-Secant method formula. 

7) Comparing the modification of Newton-Secant method with the Newton’s 

method, the Secant method, and the Newton-Secant method. It aims to know the 

effectiveness of the modification of Newton-Secant method in solving nonlinear 

equations for multiple zeros of trigonometric function. 

 The following is the result of solving 𝑓(𝑥) = (𝑐𝑜𝑠2 𝑥 + 𝑥)5 using modification of 
Newton-Secant method, Newton method, Secant method, and Newton-Secant method 
which has not been modified. 
Table 1. Solution 𝑓(𝑥) = (𝑐𝑜𝑠2 𝑥 + 𝑥)5 Use Modification of Newton-Secant Method, Newton Method, 

Secant Method, and Newton-Secant Method.   

𝒇𝒊 𝒙𝟎 Method n 𝒙𝒏 𝒇(𝒙𝒏) 𝜺 

𝒇(𝒙) = (𝒄𝒐𝒔𝟐 𝒙 + 𝒙)𝟓 

-2 

MMNS 5 -0,641714371 1,76869E-74 0,0000000000 

MNS 61 -0,641714371 8,67022E-49 0,0000000000 

MN 93 -0,641714371 1,20449E-47 0,0000000000 

MS 133 -0,641714371 -8,69088E-47 0,0000000000 

-0,8 

MMNS 4 -0,641714371 -4,17656E-74 0,0000000000 

MNS 60 -0,641714371 -1,81522E-48 0,0000000000 

MN 92 -0,641714371 -2,51216E-47 0,0000000000 

MS 131 -0,641714371 -9,67797E-47 0,0000000000 

-0,2 

MMNS 4 -0,641714371 -9,9601E-76 0,0000000000 

MNS 62 -0,641714371 3,45672E-48 0,0000000000 

MN 96 -0,641714371 1,88037E-47 0,0000000000 

MS 132 -0,641714371 -7,87677E-47 0,0000000000 

2 

MMNS 5 -0,641714371 9,07176E-75 0,0000000000 

MNS 64 -0,641714371 -9,82514E-49 0,0000000000 

MN 101 -0,641714371 1,84781E-47 0,0000000000 

MS 133 -0,641714371 -8,69088E-47 0,0000000000 

 

Information : 

𝑓𝑖  : Nonlinear equation function with  𝑖 =
1, 2, 3, …. 

𝑥0 : Initial guess or initial value 

n : Many iterations 



On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of 
Trigonometric Function 

Juhari 93 

𝑥𝑛  : Root of function 

𝑓(𝑥𝑛 ) : Function of root 𝑥𝑛  

𝜀 : Error (𝑥𝑛+1 − 𝑥𝑛 ) 

MMNS : Modification of Newton-Secant method  

MNS : Newton-Secant method 

MN : Newton method 

MS : Secant method 

 
 Based on table 1 above, the error is 0.0000000000, meaning that the error in the 
iteration value is less than the error tolerance constant used, namely 𝜀 < 10−10. 
Therefore, the iteration process stops and the approximate root is −0.641714371 which 
is the solution of 𝑓(𝑥). In the table above, it can be seen that taking four different initial 
values, namely 𝑥 = −2; 𝑥 = −0,8; 𝑥 = −0,2 and 𝑥 = 2, if the search for a solution uses 
modification of Newton-Secant method, iterations are needed more little when 
compared to Newton's method, the secant method, and the Newton-Secant method. 
Based on the many iterations, it can be said that the modification of Newton-Secant 
method is more effective in solving the nonlinear equation 𝑓(𝑥) when compared to the 
Newton method, the Secant method, and the Newton-Secant method which have not 
been modified. 
 The convergence of error values from the table of calculation results above can 
be seen in Table 2 and Table 3 below. 
Table 2. Convergence Graph Table of Error Values in the Solution of 𝑓(𝑥) Using Modification of Newton-

Secant Method and Newton-Secant Method 

𝒙𝟎 
Method 

Modification of Newton-Secant method Newton-Secant method 

-2 

 

 

-0,8 

 

 



On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of 
Trigonometric Function 

Juhari 94 

-0,2 

 

 

2 

 
 

 
Table 3. Convergence Graph Table of Error Values in the Solution of 𝑓(𝑥) Using Newton Method and 

Secant Method. 

𝒙𝟎 
Method 

Newton method Secant method 

-2 

  

-0,8 

 

 



On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of 
Trigonometric Function 

Juhari 95 

-0,2 

 

 

2 

 

 

 

CONCLUSION 

 Combining two concepts of numerical method, namely the concept of Newton's 
method and the concept of Secant method result Newton-Secant method. Then Newton-
Secant method modified by adding parameter 𝜃. So we get a new method, namely 
modification of Newton-Secant method with the following iteration formula:  

 where 

𝑥𝑛+1 = 𝑥𝑛 −
𝜃𝑓(𝑥𝑛)

𝜃𝑓(𝑥𝑛) − 𝑓(𝑥𝑛̅̅ ̅)
∙

𝑓(𝑥𝑛)

𝑓 ′(𝑥𝑛 )
  ,     𝑛 = 0, 1, 2,   .  .  .  . 

 

𝑥𝑛̅̅ ̅ = 𝑥𝑛 −
𝑓(𝑥𝑛)

𝑓 ′(𝑥𝑛 )
         and         𝜃 = (

−1 + 𝑚

𝑚
)

−1+𝑚

 

 
 Solution of 𝑓(𝑥) = (cos2 𝑥 + 5)5 use modification of Newton-Secant method with 
four different initial guess, namely 𝑥 = −2; 𝑥 = −0,8; 𝑥 = −0,2  and 𝑥 = 2 is obtained 
the root of 𝑥 approximately, namely −0,641714371 with fewer iterations when 
compared to using the Newton method, the Secant method, and the Newton-Secant 
method. Based on the problem of finding the root of the nonlinear equation 
trigonometric function 𝑓(𝑥) it can be concluded that the modification of Newton-Secant 
method is more effective than the Newton method, the Secant method, and the Newton-
Secant method that has not been modified.  

 

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