*Corresponding Author

P-ISSN: 2087-1244
E-ISSN: 2476-907X

45

ComTech: Computer, Mathematics and Engineering Applications, 13(1), June 2022, 45−52
DOI: 10.21512/comtech.v13i1.7322

The Solution of Non-Linear Equations System Containing 
Interpolation Functions by Relaxing the Newton Method

Nur Rokhman1*, Erwin Eko Wahyudi2, and Janoe Hendarto3 

1-3Department of Computer Sciences and Electronics, Faculty of Mathematics and Natural Sciences, 
Universitas Gadjah Mada

Sekip Utara, Bulaksumur, Daerah Istimewa Yogyakarta 55281, Indonesia
1nurrokhman@ugm.ac.id; 2erwin.eko.w@ugm.ac.id; 3jhendarto@ugm.ac.id

Received: 30th April 2021/ Revised: 24th June 2021/ Accepted: 25th June 2021 

How to Cite: Rokhman, N., Wahyudi, E. E., & Hendarto, J. (2022). The Solution of Non-Linear Equations System 
Containing Interpolation Functions by Relaxing the Newton Method. ComTech: Computer, Mathematics and Engineering 

Applications, 13(1), 45−52. https://doi.org/10.21512/comtech.v13i1.7322

Abstract - Many world phenomena lead to non-
linear equations systems. For some applications, the 
non-linear equations which construct the non-linear 
equations system are interpolation functions. However, 
the interpolation functions are usually not represented 
as mathematical expressions but as computer 
programs in specific programming languages. The 
research proposed using the relaxed Newton method 
to solve the non-linear equations system that contained 
interpolation functions. The interpolation functions 
were represented in the R programming language. 
Then, the experiment used the Spline interpolation 
function to construct a two-dimensional non-linear 
equations system. Eleven initial guesses, maximum 
of ten-time iterations, and 10-7 precision were applied. 
The solution of the non-linear equations system and the 
iteration needed on each initial guess were observed. 
The experiment shows that the proposed method 
works well for solving the non-linear equations system 
constructed by Spline interpolation functions. By 
observing the initial guesses used in the experiment, 
there are four possible results: true solution, spurious 
solution, false solution, and no solution. Applying 11 
initial guesses have five initial guesses resulting in true 
solutions, one initial guess in spurious solution, three 
initial guesses in false solutions, and one initial guess 
in no solution. The discussions imply that this method 
can be generalized to the three-dimensional non-linear 
equations system or higher dimensions.

Keywords:  non-linear equation, interpolation function, 
Newton method  

I. INTRODUCTION

Many world phenomena lead to the problem of 
non-linear equations systems. The methods of solving 
non-linear equations systems are very interesting. 
There are many solutions of the non-linear equations 
system which are developed from the Newton 
method. These solutions solve explicit functions 
which are represented in mathematical notations. 
On the other hand, the higher implementation of 
computer technology has raised the development of 
the mathematical function, which is represented in a 
computer program, such as interpolation functions. 
However, to the best of researchers’ knowledge, 
no method has been developed to solve non-linear 
equations systems with function represented as a 
computer program.

Interpolation can be viewed as a process 
of determining a function that passes through the 
interpolation points. In a complex computation that 
involves many interpolation points, the interpolation 
function is not presented explicitly in mathematical 
notation. The interpolation function is presented 
in computational steps as a computer program in 
such cases. The interpolation functions may work 
simultaneously and build a non-linear equations 
system. Currently, the methods to solve non-linear 
equations system work based on explicit functions. 
However, these methods cannot be used to solve non-
linear equations systems that contain interpolation 
function since the interpolation function is represented 
as computational steps. Hence, a certain method is 
needed to solve the non-linear equation system with 
the interpolation functions since the functions are 
not presented explicitly in mathematical notation but 
computational steps. 



46 ComTech: Computer, Mathematics and Engineering Applications, Vol. 13 No. 1 June 2022, 45−52

In general, the solution of a non-linear equation 
system by using the Newton method and its variants 
requires an initial value and the derivative of the non-
linear equations involved in the system. The solution 
of a non-linear equation system that does not need any 
derivative has been carried out by Rokhman (2013). 
In a non-linear equation that contains interpolation 
functions, the explicit formulation of the derivative 
cannot be obtained. Hence, the Newton method is 
unable to be used. The study of the solution of non-
linear equations which contains the interpolation 
function has been carried out by Rokhman (2017). 

The problem of solving the system of non-linear 
equations can be viewed as a matter of multi-objective 
optimization. Every equation in the system states an 
objective function that aims to minimize the difference 
between the right and left terms of the corresponding 
equation (Grosan & Abraham, 2008). A new algorithm 
based on the method of Adomian decomposition 
convergence basis for solving functional equations 
is presented. This algorithm can account for all the 
real answers of a system if a suitable and primary 
approximation is chosen (Hosseini & Kafash, 2010).

An iterative method for solving the problem 
of the non-linear equation numerically is suggested. 
A mathematical proof supports the proposed 
iteration through the n-dimensional Taylor expansion 
(Montazeri et al., 2012). Then, a new algorithm is 
proposed for the solutions of systems of non-linear 
equations which use a combination of the gradient and 
the Newton methods. A novel dynamic combinatory 
is developed to determine the contribution of the 
methods in the combination. The numerical results 
prove that the proposed combination algorithm is 
generally more robust and efficient than other methods 
on some important and difficult problems (Taheri & 
Mammadov, 2012).

Three algorithms that work based on a 
generalization of fourth-order Jarratt formula iterative 
methods for solving a system of non-linear equations 
of n-dimension with n-variables are proposed. These 
new algorithms may be viewed as extensions and 
generalizations of the existing methods for solving 
the system of non-linear equations (Khirallah & 
Hafiz, 2013). Then, a new fourth-order Jarratt-type 
method for solving systems of non-linear equations is 
proposed. The local convergence order of this method 
is proven (Ghorbanzadeh & Soleymani, 2015). A 
generalization of a family of methods for solving 
non-linear equations with unknown multiplicities to 
the system of non-linear equations is also proposed 
by using a non-zero multi-variable auxiliary function. 
Different auxiliary functions give different families 
of the iterative method to find roots with unknown 
multiplicities (Ahmad et al., 2016).

Moreover, a new two-step Newton-type method 
with third-order convergence for solving systems of 
non-linear equations is proposed. The new method 
works based on the integral interpolation of the Newton 
method. Its cubic convergence and error equation is 
proven theoretically and demonstrated numerically 

(Liu & Fang, 2015). There is also an improvement 
to the recently introduced factored method for the 
solution of non-linear equations. In this method, the 
original system is transformed by adding an offset to 
all unknowns (Ruiz-Oltra et al., 2016).

A new alternative approximation based on the 
quasi-Newton approach for solving systems of non-
linear equations using the average of midpoint and 
the Simpson quadrature is conducted. This method 
reduces the number of iterations to reach a solution 
(Mohammad & Waziri, 2015). Then, according to 
Abubakar and Waziri (2016), an enhancement of a 
three-step Newton-like method where the Jacobian 
inverse is approximated into a diagonal matrix is 
proposed. Numerical results prove that the method in 
terms of CPU time and the number of iterations is very 
encouraging.

Then, some researchers also propose a fifth-
order convergence three-step method for solving a 
system of non-linear equations. The efficiency index of 
this method is better than the other three-step methods. 
The advantages of the method lie in the feature that this 
technique not only achieves an approximate solution 
with high accuracy but also improves the calculation 
speed (Ahmadi et al., 2016).

Two multi-step derivative-free iterative methods 
for solving a system of non-linear equations are also 
presented. These methods have high computational 
efficiency and low computational cost (Wang & Fan, 
2016). It has been shown that the proposed method for 
reformulation and rearrangement of the NLE systems 
can reduce the dimension of the system, remove 
discontinuities, and considerably simplify the task 
of finding initial estimates for the unknowns so that 
solution is obtained (Shacham & Brauner, 2017).

There are also three higher-order iterative 
methods of convergence order four, five, and six for 
solving a system of non-linear equations. New weight 
functions are included in the second term of the 
Newton method to get higher-order methods. Higher-
order accuracy is achieved with only one inverse of 
the Jacobian matrix (Madhu & Jayaraman, 2017). 
Moreover, two basic methods of approximating the 
solutions of non-linear systems of algebraic equations 
are considered. The Steepest Descent method obtains 
a good and sufficient initial guess. The Broyden 
method replaces the Newton method. The obtained 
results suggest that the number is reduced compared 
to the Newton method in the same problem (Mahwash 
& Gyang, 2018).

A new iterative method based on the quasi-
Newton approach for solving systems of non-linear 
equations, especially large scale, is also proposed. It 
uses the weighted combination of the Trapezoidal and 
Simpson quadrature rules (Osinuga & Yusuff, 2018). 
According to Dwail and Shiker (2021), the Trust 
Region Method (TRM) is combined with Broyden–
Fletcher–Goldfarb–Shanno (BFGS) technique to 
find the solution to the non-linear systems. Next, 
two Broyden-like methods are developed from a 
weighted combination of quadrature rules, namely the 



47The Solution of Non-Linear..... (Nur Rokhman et al.)

Trapezoidal, Simpson 3/8, and Simpson 1/3 quadrature 
rules (Isaac et al., 2021). 

Based on Halilu and Waziri (2020), there is 
a derivative-free method for solving large-scale 
systems of non-linear equations via a double direction 
approach. The acceleration parameter used in this 
approach approximated the Jacobian matrix to form 
a derivative-free method. Then, a new line search 
algorithm is suggested for solving non-linear equations 
systems that combine a monotone technique into a 
modified line search rule. These techniques show 
that these methods solve the non-linear system with 
less time and effort without evaluating the derivative 
(Hashim et al., 2019).

Solving a non-linear equations system using the 
Newton method needs the derivative of each function 
to construct the non-linear equations system. The 
research proposes a novel way to solve a non-linear 
equations system constructed by the interpolation 
functions. The interpolation functions are represented 
in computer programs whose derivatives cannot be 
represented as explicit functions. The absence of 
derivative function causes Newton method not to 
work on solving the non-linear equations system, 
constructed by the interpolation functions.

II. METHODS

In the Newton method, a numerical solution of 
a non-linear equation f(x) = 0 is found by iterating an 
initial guess x0 that moves closer to the solution by 
using Equation (1). The iteration is stopped when the 
error tolerance (ε) is reached. Otherwise, no solution 
can be found. Relaxing the Newton method can be 
carried out by using the error tolerance (ε) to determine 
the derivative value of the function. It means that no 
function derivative is needed (Rokhman, 2011). The 
Newton method for solving a non-linear equation 
system needs derivative of the functions constructing 
the system. The derivative-free method to solve the 
non-linear equations system can be developed by 
relaxing the Newton method (Rokhman, 2013). The 
relaxed Newton method can also be used to solve a 
non-linear equation containing interpolation functions. 
An interpolation function is represented in computer 
program. In this case, no explicit function of its 
derivative like Equation (1) does not work (Rokhman, 
2017).

      (1)

A non-linear equations system with n variables 
x1, …, xn is constructed by n non-linear functions 
F1, … Fn, generally written as in Equation (2). The 
solution of the non-linear equations system is the 
value of variables x1, …, xn which are true for all non-
linear functions.

        (2)

Suppose x, F(x), F´(x) are defined as in Equation 
(3), Equation (4), and Equation (5), respectively with 

 as a partial derivative of function Fp to variable xq 
where p = 1,…n, q = 1, …n. The matrix in Equation 
(5) is known as the Jacobian matrix. Then, the Newton 
method for solving non-linear equation systems can be 
written as Equation (6) (Atkinson & Han, 2004).

          (3)

        (4)

       (5)

      (6)

The research proposes the application of 
relaxing the Newton method for solving non-linear 
equations systems which contain interpolation 
functions. A non-linear equation system that will be 
solved in the research is presented in Equation (7). 
Each f1(x) and f2(x) is the results of two-dimensional 
Spline interpolation function.

         (7)

Basically, a solution of this non-linear equations 
system is an intersection point of f1(x) and f2(x). The 
non-linear equations systems in Equation (7) can be 
rewritten, as shown in Equation (8).

       (8)

Suppose it is F1(x, y) − f1(x) − y and F2(x, y) − 
f2(x) − y. Then, the non-linear equations systems are 
a two-dimensional system. It contains two functions, 
F1(x, y) and F2(x, y). The Jacobian matrix of the non-
linear equations system can be built as presented in 
Equation (9).

     (9)



48 ComTech: Computer, Mathematics and Engineering Applications, Vol. 13 No. 1 June 2022, 45−52

Using a small value of ε, the relaxed Newton 
method can be used to construct Jacobian matrices 
of Equation (9) and gives a new Jacobian matrix as 
presented in Equation (10).

   (10)

Suppose ∂x and ∂y are the difference of x and 
y variables, respectively. The relaxing of Newton 
method for solving non-linear equation systems can 
be constructed, as shown in Equation (11).

   (11)

The initial guess for the solutions of the 
non-linear equations system is (x0, y0). Then, the 
improvement to this initial guess is carried out by using 
Equation (12). The difference of x and y variables on 
each iteration, namely ∂x and ∂y can be obtained from 
the Equation (13). These processes are repeated until 
the desired error tolerance (ε) is reached.

    (12)

An algorithm to find the solution of the non-
linear equations system, which contains the Spline 
interpolation function, can be constructed by using 
Equations (12) and (13). The algorithm inputs are two 
sets of interpolation points, a precision value called 
eps, an initial guess (x0, y0), and a maximum number 
of iterations called maxIter. When the error tolerance 
value is reached at t iteration, the output of the 
algorithm is a point (xt, yt). This point is the solution of 
the non-linear equations system containing the Spline 
interpolation function.

The first step of the algorithm constructs the 
interpolation functions. Suppose f(x) and g(x) are the 
interpolations functions. The second step calculates 
the derivatives of the f(x) and g(x) at the initial 
guess by using the relaxed Newton method. These 
values are used to construct the Jacobian matrix. The 
improvement values of the initial guess, called dx and 
dy, are calculated using the inverse of the Jacobian 
matrix and the values of the f(x) and g(x) at the initial 
guess (x0, y0). The iteration process is completed by 
improving the initial guess using dx and dy, but the 
error tolerance still needs to be checked. The solution 
is found if both dx and dy are less than eps. Otherwise, 
the solution is not yet found, and the processes are 
repeated with the new value of (x, y).

Under some conditions, the value of the 
interpolation function may be undefined, or the error 
tolerance cannot be reached after several maxIter 
iterations. The processes must be stopped, and no 
solution is found. These conditions are represented in 
the seventh and eighth steps. The complete algorithm 
is depicted in Figure 1.

      (13)

Input : two sets of interpolation points, 
        Precision as eps,
        Initial guess as (x

0
, y

0
), 

        Maximum number of itertion as maxIter 
Output: (x,y) 
Steps :

1. Do the interpolation. f(x) and g(x) 
as the result

 
t = 0
found = false

2. dfx = (f(x
t
+eps)-f(x

t
))/eps

dgx = (g(x
t
+eps)-g(x

t
))/eps

3. A = 

4. D =  = -J-1A

5.  
t = t + 1

6. If | f(x
t
)- g(x

t
)| < eps and then 

found = true
7. If f(x

t
) or g(x

t
) undefined then Break

8. Repeat from step 2 until found or (t 
>= maxIter)

9. If found then the result is(x
t
,y

t
)

Figure 1 Algorithm of Solving Non-Linear System 
Containing Interpolation Functions

The experiment is carried out by implementing 
the algorithm in the R programming language. In 
this programming language, the construction of the 
Spline interpolation function is done easily by calling 
the splinefun function. The research objects are two 
interpolation functions used by Dontchev et al. (2002). 
These functions are used to construct the non-linear 
equations system. The solutions of the system are 
the intersection points of the interpolation functions. 
Then, some initial points are used in the experiments. 
The final result and the number of iterations to obtain 
the final result of each initial point are observed.



49The Solution of Non-Linear..... (Nur Rokhman et al.)

III. RESULTS AND DISCUSSIONS

The research objects are the interpolations 
points as used in Dontchev et al. (2002). The non-linear 
equations system is constructed from the interpolation 
function of interpolation points  {(0,0, 0,0), (0,05, 0,7), 
(0,1, 1,0), (0,2, 1,0), (0,8, 0,3), (0,85, 0,05), (0,9, 0,1), 
(1,0, 1,0)} and the interpolation points {(0,0, 0,0), (0,1, 
0,9), (0,2, 0,95), (0,3, 0,9), (0,4, 0,1), (0,5, 0,05), (0,6, 
0,05), (0,8, 0,2), (1,0, 1,0)}. The interpolations are 
carried out by using the Spline interpolation function. 
Then, these interpolation functions construct the non-
linear equations system which will be solved by using 
the relaxing of the Newton method.

Suppose the first interpolation gives a function 
called f(x). The second interpolation has a function 
called g(x). The graph of these functions is shown 
in Figure 2. The non-linear equations system to be 
solved is presented in Equation (14). By observing 
the graph in Figure 2, the solutions of the non-linear 
equations system consist of five points in which the 
x value lies between 0 and 1, namely (0,0, 0,0) and 
(1,0, 1,0), which are the interpolation points, and three 
intersection points.

Figure 2 Graph of the Interpolation Functions

       (14)

Then, eleven initial guesses, namely (0,0, 0,0), 
(0,1, 1,0), (0,2, 2,0), (0,3, 3,0), (0,4, 4,0), (0,5, 5,0), 
(0,6, 6,0), (0,7, 7,0), (0,8, 8,0), (0,9, 9,0), and (1,0, 10,0) 
are used in the experiment. The maximum iteration 
is set to 10. The precision value is set to 0,0000001. 
Then, the algorithm implemented in R programming 
language is shown in Figure 3.

The proposed method is used to find the 
solutions to non-linear equations system containing 
two Spline interpolation functions in Equation (16). 
For example, there are five points of the intersection 
between f(x) and g(x). Even though the points of (0,0, 
0,0) and (1,0, 1,0) are trivial solutions, these points 
are the interpolations points of both interpolation 
functions.

The performance of relaxing the Newton 
method for solving non-linear equation system has 
been observed. Two parameters are observed: the 
iteration needed to find the solutions (n) and the 
solutions (xn, yn). The experiment results are shown 
in Table 1. The initial guesses of (0,0, 0,0) and (1,0, 
10,0) need only one iteration to get the solution since 
these initial guesses are the interpolation points of the 
interpolation functions. Then, the initial guesses of 
(0,2, 2,0) and (0,4, 4,0) have almost the same solution 
namely (0,2259318, 0,9756243) and (0,2259319, 
0,9756243) respectively. It is the first intersection 
point. However, the required iterations are different. 
The initial guess of (0,2, 2,0) requires three times 
of iterations. Meanwhile, the initial guess of (0,4, 
4,0) requires five times of iterations. The detailed 
observation in processing the initial guess of (0,2, 0,2) 
and (0,4, 0,4) is shown in Tables 2 and 3, respectively.

maxIter <- 10
eps <- 0.0000001
x1 <- c(0.0,0.05,0.1,0.2,0.8,0.85,0.9,1.0)
y1 <- c(0.0,0.7,1.0,1.0,0.3,0.05,0.1,1.0)
f <- splinefun(x1,y1)
x2 <- c(0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.8,1.0)
y2 <- c(0.0,0.9,0.95,0.9,0.1,0.05,0.05,0.2,1.0)
g <- splinefun(x2,y2)
expDataX <-c(0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0
.9,1.0)
expDataY <-c(0.0,1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0
,9.0,10.0)
for (i in 1:length(expDataX))
{
  x <- matrix(c(expDataX[i],expDataY[i]),nrow=2, 
ncol=1, byrow=TRUE)
  iter <-0
  repeat
  {
    iter <-iter + 1
    dfdx <- (f(x[1]+eps)-f(x[1]))/eps
    dfdy <- -1
    dgdx <- (g(x[1]+eps)-g(x[1]))/eps
    dgdy <- -1
    jacobi <- matrix(c(dfdx,dfdy,dgdx,dgdy),                 
nrow=2,ncol=2,byrow=TRUE)
    A <- matrix(c(f(x[1])-x[2],g(x[1])-x[2]))
    dxdy <- -solve(jacobi)%*%A
    x <- x+dxdy
    if (abs(f(x[1])-g(x[1]))<eps)
    {
      found<-TRUE
      break
    }
    if (iter >= maxIter) break
  } 
print(c(expDataX[i],expDataY[i],iter,x[1],x[2]))
}

Figure 3 R Program to Find the Solution



50 ComTech: Computer, Mathematics and Engineering Applications, Vol. 13 No. 1 June 2022, 45−52

Table 1 The Experimental Results

No (x0, y0) n (xn, yn)
1 (0,0, 0,0) 1 (0,0, 0,0)
2 (0,1, 1,0) 9 (-1,029962e-08, -7,092949e-08)
3 (0,2, 2,0) 3 (0,2259318, 0,9756243)
4 (0,3, 3,0) 4 (0,2879449, 0,9510676)
5 (0,4, 4,0) 5 (0,2259319, 0,9756243)
6 (0,5, 5,0) 4 (0,8124135, 0,2264703)
7 (0,6, 6,0) 7 (1,824736, 22,33810)
8 (0,7, 7,0) 10 (-0,0312172, -0,2897813)
9 (0,8, 8,0) 2 (0,8124135, 0,2264695)
10 (0,9, 9,0) 8 (1,824736, 22,338101)
11 (1,0, 10,0) 1 (1,0, 1,0)

Table 2 The Detail Observation of Initial Guess 
of (0,2, 0,2)

n (xn, yn) |f(xn) − g(xn)| 
1 (0,2293685, 0,9668224) 0,006472153
2 (0,2258565, 0,9756006) 0,00014464
3 (0,2259318, 0,9756243) 5,269591e-08

Table 3 The Detail Observation of Initial Guess 
of (0,4, 0,4)

n (xn, yn) |f(xn) − g(xn)|
1 (0,2468062, 0,9178059) 0,03282423
2 (0,2162326, 0,9777921) 0,01918141
3 (0,2258095, 0,9750844) 0,00023499
4 (0,2259318, 0,9756243) 1,385406e-07
5 (0,2259319, 0.9756243) 1,876277e-14

The initial guess of (0,3, 3,0) gives solution of 
(0,2879449, 0,9510676) after four times iterations. 
The detailed observation in processing the initial guess 
of (0,3, 0,3) is shown in Table 4. According to Figure 
2, this initial point finds the second intersection point. 
Table 1 shows the fact that only initial guess (0,3, 0,3) 
finds the second intersection point.

Table 4 The Detail Observation of Initial Guess 
of (0,3, 0,3)

n (xn, yn) |f(xn) − g(xn)|
1 (0,2902525, 0,9505274) 0,007918907
2 (0,2880567, 0,9510382) 0,0003654196
3 (0,2879452, 0,9510675) 9,246800e-07
4 (0,2879449, 0,9510676) 8,073542e-12

The initial guesses of (0,5, 5,0) and (0,8, 8,0) 
have almost the same solution, namely (0,8124135, 
0,2264703) and (0,8124135, 0,2264695), respectively. 
However, the required iterations are different. The 
initial guess of (0,5, 5,0) requires four times of 
iterations. Meanwhile, the initial guess of (0,8, 8,0) 
needs two times of iterations. The detailed observation 
in processing initial guesses (0,5, 5,0) and (0,8, 8,0) 
are shown in Tables 5 and 6, respectively.

Table 5 The Detail Observation of Initial Guess 
of (0,5, 0,5)

n (xn, yn) |f(xn) − g(xn)|
1 (1,4095988, 0,9933773) 4,5814418
2 (0,81692703, -7,30583402) 0,03670788
3 (0,8123981, 0,2262701) 0,0001253189
4 (0,8124135, 0,2264703) 9,604448e-11

Table 6 The Detail Observation of Initial Guess 
of (0,8, 0,8)

n (xn, yn) |f(xn) − g(xn)|
1 (0,8127117, 0,2258902) 0,002429513
2 (0,8124135, 0,2264695) 7,154819e-08

Table 5 shows the fact that the final iteration 
of the initial guess of (0,5, 0,5) finds the solution, 
but there is a point moves out from the interpolation 
range. So, the solution is not the real true solution. It 
is called a spurious solution. Table 6 shows the fact 
that the initial guess (0,8, 0,8) gives real true solution. 
There is no iteration point which moves out from the 
interpolation range by using this initial guess.

Next, initial guesses of (0,1, 1,0), (0,6, 6,0), 
and (0,9,9,0) give solutions as following (-1,029962e-
08, -7,092949e-08), (1,824736, 22,33810), and 
(1,824736, 2233810) with the final precisions 
2,729921e-08, 1,350031e-13, and 9,237056e-14, 
respectively. These are false solutions since they move 
out from the interpolation range.  However, the initial 
guess of (0,7, 7,0) has no result. The process reaches 
the maximum number of iterations but does not meet 
the required precision. The final value of this initial 
guess is (x10, y10) = −0,0312172, −0,2897831). The 
value of f(0,0312172)  is −0,7068187. The value of 
g(0,0312172) is −0,5983234. Both values give |f(x10) 
− g(x10)| > ε that means no solution is found. 

Based on the experiment results, there are 
four categories of possible results for implementing 
the relaxing Newton method for solving non-linear 
equations system. They are true solution, spurious 
solution, false solution, and no solution. The initial 
guesses of (0,0, 0,0), (0,2, 2,0), (0,3, 3,0), (0,4, 4,0), 
(0,8, 8,0), and (1,0, 10,0) result in true solutions on 



51The Solution of Non-Linear..... (Nur Rokhman et al.)

this experiment. Then, the initial guess of (5,0, 5,0) 
gives a spurious solution. On the other hand, the initial 
guesses of (0,1, 1,0), (0,6, 6,0), and (0,9, 9,0) result in 
false solutions. Last, the initial guess of (0,7, 7,0) does 
not find the solution.

The relaxing of the Newton method can also be 
implemented for solving non-linear equations system 
which contains the three-dimensional interpolation 
functions. For this case, suppose Equations (7), (8), 
and (9) are advanced by Equations (15), (16), and 
(17), respectively. The initial guess is 

      (15)

     (16)

     (17)

It applies the relaxed Newton method via 
introducing the error tolerance (ε) for approximating 
the partial derivative of F1, F2, and F3 functions to the x1 
and x2 variables. As an example, the partial derivative 

 means the ratio between the change of F1 if there 
is ε change in x2 variable and the change of x2, that is ε.  
The values of each partial derivative in Equation (17) 
can be determined using Equations (18)−(23).

   (18)

   (19)

   (20)

   (21)

   (22)

   (23)

The general form of the relaxing Newton 
method for solving non-linear equations system is 
presented in Equation (24) and Equation (25). The 
solution of Equation (25) gives the values of ∂x1, ∂x2, 

and ∂y, which are used to improve the values of  , 
, and yn, respectively.

     (24)

    (25)

IV. CONCLUSIONS

Nowadays, many functions in the computer 
program form, like the interpolation function, have been 
developed. From the discussions, it can be concluded 
that the relaxing of the Newton method can be used to 
solve the non-linear equations system which contains 
the interpolation functions. The research shows the 
fact that the solution to systems constructed by these 
functions can be found by relaxing the Newton method. 
It means, the non-linear equations system containing 
such functions can be solved easily by relaxing the 
Newton methods.  An initial guess may give a true 
solution, a spurious solution, a false solution, or no 
solution. However, there is no guarantee for an initial 
point to give the result a true solution, a spurious 
solution, a false solution, or no solution. The research 
discusses an experiment in two-dimensional data. 
Theoretically, it can be expanded to three-dimensional 
data or generalized to n-dimensional data.  

The research is limited by the absence of an 
explicit form of the interpolation function. It makes 
the properties exploration of the non-linear equations 
system containing the interpolation function cannot 
be conducted yet. Future research can be addressed to 
explore the properties of such a non-linear equations 
system.

ACKNOWLEDGEMENT

The authors are indebted to the Department of 
Computer Science and Electronics which has given 
full support to the research. Hopefully, the research 
makes a major contribution to the advancement of 
numerical solutions to non-linear equations systems.

REFERENCES

Abubakar, A. B., & Waziri, M. Y. (2016). A matrix-
free approach for solving systems of nonlinear 
equations. Journal of Modern Methods in Numerical 
Mathematics, 7(1), 1‒9. 



52 ComTech: Computer, Mathematics and Engineering Applications, Vol. 13 No. 1 June 2022, 45−52

Ahmad, F., Serra-Capizzano, S., Ullah, M. Z., & Al-
Fhaid, A. S. (2016). A family of iterative methods 
for solving systems of nonlinear equations having 
unknown multiplicity. Algorithms, 9(1), 1‒10. 
https://doi.org/10.3390/a9010005

Ahmadi, M., Esmaeili, H., & Erfanifar, R. (2016). Solving 
system of nonlinear equations by using a new three-
step method. Control and Optimization in Applied 
Mathematics, 1(2), 53–62.

Atkinson, K., & Han, W. (2004). Elementary numerical 
analysis. Wiley.

Dontchev, A. L., Qi, H. D., Qi, L., & Yin, H. (2002). A Newton 
method for shape-preserving spline interpolation. 
SIAM Journal on Optimization, 13(2), 588–602. 
https://doi.org/10.1137/S1052623401393128

Dwail, H. H., & Shiker, M. A. K. (2021). Using Trust Region 
method with BFGS technique for solving nonlinear 
systems of equations. Journal of Physics: Conference 
Series, 1818, 1‒9. https://doi.org/10.1088/1742-
6596/1818/1/012022

Ghorbanzadeh, M., & Soleymani, F. (2015). A quartically 
convergent Jarratt-type method for nonlinear system 
of equations. Algorithms, 8(3), 415–423. https://doi.
org/10.3390/a8030415

Grosan, C., & Abraham, A. (2008). A new approach 
for solving nonlinear equations systems. IEEE 
Transactions on Systems, Man, and Cybernetics 
- Part A: Systems and Humans, 38(3), 698–714. 
https://doi.org/10.1109/TSMCA.2008.918599

Hashim, K. H., Dreeb, N. K., Dwail, H. H., Mahdi, M. 
M., Wasi, H. A., Shiker, M. A. K., & Hussein, H. 
A. (2019). A new line search method to solve the 
nonlinear systems of monotone equations. Journal of 
Engineering and Applied Sciences, 14, 10080–10086. 
https://doi.org/10.36478/jeasci.2019.10080.10086

Halilu, A. S., & Waziri, M. Y. (2020). Solving systems of 
nonlinear equations using improved double direction 
method. Journal of the Nigerian Mathematical 
Society, 39(2), 287–301.

Hosseini, M. M., & Kafash, B. (2010). An efficient algorithm 
for solving system of nonlinear equations. Applied 
Mathematical Sciences, 4(3), 119–131.

Isaac, A., Stephen, T. B., & Seidu, B. (2021). A new 
Trapezoidal-Simpson 3/8 method for solving 
systems of nonlinear equations. American Journal of 
Mathematical and Computer Modelling, 6(1), 1–8. 
https://doi.org/10.11648/j.ajmcm.20210601.11

Khirallah, M. Q., & Hafiz, M. A. (2013). Solving system of 
nonlinear equations using family of Jarratt methods. 
International Journal of Differential Equations and 
Applications, 12(2), 69–83. https://doi.org/10.12732/
ijdea.v12i2.931

Liu, Z., & Fang, Q. (2015). A new Newton-type method 
with third-order for solving systems of nonlinear 
equations. Journal of Applied Mathematics and 
Physics, 3(10), 1256–1261. https://doi.org/10.4236/
jamp.2015.310154

Madhu, K., & Jayaraman, J. (2017). Some higher order 
Newton-like methods for solving system of nonlinear 
equations and its applications. International Journal 
of Applied and Computational Mathematics, 3, 
2213–2230. https://doi.org/10.1007/s40819-016-
0234-z

Mahwash, K. N., & Gyang, G. D. (2018). Numerical 
solution of nonlinear systems of algebraic equations. 
International Journal of Data Science and 
Analysis, 4(1), 20‒23. https://doi.org/10.11648/j.
ijdsa.20180401.14

Mohammad, H., & Waziri, M. Y. (2015). On Broyden-
like update via some quadratures for solving 
nonlinear systems of equations. Turkish Journal of 
Mathematics, 39, 335–345. https://doi.org/10.3906/
mat-1404-41

Montazeri, H., Soleymani, F., Shateyi, S., & Motsa, S. 
S. (2012). On a new method for computing the 
numerical solution of systems of nonlinear equations. 
Journal of Applied Mathematics, 2012, 1‒15. https://
doi.org/10.1155/2012/751975

Osinuga, I. A., & Yusuff, S. O. (2018). Quadrature based 
Broyden-like method for systems of nonlinear 
equations. Statistics, Optimization & Information 
Computing, 6(1), 130–138. https://doi.org/10.19139/
soic.v6i1.471

Rokhman, N. (2011). Secton: A combination of Newton 
Method and Secant method for solving non linear 
equations. In Seminar Nasional Aplikasi Teknologi 
Informasi (SNATI) (pp. 50–52). 

Rokhman, N. (2013). The implementation of Secton 
method for solving systems of non linear equations. 
In International Conference on Engineering and 
Technology Development (ICETD) (pp. 80–84).

Rokhman, N. (2017). Solving non-linear equations 
containing spline interpolation function by relaxing 
the Newton method. In 2017 Second International 
Conference on Informatics and Computing 
(ICIC) (pp. 1–5). IEEE. https://doi.org/10.1109/
IAC.2017.8280641

Ruiz-Oltra, J. M., Gómez-Quiles, C., & Gómez-Expósito, 
A. (2016). Offset-assisted factored solution of 
nonlinear systems. Algorithms, 9(1), 1–14. https://
doi.org/10.3390/a9010002

Shacham, M., & Brauner, N. (2017). Solving a system of 
nonlinear algebraic equations: You only get error 
messages-What to do next? Chemical Engineering 
Education, 51(2), 75–82.

Taheri, S., & Mammadov, M. (2012). Solving systems of 
nonlinear equations using a globally convergent 
optimization algorithm. Global Journal of 
Technology & Optimization, 3, 132–138.

Wang, X., & Fan, X. (2016). Two efficient derivative-free 
iterative methods for solving nonlinear systems. 
Algorithms, 9(1), 1–10. https://doi.org/10.3390/
a9010014