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// National Database of Legislation of the Republic of Uzbekistan –
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7.
Resolution No.366 of the Cabinet of Ministers of the Republic of
Uzbekistan dated October 27, 2016 "On the responsibility of heads of
ministries and departments, business associations, Chairman of the Council
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https://lex.uz/docs/3058254.
8.
Code of the Republic of Uzbekistan on Administrative Responsibility
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https://lex.uz/docs/111453.
Omonjon Hoydarov, student Termez State University
STEFFENSEN (EITKEN-STEFFENSEN) METHOD FOR SOLVING NONLINEAR
EQUATIONS
O. Hoydarov
Abstract: Solving nonlinear equations is more complicated and is a
perfectly unresolved problem in computational mathematics. This iterative
algorithm is called the Steffensen method in numerical methods. The
Steffensen method has a quadratic approximation. This method requires
calculating the value of the function twice in each iteration, in which case the
Steffensen method is less efficient than the cutters method.
Keywords: Steffensen method, iterative algorithm, approximation,
Eitken, Newton method, sequence, limit, linear approximation.
To increase the approximation speed of the test method
x
n
+ 1 = 𝜑(x
n
) n = 0,1,2, … (1)
In the expression 𝑓
′
(x
n
) Abbreviation of words that bring the harvest
closer:
𝑓
′
(x
n
) ≈
𝑓 (x
n+1
)−𝑓 (x
n
)
x
n+1
−x
n
(2)
If (1) is a left-handed approximation, then (2) is a right-handed
approximation. (2) shows that in which it has not yet been determined x
n
+1
To calculate the presence of an unknown limit, we use a simple iteration
(1.2):
x
n+1
= 𝑔(x
n
) = x
n
+ 𝑓(x
n
).
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83
As a result, we have the following approximation:
𝑓
′
(x
n
) ≈
𝑓 (x
n
+𝑓(x
n
) )−𝑓(x
n
)
𝑓(x
n
)
.
Using this expression in Newton's method, we get a new iterative
algorithm:
x
n+1
= x
n
−
𝑓(x
n
)
𝑓 (x
n
+𝑓(x
n
) )−𝑓(x
n
)
𝑓(x
n
). (3)
This iterative algorithm is called the Steffensen method in numerical
methods. The Steffensen method has a quadratic approximation, but here in
addition 𝑓 (x
n
+ 𝑓(x
n
) ) a high approximation speed is achieved by
calculating the value of the expression. .The iterative algorithm above (3)
can also be derived from the method of accelerating the convergence of
linear convergent sequences proposed by Eitken.
To do this, consider the following sequence:
z
n
= 𝑧 + 𝐶𝑞
𝑛
(4)
This is a sequence |𝑞| < 1 to z limit approached. Find the limit value of
z using simple reflections {z
n
} three in a row z
n−1
, z
n
𝑎𝑛𝑑 z
n+1
in series
z
n
−𝑧
z
n−1
−𝑧
= 𝑞 and
z
n+1
−𝑧
z
n
−𝑧
= 𝑞 this from two equations (z
n+1
− 𝑧)(z
n−1
− 𝑧) =
((z
n
− 𝑧)
2
equality. This leads to the following expression for z:
z =
z
n+1
z
n−1
− z
n
2
z
n+1
−2z
n
−z
n−1
.
Based on this result, let us consider the following Eitken proposition to
replace the {z
n
} sequence with another sequence:
ξ
n+1
=
z
n+1
z
n−1
− z
n
2
z
n+1
−2z
n
+z
n−1
. (5)
If this substitution is in any sequence in the form (4)
if we use, then at any value of n ξ
n
= 𝑧 − lim
𝑛→∞
𝑧
𝑛
equality is
appropriate. If the approximation type of the sequence {z
n
} is close to (4),
then substitution (5) gives z a new sequence that approximates z faster than
the original (even if it does not give its limit at an arbitrary value of n).
1-for example. This
𝑥
3
− 𝑥
2
− 8𝑥 + 12 = 0
approximate the double root of the equation x
r
= 2 using Newton's
method and the Eitken accelerator.
Solution. The results of the calculations are presented in the table below
with the elements of the corresponding sequences (see columns three and
four).
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84
In the third column of this table, the approximation velocity is assumed
to be a = 1, and the values of the variable C in equation (4) for each iteration
are given. The results in the table show that C changes very little during the
iterative process and is very close to C = 0.5. As a result, the hypothesis that
the rate of approach of the Newtonian method to the multiple root is linear
is proved.
Using the linear approximation {x
n
} sequence (5) to the accelerating
formula, we obtain the values of the s in the fourth column of the table ξ
n
.
By comparing the values in the second and fourth columns of the table, we
make sure that we have reached the approximation speed. Indeed, the result
obtained in the fourteenth iteration of the Newtonian method can be seen
by applying the Newtonian method and the Eitken accelerator q to the same
result in its seventh iteration. The results in the fifth column of this table
show that such an effective result was achieved not by increasing the
convergence velocity index a, but by reducing the variable C to 0.25. First of
all iterative for this x
n+1
= 𝑔(x
n
)
let's spread the right side of the formula to Taylor's row, that is
𝑔(x
n
) = 𝑔(x
r
+ (x
n
− x
r
)) = x
r
+ 𝑔
′
(x
r
)(x
n
− x
r
) + 𝑂((x
n
− x
r
)
2
),
According to this
x
n+1
− x
r
= 𝑔
′
(x
r
)(x
n
− x
r
) + 𝑂((x
n
− x
r
)
2
),
And so, e
n
= x
n
− x
r
The following approximate equation can be
written for each iteration with square accuracy:
x
n+1
− x
r
= 𝑔
′
(x
r
)(x
n
− x
r
).
Here the sequence {x
n
} can be expressed by the following formula:
x
n
≈ x
r
+ [𝑔
′
(x
r
)]
𝑛
(x
n
− x
r
)
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85
The approximation type of this sequence is the same as that of the
sequence (4). This means that the approximation sequence to the root in
simple iteration is suitable for applying the approximation acceleration
procedure.
In order to ensure that each improvement value calculated using the
approximation acceleration procedure is taken into account in subsequent
calculations, it must be taken into account immediately. This is done at each
step of the iteration as follows: Suppose that the calculations were
performed until the value of x
n
was calculated; using it we calculate two
auxiliary values 𝑥
𝑛
(1)
= 𝑔(x
n
) and
𝑥
𝑛
(2)
= 𝑔(𝑔(x
n
)). We apply the accelerator formula (5) to the three
values x, 𝑥
𝑛
(1)
, and 𝑥
𝑛
(2)
, and the result is the following
Assume that x
n
+1 approximation:
x
n
+ 1 =
x
n
𝑔 ( 𝑔 (x
n
)) − 𝑔
2
(x
n
)
𝑔 ( 𝑔 (x
n
)) − 2 g (x
n
) + x
n
It appears that Equation (3) is one of the forms of writing the Steffensen
iterative formula.
Example 2. The formula (6) is given by this
𝑥
3
− 𝑥
2
− 8𝑥 + 12 = 0
Use to find the double root of the equation.
Solution. To do this,
g(x) = x -
𝑓 ( 𝑥 )
𝑓
′
( 𝑥 )
corresponding to the Newtonian iteration
from the calculations according to formula (6)
{0.5; 1,87215909; 1,99916211; 1,99999996; 2,00000000}
we create a sequence. Comparing these values with the values of ξ
n
in
the fourth column of the table above, we can see that the efficiency is
increased by adding the accelerator to the algorithm obtained from the line,
rather than to the sequence.
The important conclusions of this work are as follows:
1. Solving nonlinear equations is more complicated and is a perfectly
unresolved problem in computational mathematics;
2. The initial problem of solving nonlinear equations is the study of the
existence, number and interval of solutions of nonlinear equations, which
are explained by solving specific examples;
3. The problem of finding the separated root of a nonlinear equation is
described in several approximate ways, explained by the solutions of
concrete examples;
Scientific research results in pandemic conditions (COVID-19)
86
4. Approximate methods of finding the roots of a nonlinear equation
have been studied from simple to complex and with their special cases,
which has made it possible to shed more light on the subject;
Thus, the problem of solving nonlinear equations depends on the type
of practical problem, the choice of the correct approximate method and the
initial condition, the effective use of these methods.
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