ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
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Выпуск журнала №-70
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369
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EQUATIONS SOLUTION HIGH ORDERLY DOG FOOD M E TODS .
CHEBYSHEV'S METHOD.
Ismailov Ahrorjon
practical
mathematics and computer science
department big teacher of physics and
mathematics sciences according to philosophy
Doctor of Philosophy (PhD)
Azimjonova Mohinur Asiljon daughter
Fergana State 3rd year university student
Annotation
:
This in the article
Chebyshev method
high orderly iterative
methods explained.Given function roots in calculation Taylor formula using reverse
function iterative solutions find process learns . In the article of the method main
principles , Taylor formula and high orderly iterative of processes work principle in
detail illuminated .
Key words :
Chebyshev method , high orderly iterations , Taylor formula ,
inverse function , function root , iterative process , mathematics methods , precision ,
iterative methods .
Introduction
. PL Chebyshev in 1933 given
(x)
f
to the function reverse was
g(y)
function Taylor formula using to describe road with high orderly iteration build
method offer It is assumed . let's do it ,
(x)
0
f
=
of the equation
x
=
root
,
a b
in
between let it lie down and
(x)
f
function and his/her enough high orderly derivatives
continuous Let it be . From now on outside this of the interval all at points
'
(x)
0
f
Let
it be . Then
'
(x)
f
this in between own gesture keeps and
(x)
f
monotonous function is
,
(y)
x
g
=
reverse to the function has will be . Reverse function
g(y)
(x)
f
of change field
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
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Выпуск журнала №-70
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370
2181-3187
,
c d
identified in is ,
(x)
f
how much continuous to derivatives has if so , that's it
continuous to derivatives has will be . Reverse function to the definition according to
(f(x))
x
g
(x [a, b])
,
(g(y))
y
f
(y [c,d])
. (1)
So,
(0)
g
=
. (2)
If
y [c, d]
If , then from Taylor's formula
(k)
(p)
1
(k)
1
(y)
( )
(0)
g(
y)
g(y)
( 1)
( 1)
!
!
p
k
p
p
k
g
g
g
y
y
k
p
−
=
=
=
− =
+
−
+ −
(3)
this on the ground
number
0
and
y
between lies . Or
y
instead of
n
x
x
=
what
putting and
(y)
x
g
=
what in mind holding ,
(k)
(p)
1
(k)
1
(y)
( )
( 1)
(x) ( 1)
(x)
!
!
p
k
p
p
k
g
g
x
f
f
k
p
−
=
= +
−
+ −
(4)
We generate . If
(k)
1
k
1
(f(x))
(x)
( 1)
(x)
!
p
k
p
k
g
x
f
k
−
=
= +
−
If we define it as , then
(x)
p
x
=
(5)
equation for
x
=
solution will be , because
(k)
1
k
1
(f( ))
( )
( 1)
( )
!
p
k
p
k
g
f
k
−
=
= +
−
=
From this
( )
0,
1,
1
j
p
j
p
=
=
−
because of
1
(
)
n
n
p
x
x
+
=
0
(
0,1, 2,...;
[ , ])
n
x
a b
=
(6)
the iterative process is of p-order. If
0
x
to close If , then (6) with defined {
n
x
}
sequence
to It is approaching . Indeed ,
'
( )
0
p
=
it was for
of so surroundings It
is found there.
'
( )
1
p
x
q
It will be . and
0
x
to enough close if {
n
x
} iterative
sequence
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-70
Часть–1_ Мая –2025
371
2181-3187
It comes close.
Now
(x)
p
of
(x)
f
and We find its explicit expression through its derivatives.
To do this, we take successive derivatives from (1):
2
3
'
'
''
'
'
''
'''
'
''
'
''
'
'''
( ( ))
( )
1,
( ( ))
( )
( ( ))
( )
0,
( ( ))
( ) 3
( ( ))
( )
( )
( ( ))
( )
0,
g f x f x
g
f x f
x
g f x f
x
g
f x f
x
g
f x f x f
x
g f x f
x
=
+
=
+
+
=
(7)
From here we go one by one.
'
( ( ))
g
f x
,
''
( ( ))
g
f x
, ...,
(
1)
( ( ))
p
g
f x
−
those and this with
together
(x)
p
what We define . (6) We make the iteration process explicit for several
specific values of p.
2
p
=
when
2
'
( )
( )
( )
f x
x
x
f x
= −
and
1
'
(
)
(
)
n
n
n
n
f x
x
x
f x
+
=
−
(8)
We will see later that this process is similar to the Newtonian process.
overlaps.
3
p
=
When (5) and (7)
''
2
3
'
'
3
( )
( )
( )
( )
( )
2[
( )]
f x
f
x f
x
x
x
f x
f x
= −
−
and
''
2
1
'
'
3
( )
( )
( )
( )
2[
( )]
n
n
n
n
n
n
n
f x
f
x
f
x
x
x
f x
f x
+
=
−
−
(9)
comes from.
4
p
=
for
2
''
2
3
''
'
'''
4
'
'
3
'
5
( )
( )
( )
( ) 3
( )
( )
( )
( )
( )
2[
( )]
12
[
( )]
n
f x
f
x f
x
f
x
f
x
f x f
x
x
x
f x
f x
f x
−
=
−
−
−
and
2
''
2
3
''
'
'''
1
'
'
3
'
5
(
)
(
)
(
)
(
) 3
(
)
(
)
(
)
(
)
2[
(
)]
12
[
(
)]
n
n
n
n
n
n
n
n
n
n
n
n
f x
f
x
f
x
f
x
f
x
f x
f
x
x
x
f x
f x
f x
+
−
=
−
−
−
(10)
We generate . These iterative processes will be iterations of order 2, 3 , and 4,
respectively.
Now
n
n
x
= −
of mistake to zero We estimate the speed of aspiration . This for
(4) in equality
n
x
x
=
, taking (6) as Considering this, we obtain the following:
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-70
Часть–1_ Мая –2025
372
2181-3187
(p)
1
( 1)
(f(x))
(x )
!
p
p
n
n
g
x
f
p
+
−
−
=
(11)
this on the ground
x
with ,
x
n
between lies ,
( )
0
f
=
that was for
'
(x )
[f( ) (fx )]
(
x ) f (x)
n
n
n
f
= −
−
= − −
(12)
(
(x)
also
with
x
n
lies between ). We substitute (12) into (11):
'
1
(f(x))
[f ( )]
!
p
p
p
n
n
g
x
p
+
=
(13)
The following
'
,
[a,b]
(f(x))
[f ( )]
!
max
p
p
x x
g
q
x
p
=
by introducing the definition, from (13)
1
p
n
n
q
+
(14)
We get an inequality. Applying this inequality sequentially, we obtain the
following:
1
1
(p 2) 1
1
...
1
1
0
0
0
(q
)
n
n
n
p
p
p
p
n
p
p
n
q
p
−
−
− +
+ + +
−
−
=
If
0
1
and
0
1
q
=
, then
1
1
n
p
p
n
−
−
(15)
becomes , which means that the iteration (6) converges extremely quickly
shows . Private without
1
10
−
and
0
1
If , for iterations (8), (9) and (10) above
, we have the following, respectively:
2
p
=
for
1
1
10
−
,
3
2
10
−
,
7
3
10
−
,
15
4
10
−
;...
3
p
=
for
1
1
10
−
,
4
2
10
−
,
13
3
10
−
,
40
4
10
−
,...
4
p
=
for
1
1
10
−
,
5
2
10
−
,
18
3
10
−
,
85
4
10
−
,...
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-70
Часть–1_ Мая –2025
373
2181-3187
So ,
0,1
when third iteration himself/herself to us necessary accuracy gives .
Conclusion :
Chebyshev method high orderly iterative from methods one Taylor
formula using functions effective analysis to do opportunity This gives method
iterations through of equations solutions fast and clear to find help The solution is p -
order accuracy and of mistake to zero aspiration of the process efficiency increases .
So so , Chebyshev method mathematician and practical in the fields effective
application possible .
Used literature
1.
Chebyshev, PL (1933). "Oh nekotorykh method solution nonlinear level ."
Trudy
Matematicheskogo Society
,.
2.
Grigorieva , NP (2010).
Method solution nonlinear uravneniy
. M.: Nauka .
3.
Durell , CV, & Robson, RJ (1950).
Advanced Calculus
. London: Blackie & Son
Ltd.
4.
Boyce, WE, & DiPrima , RC (2005).
Elementary Differential Equations and
Boundary Value Problems
. John Wiley & Sons, Inc.
5.
Press, WH, Teukolsky , SA, Vetterling , WT, & Flannery, BP (2007).
Numerical
Recipes: The Art of Scientific Computing
(3rd ed.). Cambridge University Press.
6.
Shishkin , GI, & Samarskii , AA (2006).
Numerical Methods for Grid and
Differential Equations
. Springer-Verlag .
