INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 03,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 81
FINDING THE ROOTS OF AN EQUATION USING THE CHORD METHOD
S. Khaidarova
Candidate of Technical Sciences, Associate Professor of Kokand State
University,
Uzbekistan, E-mail:
Annotation:
The article describes the chord method for approximate solution of algebraic equations.
A program for solving the problem in Pascal is provided.
Keywords and expressions:
fixed point, zero approximation
Let there be an equation
f x = 0
and the root of the equation x* lies on the segment [a,b], that
is
x
∗
[a, b]
.
To apply the chord method, the following conditions must be met:
1) The function
f x
is continuous on the interval
[a, b]
with its first and second derivatives;
2) T
he function
f x
takes opposite signs at the ends of the segment
[a, b]
, that is
f a ∙
f(b) < 0
3)
Derivatives and functions
f x
retain a certain sign on the interval
[a, b]
. This means that
the function f(x) is monotone and the root x* is unique.
The geometric meaning of the chord method is that the graph of the function f(x) on the
segment [a,b] is replaced by a chord. There can be 4 cases for the function graph:
Let's consider the first case, then the graph of the function has the following form:
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 03,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 82
Fig.1. Geometric illustration of the chord method
Let
х
*
-
be the root of the equation and
x
0
its approximate value. In order to find the
approximate value of
x
0
, the graph of the function
f (х)
on the segment
[a, b]
is replaced by the
chord AB. We write the equation of the line passing through the points
))
(
,
(
a
f
a
A
and
))
(
,
(
b
f
b
B
:
x a
b a
y
f a
f b
f a
-
-
=
-
-
( )
( )
( )
(1)
Since
x
0
[a,b]
lies on the line AB, we can find it from equation (1) by substituting the
values
x = x
0
and
y = 0
:
:
)
(
)
(
)
(
)
(
0
a
b
a
f
b
f
b
f
a
x
-
-
-
=
(2)
If the condition
f b
( )
f b
"
( )
>
0
is met, point b is fixed and then formula (2) can be used to
find the zero approximation of
x
0
.
The equation of the straight line passing through points
))
(
,
(
a
f
a
A
and
))
(
,
(
b
f
b
B
can
be written in this form:
x b
a b
y
f b
f a
f b
-
-
=
-
-
( )
( )
( )
(1
)
For
x = x
0
and we
y = 0
get:
x
b
f b
f b
f a
b a
0
= -
-
-
( )
( )
( )
(
)
(2
)
If the condition
f a
( )
f a
"
( )
>
0
is met, the point a is stationary and then formula (2
) can
be used to find the zero approximation of
x
0
.
To find the approximation x
1
, we write the equation of the line A
0
B and for x=x1 and y=0
we get:
x
x
f x
f b
f x
b x
1
0
0
0
0
=
-
-
-
( )
( )
( )
(
)
From the equation of straight line
A
1
B
at
x=x
2
and y=0 we obtain:
x
x
f x
f b
f x
b x
2
1
1
1
1
=
-
-
-
( )
( )
( )
(
)
Continuing the process, we find the n-th approximation:
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 03,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 83
x
x
f x
f b
f x
b x
n
n
n
n
n
=
-
-
-
-
-
-
-
1
1
1
1
(
)
( )
(
)
(
)
(3)
Calculations continue until the condition
x
x
h
n
n
-
-
1
is met.
There is a value of n such that
n
n
x
x
®
=
lim
*
satisfies the condition
h
x
x
n
-
*
and the
calculations stop.
The block diagram of the chord method algorithm is as follows:
1)
if point b is fixed, i.e. when the condition
f b
( )
f b
"
( )
>
0
is met, the block diagram has
the following form:
2)
if point
a
is fixed, i.e. when the condition
f a
( )
f a
"
( )
>
0
is met, the block diagram
has the following form:
Example: Find the root of the equation
f x = x
4
+ 5x
2
+ 10x − 3 = 0
on the segment
[0,1] with an accuracy of h=0.001 using the chord method.
Solution: We find the first and second derivatives:
f
'
x = 4x
3
+ 10x + 10, f
''
x = 12x
2
+ 10
f 0 f
''
0 < 0 and f 1 f
''
1 > 0
Since
f 1 f
''
1 > 0
), point B is fixed and calculations are carried out according to the
following formula:
x
x
f x
f b
f x
b x
n
n
n
n
n
=
-
-
-
-
-
-
-
1
1
1
1
(
)
( )
(
)
(
)
In this case
x
0
=a
, so we will use the first type of block diagram.
The Pascal program looks like this:
program
chord method
_
B fixed
;
var a,b,x,h,x0:real;
label
return, answer
;
function fx(x:real):real;
begin fx:=x*x*x*x+5*x*x+10*x-3
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 03,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 84
end;
begin
writeln('a='); readln(a);
writeln('b='); readln(b);
writeln('
enter precision
'); readln(h);
x:=a;
return
: x0:=x-fx(x)/(fx(b)-fx(x))*(b-x);
if abs(x0-x)<=h then goto
answer
else x:=x0; goto
return
;
answer
:writeln(x0,' ',fx(x0));
end.
After executing the program, enter the values
0,1,0.001
sequentially from the
keyboard and press the ENTER button each time. We get the following answer:
x= 0.264665, F(x) = 0.797911E-03
REFERENCES:
1. C.C. Irisqulov va b. Sonli usullar va algoritmlar. “Namangan” nashriyoti, 2013
