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CHIZIQSIZ TENGLAMALAR TAQRIBIY YECHISH METODLARI
Tayanch so‘z va iboralar: Chiziqsiz tenglama ildizlarni taqribiy topish: oddiy iteratsiya,
Nyuton, vatarlar usullari
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IV.1. Chiziqsiz tenglama ildizlarni taqribiy topish
Oddiy iteratsiya usuli
D
x x
x
x x
f
f x x
)
( ,
,0
)
( ,
,0
)
,
(
2
1
2
1
2
2
1 1
tenglamalar sistemasi
berilgan bo‘lsin.
a) Iteratsiya usuli. Tenglamalar sistemasi ekvivalent iteratsiya usulini qo‘llash
uchun qulay ko‘rinishga keltirib olinadi:
),
( ,
),
( ,
0
,0
2
1
2
2
2
1
1
1
2
1
x x
g
x
g x x
x
f
f
max
'( )
1
x D
q
g x
.
max
'( )
1
x D
q
g x
-yaqinlashish sharti,
1
( ), 2
g x g ( )
x -funksiyalar iteratsiya funksiyalari
deyiladi.
(0)
(0)
(0)
1
2
(
,
)
x
x
x
boshlang‘ich iteratsiya tanlangach, iteratsiyalar quyidagicha
quriladi:
( )
(
1)
(
1)
( )
(
1)
(
1)
1
1
1
2
2
2
1
2
(
,
),
(
,
),
1,2,.....
k
k
k
k
k
k
x
g x
x
x
g x
x
k
Qoldiq had-aniq echim bilan taqribiy echim
( )
x k
orasidagi farq norma
2
D
R
sohadagi normada quyidagicha beriladi:
( )
( )
(0)
( )
( )
(
1)
,
1
k
k
k
k
k
q
x
q
x
x
x
x
q
IV.2. Nyuton iteratsiya usuli
b) Soddalashtirilgan Nyuton usuli
(0)
(0)
(0)
(0)
1 1
1
2
2
1
1
2
(0)
(0)
(0)
1
2
(0)
(0)
(0)
(0)
1
2
1
2
2
2
1
2
(
1)
(
1)
(0)
(0)
(0)
(0)
1
1
2
2
1
1
2
1 1
1
2
1
1
(
1)
(
1)
1
2
(
1)
(
1)
(0)
(0)
2
1
2
2
2
1
2
(
,
)
(
,
)
(
,
)
,
(
,
)
(
,
)
(
,
)
(
,
)
(
,
)
(
,
(
,
)
(
,
)
k
k
k
k
k
k
f x
x
f x
x
d
d x
x
f
x
x
f
x
x
f
x
x
f x
x
f x
x
f x
d
d
f
x
x
f
x
x
(
1)
(
1)
2
(0)
(0)
(
1)
(
1)
1
2
1
2
2
1
2
,
)
(
,
)
(
,
)
k
k
k
k
x
f
x
x
f
x
x
(
1)
(
1)
( )
(
1)
( )
(
1)
1
2
1
1
2
2
(0)
(0)
,
k
k
k
k
k
k
d
d
x
x
x
x
d
d
v) Nyuton usuli
1 1
2
1
1
2
1
1 1
1
1
2
1
1
1
2
2
2
1
2
1
2
2
2
2
2
2
1
2
2
( ,
)
,
( ,
)
,
( ,
)
f
f
f
f
f
f
d
d x x
d
d x x
d
d
x x
f
f
f
f
f
f
1)
(
2
1)
(
2
( )
2
1)
(
1
1)
(
1
( )
1
)
(
,
)
(
k
k
k
k
k
k
d
d
x
x
d
d
x
x
![Ilmiybaza.uz
2
( )
(0)
2
1
2
1
1
,
,
2
max
( ) , min
( )
1.
k
k
x D
x D
M
x
q
x
q
q
m
M
f
x
f
x
m
d) Minimizatsiya masalasiga keltirish
2
2
1
2
( ):
( )
( ),min
( )
( )
0
x D
F x
f
x
f
x
F x
F
Tenglamani iteratsiya usulini qo‘llash uchun qulay ko‘rinishga keltirish.(Iteratsiya
funksiyalarini topish). Umumiy holda
( ):
n
n
f x
D
R
R
bo‘lsin.
( )
0
f x vektor
tenglamani
( )
( ),
:
n
n
x
g x
x
Lf x L R
R
,ekvivalent vektor
tenglamaga olib kelamiz, bu erda
,det( )
0.
L
ì àò ðèöà
L
Ravshanki,
1 1
1
1
( )
.
( )
( )
.
.
.
( )
.
( )
n
n
n
n
f x
f x
f
x
f
x
f
x
,
1 1
1
1
1
0
0
( )
.
( )
( )
( )
0
1
0
.
.
.
0
0
1
( )
.
( )
n
n
n
n
f x
f x
g x
E
Lf
x
L
f
x
f
x
max
'( )
1
x D
q
g x
tengsizlik bajarilishi uchun quyidagi talablarni qo‘yish mumkin:
1)
( )
0
g x
, bu erda x -biror nuqta, jumladan,
(0)
x
x
-boshlang‘ich iteratsiya.
Bu holda
1
(
1)
( )
1
( )
[
( )] ,
[
( )]
(
),
0,1,....
k
k
k
L
f
x
x
x
f
x
f x
k
va iteratsiyalar
(
1)
( )
1
( )
[
( )]
(
),
0,1,....
k
k
k
x
x
f
x
f x
k
formula asosida quriladi. Bu soddalashtirilgan Nyuton iteratsiya usuli.
2)
( ( )
)
0,
0,1,....
g x k
k
Bu holda
( )
1
(
1)
( )
( )
1
( )
[
(
)] ,
[
(
)]
(
),
0,1,....
k
k
k
k
k
L
f
x
x
x
f
x
f x
k
va iteratsiyalar
(
1)
( )
( )
1
( )
[
(
)]
(
),
0,1,....
k
k
k
k
x
x
f
x
f x
k
formula asosida quriladi. Bu Nyuton iteratsiya usuli.
Masalaning qo‘yilishi. Ushbu sistema yechilsin :](/media/image/chiziqsiz-tenglamalar-taqribiy-yechish-metodlari1097page_3.png "Ilmiybaza.uz
2
( )
(0)
2
1
2
1
1
,
,
2
max
( ) , min
( )
1.
k
k
x D
x D
M
x
q
x
q
q
m
M
f
x
f
x
m
d) Minimizatsiya masalasiga keltirish
2
2
1
2
( ):
( )
( ),min
( )
( )
0
x D
F x
f
x
f
x
F x
F
Tenglamani iteratsiya usulini qo‘llash uchun qulay ko‘rinishga keltirish.(Iteratsiya
funksiyalarini topish). Umumiy holda
( ):
n
n
f x
D
R
R
bo‘lsin.
( )
0
f x vektor
tenglamani
( )
( ),
:
n
n
x
g x
x
Lf x L R
R
,ekvivalent vektor
tenglamaga olib kelamiz, bu erda
,det( )
0.
L
ì àò ðèöà
L
Ravshanki,
1 1
1
1
( )
.
( )
( )
.
.
.
( )
.
( )
n
n
n
n
f x
f x
f
x
f
x
f
x
,
1 1
1
1
1
0
0
( )
.
( )
( )
( )
0
1
0
.
.
.
0
0
1
( )
.
( )
n
n
n
n
f x
f x
g x
E
Lf
x
L
f
x
f
x
max
'( )
1
x D
q
g x
tengsizlik bajarilishi uchun quyidagi talablarni qo‘yish mumkin:
1)
( )
0
g x
, bu erda x -biror nuqta, jumladan,
(0)
x
x
-boshlang‘ich iteratsiya.
Bu holda
1
(
1)
( )
1
( )
[
( )] ,
[
( )]
(
),
0,1,....
k
k
k
L
f
x
x
x
f
x
f x
k
va iteratsiyalar
(
1)
( )
1
( )
[
( )]
(
),
0,1,....
k
k
k
x
x
f
x
f x
k
formula asosida quriladi. Bu soddalashtirilgan Nyuton iteratsiya usuli.
2)
( ( )
)
0,
0,1,....
g x k
k
Bu holda
( )
1
(
1)
( )
( )
1
( )
[
(
)] ,
[
(
)]
(
),
0,1,....
k
k
k
k
k
L
f
x
x
x
f
x
f x
k
va iteratsiyalar
(
1)
( )
( )
1
( )
[
(
)]
(
),
0,1,....
k
k
k
k
x
x
f
x
f x
k
formula asosida quriladi. Bu Nyuton iteratsiya usuli.
Masalaning qo‘yilishi. Ushbu sistema yechilsin :")
Ilmiybaza.uz
2
( )
(0)
2
1
2
1
1
,
,
2
max
( ) , min
( )
1.
k
k
x D
x D
M
x
q
x
q
q
m
M
f
x
f
x
m
d) Minimizatsiya masalasiga keltirish
2
2
1
2
( ):
( )
( ),min
( )
( )
0
x D
F x
f
x
f
x
F x
F
Tenglamani iteratsiya usulini qo‘llash uchun qulay ko‘rinishga keltirish.(Iteratsiya
funksiyalarini topish). Umumiy holda
( ):
n
n
f x
D
R
R
bo‘lsin.
( )
0
f x vektor
tenglamani
( )
( ),
:
n
n
x
g x
x
Lf x L R
R
,ekvivalent vektor
tenglamaga olib kelamiz, bu erda
,det( )
0.
L
ì àò ðèöà
L
Ravshanki,
1 1
1
1
( )
.
( )
( )
.
.
.
( )
.
( )
n
n
n
n
f x
f x
f
x
f
x
f
x
,
1 1
1
1
1
0
0
( )
.
( )
( )
( )
0
1
0
.
.
.
0
0
1
( )
.
( )
n
n
n
n
f x
f x
g x
E
Lf
x
L
f
x
f
x
max
'( )
1
x D
q
g x
tengsizlik bajarilishi uchun quyidagi talablarni qo‘yish mumkin:
1)
( )
0
g x
, bu erda x -biror nuqta, jumladan,
(0)
x
x
-boshlang‘ich iteratsiya.
Bu holda
1
(
1)
( )
1
( )
[
( )] ,
[
( )]
(
),
0,1,....
k
k
k
L
f
x
x
x
f
x
f x
k
va iteratsiyalar
(
1)
( )
1
( )
[
( )]
(
),
0,1,....
k
k
k
x
x
f
x
f x
k
formula asosida quriladi. Bu soddalashtirilgan Nyuton iteratsiya usuli.
2)
( ( )
)
0,
0,1,....
g x k
k
Bu holda
( )
1
(
1)
( )
( )
1
( )
[
(
)] ,
[
(
)]
(
),
0,1,....
k
k
k
k
k
L
f
x
x
x
f
x
f x
k
va iteratsiyalar
(
1)
( )
( )
1
( )
[
(
)]
(
),
0,1,....
k
k
k
k
x
x
f
x
f x
k
formula asosida quriladi. Bu Nyuton iteratsiya usuli.
Masalaning qo‘yilishi. Ushbu sistema yechilsin :
Ilmiybaza.uz
f(x,y)=x*x+y*y-1=0,g(x,y)=x*x-y=0.
Birinchi tenglama birlik aylana, ikkinchi tenglama kvadratik paraboladir. Ular 1-
va 4- choraklarda kesishadi va 2 ta haqiqiy ildizlarga ega. Uni Mathcad da grafik
chizib ham ko‘ramiz.
f x
( )
1
x2
1
2
g x
( )
x2
x
4
4
y
10
10
5
0
5
0
10
20
f x
( )
g x
( )
x
Masalaning Mathcad dasturida yechilishi.
a) Given ...Find bloki yordamida yechish
x
1
y
0
Given
x2
y2
1
x2
y
0
r
Find x y
(
)
r
0.7861512594
0.6180339205
x
1
y
0
Given
x2
y2
1
x2
y
0
r
Find x y
(
)
r
0.7861512594
0.6180339205
b) iteratsiya usulini Mathcad da tashkil etish
![Ilmiybaza.uz
ORIGIN
1
g x
( )
1
0.5 cos x2
sin x1
1
1.2
k
1 5
x
1
0.4
0.5
x
k 1
g x
k
x
0.4
0.5
0.5612087191
0.21455027
0.5114638779
0.2000459608
0.5099712771
0.2017596534
0.5101422642
0.2018492731
0.5101512458
0.2018388938
x
1
0.6
0.1
x
k 1
g x
k
x
0.6
0.1
0.5024979174
0.8105816577
0.6554614854
0.2023314299
0.5101996345
0.2035819541
0.5103256662
0.2018354178
0.510149857
0.2018277933
c) Nyuton usulini Mathcad da tashkil etish
s) Boshlang‘ich qiymat x0=[-1,0.5]
Nyuton va soddalashtirilgan Nyuton usullari
Nyuton usuli
Sodda Nyuton usuli
g x
( )
x1
2
x2
2
1
x1
2
x2
J x
( )
2 x1
2 x1
2 x2
1
k
1 5
x 1
1
0.5
y 1
x 1
x k 1
x k
J x k
1
g x k
y k 1
y k
J y 1
1
g y k
x
1
0.5
0.8125
0.625
0.786592
0.618056
0.786152
0.618034
0.786151
0.618034
0.786151
0.618034
ORIGIN
1](/media/image/chiziqsiz-tenglamalar-taqribiy-yechish-metodlari1097page_5.png "Ilmiybaza.uz
ORIGIN
1
g x
( )
1
0.5 cos x2
sin x1
1
1.2
k
1 5
x
1
0.4
0.5
x
k 1
g x
k
x
0.4
0.5
0.5612087191
0.21455027
0.5114638779
0.2000459608
0.5099712771
0.2017596534
0.5101422642
0.2018492731
0.5101512458
0.2018388938
x
1
0.6
0.1
x
k 1
g x
k
x
0.6
0.1
0.5024979174
0.8105816577
0.6554614854
0.2023314299
0.5101996345
0.2035819541
0.5103256662
0.2018354178
0.510149857
0.2018277933
c) Nyuton usulini Mathcad da tashkil etish
s) Boshlang‘ich qiymat x0=[-1,0.5]
Nyuton va soddalashtirilgan Nyuton usullari
Nyuton usuli
Sodda Nyuton usuli
g x
( )
x1
2
x2
2
1
x1
2
x2
J x
( )
2 x1
2 x1
2 x2
1
k
1 5
x 1
1
0.5
y 1
x 1
x k 1
x k
J x k
1
g x k
y k 1
y k
J y 1
1
g y k
x
1
0.5
0.8125
0.625
0.786592
0.618056
0.786152
0.618034
0.786151
0.618034
0.786151
0.618034
ORIGIN
1")
Ilmiybaza.uz
ORIGIN
1
g x
( )
1
0.5 cos x2
sin x1
1
1.2
k
1 5
x
1
0.4
0.5
x
k 1
g x
k
x
0.4
0.5
0.5612087191
0.21455027
0.5114638779
0.2000459608
0.5099712771
0.2017596534
0.5101422642
0.2018492731
0.5101512458
0.2018388938
x
1
0.6
0.1
x
k 1
g x
k
x
0.6
0.1
0.5024979174
0.8105816577
0.6554614854
0.2023314299
0.5101996345
0.2035819541
0.5103256662
0.2018354178
0.510149857
0.2018277933
c) Nyuton usulini Mathcad da tashkil etish
s) Boshlang‘ich qiymat x0=[-1,0.5]
Nyuton va soddalashtirilgan Nyuton usullari
Nyuton usuli
Sodda Nyuton usuli
g x
( )
x1
2
x2
2
1
x1
2
x2
J x
( )
2 x1
2 x1
2 x2
1
k
1 5
x 1
1
0.5
y 1
x 1
x k 1
x k
J x k
1
g x k
y k 1
y k
J y 1
1
g y k
x
1
0.5
0.8125
0.625
0.786592
0.618056
0.786152
0.618034
0.786151
0.618034
0.786151
0.618034
ORIGIN
1
