CHIZIQLI TENGLAMALAR SISTEMACINI YECHISH

Загружено в:

08.08.2023

Скачано:

0

Размер:

146.291015625 KB

Скачать

Reja:
Kirish.
1.Chiziqli tenglamalar sistemasini Gauss usulida hisoblash
2.Chiziqli tenglamalar sistemasini Kramer usulida hisoblash
3.Matritsa tenglamasini teskari matritsa toppish usulida
hisoblash
Xulosa .
Foydalanilgan adabiyotlar.

1. CHIZI Q LI TENGLAMALAR SISTEMACINI Y ECHISH
Chiziqli tenglamalar sistemasini Gauss, Kramer va teskari matritsa usullari bilan y echimini topish
masala la rini ko‘ramiz.
1.1-masala. Quyidagi uch noma’lumli chiziqli tenglamalar sistemasi ning yechimini:
1) Gauss usuli, 2) Kramer, 3) Matritsa usulida toping. 


  
  
  
39 16 25 5
18 12 14 3
0 13 7 2
3 2 1
3 2 1
3 2 1
x x x
x x x
x x x
(1)
1.1. Gauss usulida y echish.
Yetakchi tenglama uchun birinchi tenglamani olamiz. Bu tenglamadan etakchi noma’lum
uchun x
1 va a
11 ≠0 ni etakchi element uchun tanlaymiz. Birinchi tenglama dagi x
1 ning koeffitsenti a
11 ni 1
ga aylantirish uchun birinchi tenglamani ng barcha qo‘shiluvchilarini a
11 ≠0 ga b o‘ lamiz . Ҳosil bo‘lgan
tenglamadan foydalanib ikkinchi va uchinchi tenglmalardan x
1 nomahlumni yo‘qotish yoki uning
koeffitsentini nolg‘ga aylantirish uchun etakchi tenglamani –3 ga ko‘paytirib 2- tenglamaga qo‘shamiz,
so‘ngra etakchi tenglamani –5 ko‘paytirib 3- teglamaga qo‘shamiz. Natijada quyidagicha sistemaga
kelamiz:








 
 
  
39 2
33
2
15
18 2
15
2
7
0 13 7 2
3 2
3 2
3 2 1
x x
x x
x x x
Bu tenglamlar sistemasida etakchi tenglama uchu 2- tenglamani olamiz. Unda 7/2 koeffitsient l i x
2








 
 
  
7
3
7
3
18 2
15
2
7
0 13 7 2
332 321
x
x x
x x x
bu sistemaning 3- tenglamasidan x
3 nomahlumni topamiz. x
3 asosida 2- tenglamadan x
2 ni topamiz. x
3 , x
2
lar asosida 1- tenglamadan x
1 ni topamiz.
x
3 = - 1
x
2 =
3 )2
15 18(7
2 ) 2
15 18(7
2
3     x
x
1 =
4 ))1 ( 13 3 7(
2
1 ) 13 7(
2
1
3 2         x x
Demak sistema echimi: x
1 =-4, x
2 =3, x
3 = -1

Maple12 dasturida masalani echish.
Uch noma’lumli chiziqli tenglamalar sistemasini oddiy va Gauss usulida echish( 1 . 1 -
masala).
1 2 3
1 2 3
1 2 32 7 13 0,
3 14 12 18,
5 25 16 39,
ő ő ő
ő ő ő
ő ő ő
   
    
    
Maple7 dasturida masalalarni echishdagi amallarni bajarish uchun ishchi oynada > belgidan
so‘ng kerakli buyruqni yozib Enter tugmasini bosish kerak.
1. Oddiy usulida echish ( Gauss.mw ).
> solve( {2*x + 7*y + 13*z = 0, 3*x + 14*y + 12*z =18, 5*x + 25*y +16*z =39}, [x, y, z]);
[x=-4, y=3, z=-1]
2. Gauss usulida uch noma’lumli chiziqli tenglamalar sistemasini echish .
> with(LinearAlgebra):
A := <<2,3,5>|<7,14,25>|<13,12,16>>;
2 7 13
: 3 14 12
5 25 16
A
 
  
  
> B := <0,18,39>;
0
: 18
39
b
 
  
  
> GaussianElimination(A);
2 7 13
7 15 0 3 2
3 0 0 7
            
> GaussianElimination(A,'method'='FractionFree');
2 7 13
0 7 15
0 0 3
 
    
    
>ReducedRowEchelonForm(<A|b>);
1 0 0 4
0 1 0 3
0 0 1 1
  
 
 
    

3. To‘rt noma’lumli chiziqli tenglamalar sistemasini Maple12 dasturida echish
1) Oddiy usulida echish
> sys:=({1*x1-5*x2-1*x3+3*x4=-5,2*x1+3*x2+1*x3-1*x4=4, 3*x1-2*x2+3*x3+4*x4=-
1,5*x1+3*x2+2*x3+2*x4=0}):
> solve(sys,{x1,x2,x3,x4});{x4 = K 3,x2 = K 1,x1 = 1,x3 = 2}
2) Gauss usulida echish






0.= 2x+ 2x+ 3x+ 5x
-1,= 4x+ 3x+ 2x- 3x
4,= x- x3+ 3x+ 2x
-5,= 3x+ x- 5x- x
4 3 2 1
4 3 2 1
4 2 1
4 3 2 1
> with(LinearAlgebra):
A := <<1,2,3,5>|<-5,3,-2,3>|<-1,1,3,2>|<3,-1,4,2>>;
1 5 1 3
2 3 1 1
:
3 2 3 4
5 3 2 2A  
 
 

 

 

 
 
> b := <-5,4,-1,0>;
5
4 : 1
0
B
 
 
    
  
> GaussianElimination(A,'method'='FractionFree');













 
39/ 67 0 0 0
2 3 0 0
7 3 13 0
3 1 5 1
> ReducedRowEchelonForm(<A|b>);














3 0 0 0
2 1 0 0
1 0 1 0
1 0 0 1

1. 2. Kramer q oidasi yordamida echish.
Berilgan tenglamalar sistema nomahlumlarning koeffitsientlari va ozod hadalari yordamida
determinantlarni tuzamiz va ularni hisoblashning uchburchak yoki Sarrus usullaridan foydalanamiz. Biz
(1) tenglamlar sistemasining determinantlarini tuzib, uchburchak usulida hisolab uni son qiymatlarni
topamiz.
16255 21143 1375

3 336 600 910 420 975 448
16 7 3 25 2 12 13 14 5 5 12 7 13 25 3 26 14 2
      
                  
12 2016 0 7098 5850 3276 0
16 25 39
12 14 18
13 7 0
1         x
99360117015210576
163916 12183 1302
2  x
x
3 =
3 819 900 0 630 0 1092
39 25 5
18 14 3
0 7 2
      
Maple12 dasturida masalani echish.
> with(Student[LinearAlgebra]):
> d := <<2,3,5>|<7,14,25>|<13,12,16>>;
> d:=Determinant(d);













 
39/ 67 0 0 0
2 3 0 0
7 3 13 0
3 1 5 1
> ReducedRowEchelonForm(<A|b>);














3 0 0 0
2 1 0 0
1 0 1 0
1 0 0 1

213 ) 325 112(
16 25
13 7 )1( 12
21       A
,
33 65 32
16 5
13 2
)1 ( 22 22        A

15 ) 35 50(
25 5
7 2 )1 ( 32 23       A
3-ustun elementlari:

98 182 84
12 14
13 7
)1 ( 13
31       A
,

15 ) 39 24(
12 3
13 2
)1 ( 23 32       A

7 21 28
14 3
7 2
)1 ( 33 33       A
A matrtitsaga teskari A -1
matritsani yozamiz:




 

7155 153312 9821376
31
1
A

B A X   1 tenglama echimini topamiz:






















 
 









1
3
4
39
18
0
37 5 35
5 11 4
3 98 71 2
3
2
1
x
x
x
X
x
1 = -4, x
2 = 3, x
3 = -1
Maple12 dasturida masalani echish.
> with(Student[LinearAlgebra]):
> A := <<2,3,5>|<7,14,25>|<13,12,16>>;







16 25 5
12 14 3
18 7 2
A
> B := <<0,18,39>>;











39
18
0
:B
> X:=A^(-1).B;

X : =











1
3
4
1. 4 . Matritsaviy tenglamani echish
Quyi dagi matritsaviy tenglamani undaagi koeffitsent matritsaga teskari matritsa topish bilan
ech amiz.

















5 5 2
2 4 2
6 4 13
0 1 1
2 1 0
1 3 1
Ő
Yuqorida keltirilgan formula (*) va (**) formulalar
BXA 
,
B A X   1
asosida echish uchun oldin
1A ni topamiz va unga B matritsani ko‘paytiramiz. Bu amalni Maple 7
dasturida quyidagicha bajaramiz:
> with(Student[LinearAlgebra]):
> A := <<1,0,1>|<3,1,-1>|<-1,-2,0>>;









0 1 1
2 1 0
1 3 1
:A
> A -1
:=A^(-1);






 
 

 
7
1
7
4
7
1
7
2
7
1
7
2
7
5
7
1
7
2
:1A
> B := <<13,2,-2>|<-4,-4,5>|<6,2,5>>;













5 5 2
2 4 2
6 4 13
:B
> X:=A^(-1).B;











 
1 1 1
0 2 4
5 3 2
: X
1. CHIZI Q LI TENGLAMALAR SISITEMACINI ECHISH
Chiziqli tenglamalar sistemasini Gauss, Kramer va teskari
matritsa usullari bilan echimini topish masalarini
ko‘ramiz.

1.1-masala. Quyidagi uch nomahlumli chiziqli
tenglamalar sistemasi ning echimini:
1) Gauss usuli, 2) Kramer, 3) Matritsa usulida
toping.
11. 1. {2x
1
−x
2
+x
3
−x
4
=1¿{2x
1
−x
2
−3x
4
=2¿{3x
1
−x
3
+x
4
=−3¿¿¿¿
2.
{2x
1
+x
2
−x
3
=1¿{x
1
+x
2
+x
3
=6¿¿¿¿

3)
(
5 −1 3
0 2 −1
−2 −1 0)
Х=
(
3 7 −2
1 1 −2
0 1 3 )

Maple12 dasturida masalani yechish.
To’rt noma’lumli chiziqli tenglamalar sistemasini
oddiy va Gauss usulida y echish( 1 . 1 - masala).
11. 1.
shartlar orqali javoblarga erishamiz,
> with(LinearAlgebra):
> A:=<<4,2,3,2>|<-2,-1,0,2>|<1,1,-1,-2>|<-4,-1,1,5>>;
:= A 







 







4 -2 1 -4
2 -1 1 -1
3 0 -1 1
2 2 -2 5
> B:=<3,1,-3,-6>;
:= B 







 







3
1
-3
-6
> GaussianElimination(A);














 












4 -2 1 -4
0 3
2 -7
4 4
0 0 1
2 1
0 0 0 -3
> GaussianElimination(A,'method'='FractionFree');








 







4 -2 1 -4
0 6 -7 16
0 0 3 6
0 0 0 -9
> ReducedRowEchelonForm(<A|B>);





1 0 0 0 0
0 1 0 0 2
0 0 1 0 5
3
0 0 0 1 -4
3
>
1. Maple 12,dast urida misollarni Kramer usulida y echish.
Krameer usulida dast avval determenant topiladi va x1
matritsa kirgizladi va x1 ning xam diterminanti topiladi va x4
gacha giterminanti topiladi va
Orqali x,y,z topiladi,

> d:=<<2,5,4>|<3,-2,-1>|<-3,2,1>>; := d






2 3 -3
5 -2 2
4 -1 1
> d:=Determinant(d);
:= d 0
> dx1:=<<0,20,1>|<-8,5,0>|<0,-2,6>>;
:= dx1 




 




0 -8 0
20 5 -2
1 0 6
> d1:=Determinant(dx1);
:= d1 976
> dx2:=<<1,5,0>|<5,2,7>|<-8,0,2>>;
>
:= dx2






1 5 -8
5 2 0
0 7 2
> d2:=Determinant(dx2);
:= d2 -326
> dx3:=<<5,0,9>|<-7,0,6>|<-1,6,0>>;
:= dx3






5 -7 -1
0 0 6
9 6 0
> d3:=Determinant(dx3);
:= d3 -558
> x:=d1|d;y:=d2|d;z:=d3|d;

3. Matritsaviy tenglamani echish

Quyidagi matritsaviy tenglamani undaagi koeffitsent
matritsaga teskari matritsa topish bilan yechamiz.
A = X=

,
asosida yechish uchun oldin ni topamiz va unga X matritsani
ko‘paytiramiz. Bu amalni Maple 12 dasturida quyidagicha bajaramiz:
Dastavval A matritsa yozib olinadi.
>
> A:=<<1,2,-1>|<-1,0,2>|<-8,2,3>>;
:= A 




 




1 -1 -8
2 0 2
-1 2 3
> A^(-1);













 












1
7 13
28 1
14
2
7 5
28 9
14
-1
7 1
28 -1
14
> X:=<<-1,0,2>|<0,-2,0>|<1,4,7>>;
:= X 




 




-1 0 1
0 -2 4
2 0 7
> B:=A^(-1).X;

$:= B                            0 -13 14 5 2 1 -5 14 11 2 0 -1 14 -1 2 Oxirgi qiladigan ishimiz B ni topishimiz uchun M va X ni ko’paytirib olamiz. Misollar . 1-misol > eq:={x^2-y^2=0,4-x*y=0}; := eq { } ,  4 xy 0  x2 y2 0 > s:=solve(eq,{x,y}); s { } , x 2 ( ) RootOf ,  _Z 2 1  label _L1 y 2 ( ) RootOf ,  _Z 2 1  label _L1 , := { } , x 2 y 2 { } , x -2 y -2 , > 2-MISOL > restart; > eq:={x+y=4,1/x+1/y=1}; := eq { },x y 4 1 x 1 y 1 > s:=solve(eq,{x,y}); := s , { } , x 2 y 2 { } , x 2 y 2 > 3-misol > eq:={2*x^2-y^2=46,x*y=10}; := eq { } ,  xy 10   2x2 y2 46 > s:=solve(eq,{x,y}); s { } , x 5 y 2 { } , x -5 y -2 , , := { } , x  ( ) RootOf ,  _Z 2 2  label _L1 y 5 ( ) RootOf ,  _Z 2 2  label _L1 > Xulosa$

Chiziqli tenglamalarni maple dasturida ishlashni kurib
chiqdim , bu tenglamalarni Gauss usuli, 2) Kramer, 3) Matritsa
usulida misollarni ishlab chiqdim.
Ushbu amaliy mashg’ulotda Maple matematik paketi yordamida
tenglamalarni va tenglamalar sistemasini yechishning bir nechta usulini
o’rgandim.Bu amaliy mashg’ulotda matritsaviy tenglamalarni Maple
matematik paketi yordamida yechishni ko’rib chiqdim . Ushbu
laboratoriya ishida Maple matematik paketi yordamida tenglamalarni va
tenglamalar sistemasini yechishning bir nechta usuli o’rgandim.
Matritsali tenglamani yechish o’rgandim.

Foydalanilgan adabiyotlar
1. Матросов А. Решение задач математики и механики в среде
Maple 6. СПб.: Питер, 2000.
2. В.З. АЛАДЬЕВ. Основы программирования в Maple. Таллинн,
2006.
3. Основы использования математического пакета Maple в
моделировании: Учебное пособие / Международный институт
компьютерных технологий. Липецк, 2006. 119с.
4. Дьяконов В. Maple 6. Учебный курс СПб.: Питер, 2001.
2. https://arxiv.uz/ru/documents/referatlar/algebra/maple-tizimida-
tenglama-va-tengsizliklarni-tenglama-va-tengsizliklar-sistemasini-
yechish
3. https://hozir.org/mavzu-mapleda-tenglamalar-sistemasi-va-
tenglamalarni-yechish.html
4. Матросов А. Решение задач математики и механики в среде
Maple 6.
СПб.: Питер, 2000.
5. В.З. АЛАДЬЕВ. Основы программирования в Maple. Таллинн,
2006.
3. Основы использования математического пакета Maple в
моделировании:
Учебное пособие / Международный институт компьютерных
технологий.
Липецк, 2006. 119с.
6. Дьяконов В. Maple
7. Учебный курс СПб.: Питер, 2001.
8. Аладьев В.З., Лиопо В.А., Никитин А.В. Математический пакет
Maple в
физическом моделировании.- Гродно: Гродненский госу-
дарственный
университет им. Янки Купалы , 2002, 416 с.

9. O’runbayev E., Murodov F. Kompyuter algebrasi tizimlarining
amaliy
tadbiqlari. –SamDU nashri – Samarqand, 2003, 96 s.

Reja: Kirish. 1.Chiziqli tenglamalar sistemasini Gauss usulida hisoblash 2.Chiziqli tenglamalar sistemasini Kramer usulida hisoblash 3.Matritsa tenglamasini teskari matritsa toppish usulida hisoblash Xulosa . Foydalanilgan adabiyotlar.

1. CHIZI Q LI TENGLAMALAR SISTEMACINI Y ECHISH Chiziqli tenglamalar sistemasini Gauss, Kramer va teskari matritsa usullari bilan y echimini topish masala la rini ko‘ramiz. 1.1-masala. Quyidagi uch noma’lumli chiziqli tenglamalar sistemasi ning yechimini: 1) Gauss usuli, 2) Kramer, 3) Matritsa usulida toping.             39 16 25 5 18 12 14 3 0 13 7 2 3 2 1 3 2 1 3 2 1 x x x x x x x x x (1) 1.1. Gauss usulida y echish. Yetakchi tenglama uchun birinchi tenglamani olamiz. Bu tenglamadan etakchi noma’lum uchun x 1 va a 11 ≠0 ni etakchi element uchun tanlaymiz. Birinchi tenglama dagi x 1 ning koeffitsenti a 11 ni 1 ga aylantirish uchun birinchi tenglamani ng barcha qo‘shiluvchilarini a 11 ≠0 ga b o‘ lamiz . Ҳosil bo‘lgan tenglamadan foydalanib ikkinchi va uchinchi tenglmalardan x 1 nomahlumni yo‘qotish yoki uning koeffitsentini nolg‘ga aylantirish uchun etakchi tenglamani –3 ga ko‘paytirib 2- tenglamaga qo‘shamiz, so‘ngra etakchi tenglamani –5 ko‘paytirib 3- teglamaga qo‘shamiz. Natijada quyidagicha sistemaga kelamiz:                39 2 33 2 15 18 2 15 2 7 0 13 7 2 3 2 3 2 3 2 1 x x x x x x x Bu tenglamlar sistemasida etakchi tenglama uchu 2- tenglamani olamiz. Unda 7/2 koeffitsient l i x 2                7 3 7 3 18 2 15 2 7 0 13 7 2 332 321 x x x x x x bu sistemaning 3- tenglamasidan x 3 nomahlumni topamiz. x 3 asosida 2- tenglamadan x 2 ni topamiz. x 3 , x 2 lar asosida 1- tenglamadan x 1 ni topamiz. x 3 = - 1 x 2 = 3 )2 15 18(7 2 ) 2 15 18(7 2 3     x x 1 = 4 ))1 ( 13 3 7( 2 1 ) 13 7( 2 1 3 2         x x Demak sistema echimi: x 1 =-4, x 2 =3, x 3 = -1

Maple12 dasturida masalani echish. Uch noma’lumli chiziqli tenglamalar sistemasini oddiy va Gauss usulida echish( 1 . 1 - masala). 1 2 3 1 2 3 1 2 32 7 13 0, 3 14 12 18, 5 25 16 39, ő ő ő ő ő ő ő ő ő               Maple7 dasturida masalalarni echishdagi amallarni bajarish uchun ishchi oynada > belgidan so‘ng kerakli buyruqni yozib Enter tugmasini bosish kerak. 1. Oddiy usulida echish ( Gauss.mw ). > solve( {2*x + 7*y + 13*z = 0, 3*x + 14*y + 12*z =18, 5*x + 25*y +16*z =39}, [x, y, z]); [x=-4, y=3, z=-1] 2. Gauss usulida uch noma’lumli chiziqli tenglamalar sistemasini echish . > with(LinearAlgebra): A := <<2,3,5>|<7,14,25>|<13,12,16>>; 2 7 13 : 3 14 12 5 25 16 A         > B := <0,18,39>; 0 : 18 39 b         > GaussianElimination(A); 2 7 13 7 15 0 3 2 3 0 0 7              > GaussianElimination(A,'method'='FractionFree'); 2 7 13 0 7 15 0 0 3             >ReducedRowEchelonForm(<A|b>); 1 0 0 4 0 1 0 3 0 0 1 1            

3. To‘rt noma’lumli chiziqli tenglamalar sistemasini Maple12 dasturida echish 1) Oddiy usulida echish > sys:=({1*x1-5*x2-1*x3+3*x4=-5,2*x1+3*x2+1*x3-1*x4=4, 3*x1-2*x2+3*x3+4*x4=- 1,5*x1+3*x2+2*x3+2*x4=0}): > solve(sys,{x1,x2,x3,x4});{x4 = K 3,x2 = K 1,x1 = 1,x3 = 2} 2) Gauss usulida echish       0.= 2x+ 2x+ 3x+ 5x -1,= 4x+ 3x+ 2x- 3x 4,= x- x3+ 3x+ 2x -5,= 3x+ x- 5x- x 4 3 2 1 4 3 2 1 4 2 1 4 3 2 1 > with(LinearAlgebra): A := <<1,2,3,5>|<-5,3,-2,3>|<-1,1,3,2>|<3,-1,4,2>>; 1 5 1 3 2 3 1 1 : 3 2 3 4 5 3 2 2A                  > b := <-5,4,-1,0>; 5 4 : 1 0 B             > GaussianElimination(A,'method'='FractionFree');                39/ 67 0 0 0 2 3 0 0 7 3 13 0 3 1 5 1 > ReducedRowEchelonForm(<A|b>);               3 0 0 0 2 1 0 0 1 0 1 0 1 0 0 1

1. 2. Kramer q oidasi yordamida echish. Berilgan tenglamalar sistema nomahlumlarning koeffitsientlari va ozod hadalari yordamida determinantlarni tuzamiz va ularni hisoblashning uchburchak yoki Sarrus usullaridan foydalanamiz. Biz (1) tenglamlar sistemasining determinantlarini tuzib, uchburchak usulida hisolab uni son qiymatlarni topamiz. 16255 21143 1375  3 336 600 910 420 975 448 16 7 3 25 2 12 13 14 5 5 12 7 13 25 3 26 14 2                           12 2016 0 7098 5850 3276 0 16 25 39 12 14 18 13 7 0 1         x 99360117015210576 163916 12183 1302 2  x x 3 = 3 819 900 0 630 0 1092 39 25 5 18 14 3 0 7 2        Maple12 dasturida masalani echish. > with(Student[LinearAlgebra]): > d := <<2,3,5>|<7,14,25>|<13,12,16>>; > d:=Determinant(d);                39/ 67 0 0 0 2 3 0 0 7 3 13 0 3 1 5 1 > ReducedRowEchelonForm(<A|b>);               3 0 0 0 2 1 0 0 1 0 1 0 1 0 0 1

CHIZIQLI TENGLAMALAR SISTEMACINI YECHISH

08.08.2023

0

146.291015625 KB

Похожие