3-Mavzu: Mapleda algebra va sonlar nazariyasi masalalarini yechish. Matrisalar ustida amallarni bajarish

3-Mavzu: Mapleda algebra va sonlar nazariyasi masalalarini yechish. Matrisalar ustida amallarni bajarish.

Chiziqli tenglamalar sistemasini Gauss, Kramer va teskari matritsa usullari bilan echimini topish masalarini ko‘ramiz.

1.1-masala. Quyidagi uch nomahlumli chiziqli tenglamalar sistemasining echimini:
1) Gauss usuli, 2) Kramer, 3) Matritsa usulida toping.
(1)
Echish:

1. 
   Gauss usulida echish.
   

Etakchi tenglama uchun birinchi tenglamani olamiz. Bu tenglamadan etakchi nomahlum uchun x₁ va a₁₁≠0 ni etakchi element uchun tanlaymiz. Birinchi tenglamadagi x₁ ning koeffitsenti a₁₁ ni 1 ga aylantirish uchun birinchi tenglamaning barcha qo‘shiluvchilarini a₁₁≠0 ga bo‘lamiz. Ҳosil bo‘lgan tenglamadan foydalanib ikkinchi va uchinchi tenglmalardan x₁ 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:

Bu tenglamlar sistemasida etakchi tenglama uchu 2- tenglamani olamiz. Unda 7/2 koeffitsientli x₂ nomahlumni etakchi element uchun olib, 2- tenglamani 7/2 ga bo‘lib hosil bo‘lgan etakchi tenglamani –15 ga ko‘paytirib 3- tenglamaga qo‘shamiz:

bu sistemaning 3- tenglamasidan x₃ nomahlumni topamiz. x₃ asosida 2- tenglamadan x₂ ni topamiz. x₃, x₂ lar asosida 1- tenglamadan x₁ ni topamiz.

x₃ = -1

x₂ =
x₁=

Demak sistema echimi: x₁=-4, x₂=3, x₃= -1

Maple12 dasturida masalani echish.
Uch noma’lumli chiziqli tenglamalar sistemasini oddiy va Gauss usulida echish(1.1- masala).

Maple12 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>>;

> B := <0,18,39>;

> GaussianElimination(A);

> GaussianElimination(A,'method'='FractionFree');

>ReducedRowEchelonForm();

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});

2) Gauss usulida echish

> with(LinearAlgebra):
A := <<1,2,3,5>|<-5,3,-2,3>|<-1,1,3,2>|<3,-1,4,2>>;



> b := <-5,4,-1,0>;

> GaussianElimination(A,'method'='FractionFree');



> ReducedRowEchelonForm();



1.2. Kramer qoidasi 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.

₃ =


Maple12 dasturida masalani echish.


> with(Student[LinearAlgebra]):
> d := <<2,3,5>|<7,14,25>|<13,12,16>>;
> d:=Determinant(d);


> dx1:=<<0,18,39>|<7,14,25>|<13,12,16>>;
> d1:=Determinant(dx1);


> dx2 := <<2,3,5>|<0,18,39>|<13,12,16>>;
> d2:=Determinant(dx2);


> dx3 := <<2,3,5>|<7,14,25>|<0,18,39>>;
> d3:=Determinant(dx3);


Kramer koidasiga asosnan sistema echimini topmamiz:
, ,
> x:=d1/d;y:=d2/d;z:=d3/d;