Chiziqli algebraik tenglamalar tizimini yechishning Gauss usuli uchun dastur ishlab chiqish
![void gaussElimination(vector<vector<double>> &a, vector<double> &b,
vector<double> &x, int n) {
for (int i = 0; i < n; i++) {
// Maksimal elementni qatorni tanlash
double maxEl = abs(a[i][i]);
int maxRow = i;
for (int k = i + 1; k < n; k++) {
if (abs(a[k][i]) > maxEl) {
maxEl = abs(a[k][i]);
maxRow = k;
}
}
// Maksimal elementni tanlangan qatorni birinchi qator bilan
almashtiramiz
for (int k = i; k < n; k++) {
swap(a[maxRow][k], a[i][k]);
}
swap(b[maxRow], b[i]);
// Diagonal elementlarni 1 ga keltiramiz
for (int k = i + 1; k < n; k++) {
double c = -a[k][i] / a[i][i];
for (int j = i; j < n; j++) {
if (i == j) {
a[k][j] = 0;
} else {
a[k][j] += c * a[i][j];
}
}
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![b[k] += c * b[i];
}
}
// Yechimni hisoblaymiz
for (int i = n - 1; i >= 0; i--) {
x[i] = b[i] / a[i][i];
for (int k = i - 1; k >= 0; k--) {
b[k] -= a[k][i] * x[i];
}
}
}
int main() {
int n;
cout << "Tenglamalar sonini kiriting: ";
cin >> n;
vector<vector<double>> a(n, vector<double>(n));
vector<double> b(n);
vector<double> x(n);
cout << "Koeffitsientlar matritsasini kiriting:\n";
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
cin >> a[i][j];
}
}
cout << "Erkin hadlar vektorini kiriting:\n";
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![for (int i = 0; i < n; i++) {
cin >> b[i];
}
gaussElimination(a, b, x, n);
cout << "Yechim:\n";
for (int i = 0; i < n; i++) {
cout << "x" << i + 1 << " = " << setprecision(6) << x[i] << endl;
}
return 0;
}
Yuqoridagi kod Gauss eliminatsiya usulini foydalanib chiziqli tenglamalar tizimini
yechish uchun yozilgan. Bu usul matematikada keng qo‘llanilib, koeffitsientlar matritsasi
va erkin hadlar vektoridan foydalanib, chiziqli tenglamalar tizimini yechishga yordam
beradi.
Kodni tushuntirish:
Kutubxonalar:
cpp
#include <iostream>
#include <vector>
#include <iomanip>
#include <cmath>
iostream : Konsolga kiritish va chiqarish uchun.
vector : Dinamik massivlarni (vektorlarni) ishlatish uchun.
iomanip : Formatlangan chiqarish uchun (masalan, o‘nli kasr sonlar uchun aniq
format).
cmath : Matematik funksiyalar uchun (masalan, abs() funksiyasi).
Bu funksiyada koeffitsientlar matritsasi (a), erkin hadlar vektori (b), yechim vektori
(x) va tenglamalar soni (n) qabul qilinadi. Quyidagi asosiy bosqichlardan iborat:
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![Dastur natijasi bergan yechim :
x1 = 1
x2 = 1
x3 = 2
Keling, berilgan tenglamalar sistemasini Gauss usuli yordamida qadam-baqadam
yechamiz.
Ushbu sistemani kengaytirilgan matritsa shakliga keltiramiz:
1-qadam: Yuqori uchburchak matritsaga aylantirish
1-pivot elementi: 1
Birinchi qatordagi pivot element - 1.
2-qatorni yangilash:
2-qatorni 1-qatorning 2-multiplikatori bilan kamaytiramiz:
3-qatorni yangilash:
3-qatorni 1-qatorning 3-multiplikatori bilan kamaytiramiz:
2-pivot elementi: 2 (R3[2,2] elementi)
3-qatorni yangilash:
3-qatorni 2-qatorning -2/1-multiplikatori bilan kamaytiramiz:
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Chiziqli algebraik tenglamalar tizimini yechishning Gauss usuli uchun dastur ishlab chiqish