Chiziqli tenglamalar sistemalarini Zeydel usuli uchun dastur ishlab chiqish
![A: koeffitsient matritsasi (n x n)
b: o'ng tomonli vektor (n x 1)
x: dastlabki taxmin vektori (n x 1)
max_iterations: takrorlashlarning maksimal soni
tolerance: konvergentsiyaga tolerantlik
Output:
x: yechim vektori (n x 1)
Procedure SeidelMethod(A, b, x, max_iterations, tolerance):
1. n = number of rows/columns in A
2. iteration = 0
3. error = infinity
4. while iteration < max_iterations and error > tolerance:
iteration = iteration + 1
x_old = copy of x
error = 0
for i = 1 to n do:
5. sum = 0
6. for j = 1 to n do:
if j ≠ i then
7. sum = sum + A[i][j] * x[j]
8. end for
9. x[i] = (b[i] - sum) / A[i][i]
10. error = max(error, abs(x[i] - x_old[i]))
11. end for](https://docx.uz/documents/6a279686-9012-4d35-8e01-3647a124faff/page_10.png?v=1)
![while iteration < max_iterations and error > tolerance:
iteration += 1
x_old = np.copy(x)
error = 0
for i in range(n):
sum = 0
for j in range(n):
if j != i:
sum += A[i][j] * x[j]
x[i] = (b[i] - sum) / A[i][i]
error = max(error, abs(x[i] - x_old[i]))
return x
# Natija:
A = np.array([[10, 2, 1], [1, 5, 1], [2, 3, 10]])
b = np.array([7, -8, 6])
x = np.zeros_like(b)
max_iterations = 100
tolerance = 1e-6
solution = seidel_method(A, b, x, max_iterations, tolerance)
print("Solution:", solution)
Ushbu Python misoli matritsa operatsiyalari uchun NumPy yordamida Zeydel
usulini qanday amalga oshirishni ko'rsatadi. Kirish matritsalarini sozlang A va b,
maksimal takrorlashlar soni va tolerantlik bilan bir qatorda, muayyan muammo
talablariga mos keladi.
C++ da Zeydel usulini amalga oshirishning qisqacha misoli:
#include <iostream>](https://docx.uz/documents/6a279686-9012-4d35-8e01-3647a124faff/page_12.png?v=1)
![#include <vector>
#include <cmath>
using namespace std;
vector<double> seidel_method(const vector<vector<double>>& A, const
vector<double>& b, vector<double> x, int max_iterations, double tolerance) {
int n = b.size();
vector<double> x_old(n);
double error = tolerance + 1;
int iteration = 0;
while (iteration < max_iterations && error > tolerance) {
x_old = x;
error = 0.0;
for (int i = 0; i < n; ++i) {
double sum = 0.0;
for (int j = 0; j < n; ++j) {
if (j != i) {
sum += A[i][j] * x[j];
}
}
x[i] = (b[i] - sum) / A[i][i];
error = max(error, abs(x[i] - x_old[i]));
}
iteration++;
}
return x;
}
int main() {](https://docx.uz/documents/6a279686-9012-4d35-8e01-3647a124faff/page_13.png?v=1)
![vector<vector<double>> A = {{10, 2, 1}, {1, 5, 1}, {2, 3, 10}};
vector<double> b = {7, -8, 6};
vector<double> x(b.size(), 0);
int max_iterations = 100;
double tolerance = 1e-6;
vector<double> solution = seidel_method(A, b, x, max_iterations, tolerance);
cout << "Solution vector:" << endl;
for (int i = 0; i < solution.size(); ++i) {
cout << "x[" << i << "] = " << solution[i] << endl;
}
return 0;
}
qisqacha mazmuni:
• Sarlavhalar : Kirish/chiqarish va ma'lumotlar tuzilmalari uchun zarur bo'lgan
C++ standart sarlavhalarini ( iostream, vector, ) cmath o'z ichiga oladi.
• Funktsiya seidel_method :
• Koeffitsient A matritsasini oladi, b o'ng tomonli vektor, x dastlabki taxmin
vektori, maksimal takrorlash soni va yaqinlashuvga tolerantlik.
• Konvergentsiya mezonlari bajarilmaguncha yoki maksimal iteratsiyaga
erishilgunga qadar Zeydel iteratsiyasini bajaradi.
• x har bir komponentni yangilaydi. Gauss-Zeydel formulasidan foydalanib,
ketma-ket takrorlashlar orasidagi maksimal xatoni hisoblab chiqadi.
• Yechim vektorini qaytaradi.
• Asosiy funksiya ( main) :](https://docx.uz/documents/6a279686-9012-4d35-8e01-3647a124faff/page_14.png?v=1)
![for i = 1:n
sum1 = sum(A(i, 1:i-1) .* x(1:i-1)');
sum2 = sum(A(i, i+1:n) .* x_old(i+1:n)');
x(i) = (b(i) - sum1 - sum2) / A(i, i);
end
if norm(x - x_old, inf) < tol
fprintf('Converged in %d iterations\n', k);
return;
end
end
fprintf('Maximum iterations reached\n');
end
A = [4, -1, 0, 0; -1, 4, -1, 0; 0, -1, 4, -1; 0, 0, -1, 3];
b = [15; 10; 10; 10];
x0 = zeros(size(b));
x = gauss_seidel(A, b, x0)
disp('Solution:')
disp(x)
Ushbu ilovalar o'xshash tuzilishga ega:](https://docx.uz/documents/6a279686-9012-4d35-8e01-3647a124faff/page_16.png?v=1)
![vector<double> seidel_method(const vector<vector<double>>& A, const
vector<double>& b, vector<double> x, int max_iterations, double tolerance,
double omega)
{
int n = b.size();
vector<double> x_old(n);
double error = tolerance + 1;
int iteration = 0;
while (iteration < max_iterations && error > tolerance) {
x_old = x;
error = 0.0;
for (int i = 0; i < n; ++i) {
double sum = 0.0;
for (int j = 0; j < n; ++j) {
if (j != i) {
sum += A[i][j] * x[j];
}
}
x[i] = (1 - omega) * x[i] + (omega / A[i][i]) * (b[i] - sum);
error = max(error, abs(x[i] - x_old[i]));
}
iteration++;
}](https://docx.uz/documents/6a279686-9012-4d35-8e01-3647a124faff/page_25.png?v=1)
![return x;
}
int main() {
vector<vector<double>> A = {{10, 2, 1}, {1, 5, 1}, {2, 3, 10}};
vector<double> b = {7, -8, 6};
vector<double> x(b.size(), 0);
int max_iterations = 100;
double tolerance = 1e-6;
double omega = 1.2;
vector<double> solution = seidel_method(A, b, x, max_iterations, tolerance,
omega);
cout << "Solution vector:" << endl;
for (int i = 0; i < solution.size(); ++i) {
cout << "x[" << i << "] = " << solution[i] << endl;
}
return 0;
}
Zeydel usulida (yoki har qanday iterativ usulda) samaradorlikni oshirish va
xatolarni bartaraf etish algoritmik takomillashtirish, ehtiyotkorlik bilan dasturlash
amaliyoti va raqamli jihatlarni hisobga olish kombinatsiyasini o'z ichiga oladi.
Ushbu strategiyalarni qo'llash orqali siz keng ko'lamli chiziqli tizimni hal qilish
vazifalarini yaxshiroq hal qilish uchun amalga oshirishning ishlashi, ishonchliligi
va aniqligini optimallashtirishingiz mumkin.
Zeydel usulining ko'p qirraliligi va samaradorligi uni turli ilmiy, muhandislik
va hisoblash fanlari bo'yicha qimmatli vositaga aylantiradi. Uning keng miqyosli](https://docx.uz/documents/6a279686-9012-4d35-8e01-3647a124faff/page_26.png?v=1)
Chiziqli tenglamalar sistemalarini Zeydel usuli uchun dastur ishlab chiqish