Chiziqli algebraik tenglamalar tizimini yechishning bosh elementlar usuli uchun dastur ishlab chiqish
![double determinant (const vector<vector<double>>& matrix) {
int n = matrix.size ();
vector<vector<double>> temp = matrix;
double det = 1;
for (int i = 0; i < n; i++) {
int maxRow = i;
for (int k = i + 1; k < n; k++) {
if (abs(temp[k][i]) > abs(temp[maxRow][i])) { // std::abs dan foydalanish
maxRow = k;
}
}
if (i != maxRow) {
swap(temp[i], temp[maxRow]);
det = -det;
}
det *= temp[i][i];
for (int k = i + 1; k < n; k++) {
double factor = temp[k][i] / temp[i][i];
for (int j = i; j < n; j++) {
temp[k][j] -= factor * temp[i][j];
}
}
}
return det;
}
// Kramer usuli yordamida chiziqli tenglamalar tizimini yechish
vector<double> cramer(vector<vector<double>> A, vector<double> B) {
int n = A.size();
double detA = determinant(A);
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![if (detA == 0) {
cerr << "System has no unique solution" << endl;
exit(1);
}
vector<double> X(n);
for (int i = 0; i < n; i++) {
vector<vector<double>> Ai = A;
for (int j = 0; j < n; j++) {
Ai[j][i] = B[j];
}
X[i] = determinant(Ai) / detA;
}
return X;
}
int main () {
vector<vector<double>> A = {
{2, -1, 0},
{-1, 2, -1},
{0, -1, 2}
};
vector<double> B = {1, 0, 1};
vector<double> X = cramer(A, B);
for (int i = 0; i < X.size(); i++) {
cout << "x[" << i << "] = " << X[i] << endl;
}
return 0;
}
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![Aniqroq tushuntirish maqsadida, quyidagi C++ kodlarida Gauss eliminatsiya usulini
amalga oshirishni ko'rsataman. Bu dastur matritsa va burchaklarni o'zlashtirib, Gauss
eliminatsiya usuli orqali noma'lumlar qiymatlarini topadi va chiqaradi.
#include <iostream>
#include <vector>
#include <cmath>
using namespace std;
// Gauss eliminatsiya usuli yordamida chiziqli tenglamalar tizimini yechish
void gaussElimination(vector<vector<double>>& A, vector<double>& b) {
int n = A.size();
// Uchburchak ko'rinishga keltirish
for (int i = 0; i < n; i++) {
// Pivot elementini topish
int maxRow = i;
for (int k = i + 1; k < n; k++) {
if (fabs(A[k][i]) > fabs(A[maxRow][i])) {
maxRow = k;
}
}
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![// Qatorlarni almashtirish
swap(A[maxRow], A[i]);
swap(b[maxRow], b[i]);
// Qatorlarni modifikatsiya qilish
for (int k = i + 1; k < n; k++) {
double factor = A[k][i] / A[i][i];
for (int j = i; j < n; j++) {
A[k][j] -= factor * A[i][j];
}
b[k] -= factor * b[i];
}
}
// Orqadan o'rinli yechim topish
vector<double> x(n);
for (int i = n - 1; i >= 0; i--) {
x[i] = b[i];
for (int j = i + 1; j < n; j++) {
x[i] -= A[i][j] * x[j];
}
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![x[i] /= A[i][i];
}
// Yechimni chiqarish
for (int i = 0; i < n; i++) {
cout << "x[" << i << "] = " << x[i] << endl;
}
}
int main () {
vector<vector<double>> A = {
{2, -1, 0},
{-1, 2, -1},
{0, -1, 2}
};
vector<double> b = {1, 0, 1};
gauss Elimination (A, b);
return 0;
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![vector<vector<double>> elements;
public:
// Constructor
Matrix(vector<vector<double>> elems) : elements(elems) {}
// Getter method to access elements
double get (int row, int col) const {
return elements[row][col];
}
// Setter method to modify elements
void set (int row, int col, double value) {
elements[row][col] = value;
}
// Method to get the number of rows
int nomer Rows () const {
return elements.size ();
}
// Method to get the number of columns
int numCols () const {
if (elements.empty ()) {
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![return 0;
}
return elements[0].size();
}
// Method to print the matrix
void print() const {
for (int i = 0; i < elements.size(); ++i) {
for (int j = 0; j < elements[i].size(); ++j) {
cout << elements[i][j] << " ";
}
cout << endl;
}
}
};
// Yakobi usulini amalga oshiruvchi funksiya
vector<double> jacobiMethod(const Matrix& A, const vector<double>& B, int
maxIterations) {
int n = A.numRows();
vector<double> X(n, 0); // Boshlang'ich qiymatlar X = [0, 0, ..., 0]
vector<double> newX(n, 0); // Yangi qiymatlar
for (int iter = 0; iter < maxIterations; ++iter) {
for (int i = 0; i < n; ++i) {
double sum = 0.0;
for (int j = 0; j < n; ++j) {
if (j != i) {
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![sum += A.get(i, j) * X[j];
}
}
newX[i] = (B[i] - sum) / A.get(i, i);
}
// Farqni hisoblash
double maxDiff = 0.0;
for (int i = 0; i < n; ++i) {
maxDiff = max (maxDiff, abs(newX[i] - X[i]));
}
// Farq epsilon dan katta bo'lsa, natijani qaytarish
if (maxDiff < EPSILON) {
return newX;
}
// Yangi qiymatlarni o'zgartirish
X = newX;
}
// MaxIterations ga yetganda natija qaytariladi
return newX;
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![1. Qo'shish (+) : Vektorlar va matrislar orasida qo'shish amaliyatini bajaradi.
2. Ayirish (-) : Vektorlar va matrislar orasida ayirish amaliyatini bajaradi.
3. Ko'paytirish (*) : Skalarni vektor yoki matrisga ko'paytirish amaliyatini
bajaradi.
4. Bo'lish (/) : Vektorni skalar (son) bilan bo'lish amaliyatini bajaradi.
C++ dasturlash tili operatorlar yukori amallarni o'z ichiga olgan xususiy funksiyalar
orqali yozilgan klasslar yoki strukturalar bilan birga ishlatiladi. Bu usul obyektlarni
standart algebraik amallarni bajarish uchun tayyorlaydi va dasturlashda matematik
amallarni bajarishni osonlashtiradi.
Masalan, quyidagi C++ kodida, vektorlar uchun qo'shish (+) va ko'paytirish (*)
operatorlaridan foydalanish ko'rsatilgan:
#include <iostream>
#include <vector>
using namespace std;
class Vector {
private:
vector<double> elements;
public:
// Constructor
Vector(vector<double> elems): elements(elems) {}
// Getter method to access elements
double get (int index) const {
return elements[index];
}
// Setter method to modify elements
void set (int index, double value) {
elements[index] = value;
}
// Method to get the size of the vector
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![int size () const {
return element.size ();
}
// Operator overloading for vector addition
Vector operator+ (const Vector& other) const {
if (size ()!= other.size()) {
cerr << "Vector sizes don't match for addition" << endl;
exit (1);
}
vector<double> result;
for (int i = 0; i < size (); ++i) {
result.push_back(elements[i] + other.elements[i]);
}
return Vector(result);
}
// Operator overloading for scalar multiplication
Vector operator*(double scalar) const {
vector<double> result;
for (int i = 0; i < size (); ++i) {
result.push_back(elements[i] * scalar);
}
return Vector(result);
}
};
int main () {
Vector vec1({1, 4, 3});
Vector vec2({4, 5, 6});
// Vector addition
Vector resultAdd = vec1 + vec2;
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Chiziqli algebraik tenglamalar tizimini yechishning bosh elementlar usuli uchun dastur ishlab chiqish