Editorial StaffLU decomposition is a widely used numerical method in linear algebra for solving systems of linear equations.
Here, we will explore the LU decomposition method and provide a comprehensive C++ program that allows users to input their own matrix.
Understanding this powerful technique can greatly enhance your problem-solving skills in various domains, such as engineering, physics, and computer science.
What is LU Decomposition?
LU decomposition factorizes a square matrix into two triangular matrices: an upper triangular matrix (U) and a lower triangular matrix (L).
The decomposition of a matrix A can be represented as A = LU, where L contains the lower triangular elements, U contains the upper triangular elements, and L has ones on its main diagonal.
LU decomposition is useful for solving linear equations, matrix inversion, and calculating determinants.
Implementing LU Decomposition
To implement LU decomposition in C++, you can use libraries like Eigen or implement it from scratch. In this article, we’ll demonstrate a basic implementation from scratch.
#include <iostream>
#include <vector>
// Perform LU decomposition of a square matrix
void LU_Decomposition(const std::vector<std::vector<double>>& A, std::vector<std::vector<double>>& L, std::vector<std::vector<double>>& U) {
int n = A.size();
// Initialize L and U matrices
L.resize(n, std::vector<double>(n, 0.0));
U.resize(n, std::vector<double>(n, 0.0));
// Perform LU decomposition
for (int i = 0; i < n; i++) {
// Calculate upper triangular matrix (U)
for (int k = i; k < n; k++) {
double sum = 0.0;
for (int j = 0; j < i; j++)
sum += L[i][j] * U[j][k];
U[i][k] = A[i][k] - sum;
}
// Calculate lower triangular matrix (L)
for (int k = i; k < n; k++) {
if (i == k)
L[i][i] = 1.0;
else {
double sum = 0.0;
for (int j = 0; j < i; j++)
sum += L[k][j] * U[j][i];
L[k][i] = (A[k][i] - sum) / U[i][i];
}
}
}
}
int main() {
// Get the matrix size from the user
int n;
std::cout << "Enter the size of the square matrix: ";
std::cin >> n;
// Get the matrix elements from the user
std::vector<std::vector<double>> A(n, std::vector<double>(n, 0.0));
std::cout << "Enter the elements of the matrix:\n";
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
std::cin >> A[i][j];
}
}
// Perform LU decomposition
std::vector<std::vector<double>> L, U;
LU_Decomposition(A, L, U);
// Print L and U matrices
std::cout << "L matrix:\n";
for (const auto& row : L) {
for (double val : row)
std::cout << val << " ";
std::cout << std::endl;
}
std::cout << "U matrix:\n";
for (const auto& row : U) {
for (double val : row)
std::cout << val << " ";
std::cout << std::endl;
}
return 0;
}
