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Determinant of N x N matrix using multi-threading
  • Last Updated : 20 Mar, 2018

Given a matrix of N x N, task is to find the determinant of the matrix using multi-threading.

Examples :

Input : mat = {{0, 4, 0, -3},
               {1, 1, 5, 2},
               {1, -2, 0, 6},
               { 3, 0, 0, 1}}
Output : -250

Input :  mat = {{1, 0, 2, -1},
                {3, 0, 0, 5},
                {2, 1, 4, -3},
                {1, 0, 5, 0}}
Output: 30

Approach :
It is known that finding the determinant of a matrix can be very costly in the amount of time it takes. So, computing things parallely may lead to a comparatively less costly program. A code can be parallelized using threads, which is light weight process and divides a single flow program into a programs with multiple flows. As for finding the determinant of a matrix, first find the determinant of the submatrices of the matrix, distribute the task of finding the determinant of submatrices to the threads, and these threads will run parallelly resulting in less time execution time than the sequential method.

Note :The code is Linux specific.




// CPP program for finding determinant matrix
// with parallalizing the code
#include <iostream>
#include <vector>
#include <pthread.h>
#define size 4
  
using namespace std;
  
// matrix whoe's determinant is required
int mat[][size] = { { 0, 4, 0, -3 },
                    { 1, 1, 5, 2 },
                    { 1, -2, 0, 6 },
                    { 3, 0, 0, 1 } };
int det[size];
  
// declaring variable for storing thread id
pthread_t thread[size];
  
// function for finding determinant
int determinant(vector<vector<int> > mat2, int s)
{
  
    if (s == 2) {
          
        // if size of matrix is 2X2
        // then returning the determinant
        return mat2[0][0] * mat2[1][1] - 
               mat2[0][1] * mat2[1][0];
    }
    else {
          
        // else dividing the matrix in smaller part.
        vector<vector<int> > mat1(s - 1), 
                      mat3(s - 1), mat4(s - 1);
        int k, l, m, i, j;
          
        for (i = 0; i < s - 1; i++) {
            mat1[i] = vector<int>(s - 1);
            mat3[i] = vector<int>(s - 1);
            mat4[i] = vector<int>(s - 1);
        }
          
        for (i = 1; i < s; i++) {
            k = 0;
            l = 0;
            m = 0;
            for (j = 0; j < s; j++) {
                if (j != 0) {
                    mat1[i - 1][k] = mat2[i][j];
                    k++;
                }
                if (j != 1) {
                    mat3[i - 1][l] = mat2[i][j];
                    l++;
                }
                if (j != 2) {
                    mat4[i - 1][m] = mat2[i][j];
                    m++;
                }
            }
        }
  
        return mat2[0][0] * determinant(mat1, s - 1) - 
               mat2[0][1] * determinant(mat3, s - 1) + 
               mat2[0][2] * determinant(mat4, s - 1);
    }
}
  
// function for finding determinant using first row
// with each element of row a thread is associated.
void* createTd(void* arg)
{
    int *ar = (int *)arg, i, j, k;
    vector<vector<int> > mat2(size - 1);
  
    for (i = 0; i < size - 1; i++)
        mat2[i] = vector<int>(size - 1);
          
    // extracting the matrix smaller by size one.
    // for finding the determinant.
    for (i = 1; i < size; i++) {
        k = 0;
        for (j = 0; j < size; j++) {
            if (j != (*ar)) {
                mat2[i - 1][k] = mat[i][j];
                k++;
            }
        }
    }
      
    // calling determinant function
    det[*ar] = det[*ar] * determinant(mat2, size - 1);
}
  
// driver function
int main()
{
    int i, j, detfin = 0;
    int p[size];
      
    // storing the first row in a array
    // for later multipying with the determinant
    // of smaller matrix
    for (i = 0; i < size; i++)
        det[i] = mat[0][i];
          
    // creating thread
    for (i = 0; i < size; i++) {
        p[i] = i;
        pthread_create(&thread[i], NULL, &createTd, (void*)&p[i]);
    }
      
    // waiting for all the threads to join
    pthread_join(thread[0], NULL);
    pthread_join(thread[1], NULL);
    pthread_join(thread[2], NULL);
    pthread_join(thread[3], NULL);
    for (i = 0; i < size; i++) {
        if (i % 2 == 0)
            detfin += det[i];
        else
            detfin -= det[i];
    }
      
    cout << detfin << endl;
      
    return 0;
}

Output:

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