couldn't find the determinant of n*n matrix

Question

I have written the following code to calculate the determinant of a N*N matrix. it works perfectly for the matrix 4*4 and 5*5. but it could not find the determinant of the 40*40 matrix named Z_z. The elements of matrix Z_z are presented herein.

#include <iostream>
int const N_M=40;
void getCofactor(double A[N_M][N_M], double A_rc[N_M][N_M], int r,int c, int n) 
{ int i = 0, j = 0; 
// Looping for each element of the matrix 
for (int row = 0; row < n; row++) 
{ 
    for (int col = 0; col < n; col++) 
    { 
        //  Copying into temporary matrix only those element 
        //  which are not in given row and column 
        if (row != r && col != c) 
        { 
            A_rc[i][j] = A[row][col]; 
            j=j+1;
            // Row is filled, so increase row index and 
            // reset col index 
            if (j == n - 1) 
            { 
                j = 0; 
                i=i+1; 
            } 
        } 
    } 
} 
}

double determinant(double A[N_M][N_M], int n) 
{ double D = 0.0; // Initialize result

//  Base case : if matrix contains single element
if (n==1) return A[0][0];
else if (n == 2) return (A[0][0]*A[1][1])-(A[0][1]*A[1][0]); 
else {
double sub_Matrix_A_0c[N_M][N_M]; // To store cofactors 

 // Iterate for each element of first row 
for (int c = 0; c < n; c++) 
{ 
    // Getting Cofactor of A[0][f] 
    getCofactor(A, sub_Matrix_A_0c, 0, c, n); 
    D += pow(-1.0,c) * A[0][c] * determinant(sub_Matrix_A_0c, n - 1); 


} 

return D;} 
}

int main () {
double Z_z[N_M][N_M]=

{{-64,16,-4,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {-43.7213019529827,12.4106746539480,-3.52287874528034,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,-43.7213019529827,12.4106746539480,-3.52287874528034,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,-27,9,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,-27,9,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,-16.0579142269798,6.36491716338729,-2.52287874528034,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,-16.0579142269798,6.36491716338729,-2.52287874528034,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,-8,4,-2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8,4,-2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3.53179897265895,2.31915967282662,-1.52287874528034,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3.53179897265895,2.31915967282662,-1.52287874528034,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-0.142956190020121,0.273402182265940,-0.522878745280338,1,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-0.142956190020121,0.273402182265940,-0.522878745280338,1,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2.20222621658426,1.69267904961742,1.30102999566398,1}, {37.2320239618439,-7.04575749056068,1,0,-37.2320239618439,7.04575749056068,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,27,-6,1,0,-27,6,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,19.0947514901619,-5.04575749056068,1,0,-19.0947514901619,5.04575749056068,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,12,-4,1,0,-12,4,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6.95747901847985,-3.04575749056068,1,0,-6.95747901847985,3.04575749056068,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,-2,1,0,-3,2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.820206546797821,-1.04575749056068,1,0,-0.820206546797821,1.04575749056068,-1,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,2,1,0,-3,-2,-1,0}, {-21.1372724716820,2,0,0,21.1372724716820,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,-18,2,0,0,18,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,-15.1372724716820,2,0,0,15.1372724716820,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,-12,2,0,0,12,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-9.13727247168203,2,0,0,9.13727247168203,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-6,2,0,0,6,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3.13727247168203,2,0,0,3.13727247168203,-2,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,-2,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,2,0,0,-6,-2,0,0}, {24,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-7.80617997398389,-2,0,0}};

double det=determinant(Z_z, 40); 
cout<<det;

system ("pause");
return 0;}

Answer 1

You recursively call determinant() functuon n times at the first stage, then n - 1 times for each of n calls, etc. So total number of call would be closed to n! (factorial).

When n = 4 or n = 5 the number of calls is still acceptable, but at n = 40 you try to make 40! = 815915283247897734345611269596115894272000000000 virtual calls to say nothing about so many operations of any kind. I don't think you can find a machine to handle that.