Calculate determinant of matrix using an iterative method - c++

Question

I wrote C++ codes that calculates the determinant of matrix using the recursive method.
Now, I would like to rewrite that recursive method using an iterative method. But I cannot figure out how to accomplish that.
The question is: How to rewrite the specific recursive method (int Determinant(int *&, const int)) into an iterative method?
Here are my C++ codes:
// Determinant C++
#include <iostream>
#include <ctime>
using namespace std;
void RandInitArray(int *, const int, int = 0, int = 20);
void Display(int *, const int);
int CalculateDeterminant(int *&, const int);
int Determinant(int *&, const int);
int main()
{
start = clock();
srand((unsigned int)time(NULL));
int N = 12; // Size of matrix
int * S = new int[N * N];
int a(-10), b(10), det;
RandInitArray(S, N, a, b);
cout.precision(4);
Display(S, N);
det = CalculateDeterminant(S, N);
cout << "\nDeterminant = " << det << "\n\n";
cin.get();
return 0;
}
void RandInitArray(int * arr, const int N, int a, int b)
{
for (int i = 0; i < N * N; i++)
arr[i] = rand() % (b - a + 1) + a;
}
void Display(int *arr, const int N)
{
for (int i = 0; i < N * N; i++)
cout << arr[i] << ((i + 1) % N ? "\t" : "\n");
}
int CalculateDeterminant(int *& S, const int N)
{
int rez;
if (N < 1)
cout << "Size of matrix must be positive\n";
else if (N == 1)
rez = *S;
else if (N == 2)
rez = S[0] * S[3] - S[1] * S[2];
else if (N == 3)
rez = S[0] * S[4] * S[8] + S[1] * S[5] * S[6] + S[2] * S[3] * S[7] -
S[2] * S[4] * S[6] - S[1] * S[3] * S[8] - S[0] * S[5] * S[7];
else
rez = Determinant(S, N);
return rez;
}
int Determinant(int *& S, const int N)
{
int sign(1), det(0), res, M(N - 1);
int * _S;
for (int k = 0; k < N; k++)
{
_S = new int[M * M];
int ind = 0;
for (int i = N; i < N * N; i++)
{
if (i % N != k)
_S[ind++] = S[i];
}
if (M == 3)
{
res = S[k] == 0 ? 0 : _S[0] * _S[4] * _S[8] + _S[1] * _S[5] * _S[6] + _S[2] * _S[3] * _S[7] -
_S[2] * _S[4] * _S[6] - _S[1] * _S[3] * _S[8] - _S[0] * _S[5] * _S[7];
delete[] _S;
}
else
res = S[k] == 0 ? 0 : Determinant(_S, N - 1);
det += S[k] * sign * res;
sign *= -1;
}
delete[] S;
return det;
}
I know this is not the smartest way to calculate determinant of a large matrix. I can simplify matrix using matrix transformations and drastically speed up calculations.
Despite that any help or hint would be greatly appreciated.

Answer 1

You should sum up a total of N! product terms, each of which is a unique product N elements none of which reside on same row nor colomn: Sorted in row order, colomn indices are permutations of range [1··N), with odd permutations negated. You can use std::next_permutation to calculate permutation and invert sign per iteration.