The problem is to compute the determinants of thousands of small complex matrices (`5x5`

, `4x4`

, and `3x3`

) within a device function, that is on each thread. Here is a working example:

```
#include <stdio.h>
#include <complex.h>
#include <cstring>
#include <cuda.h>
#include <cuComplex.h>
#include <cuda_runtime.h>
#include <device_launch_parameters.h>
#define N 5
__device__ cuDoubleComplex det_4x4(cuDoubleComplex *matrix) {
cuDoubleComplex det = make_cuDoubleComplex(0.0, 0.0);
cuDoubleComplex det1 = make_cuDoubleComplex(0.0, 0.0);
cuDoubleComplex det2 = make_cuDoubleComplex(0.0, 0.0);
cuDoubleComplex det3 = make_cuDoubleComplex(0.0, 0.0);
cuDoubleComplex det4 = make_cuDoubleComplex(0.0, 0.0);
// Calculate the determinant using cofactor expansions
det1 = cuCmul(matrix[0],
cuCadd( cuCsub( cuCmul( matrix[5],
cuCsub( cuCmul(matrix[10], matrix[15]),
cuCmul(matrix[11], matrix[14]) ))
, cuCmul(matrix[6],
cuCsub( cuCmul(matrix[9], matrix[15]),
cuCmul(matrix[11], matrix[13]) ) ) )
, cuCmul(matrix[7],
cuCsub( cuCmul(matrix[9], matrix[14]),
cuCmul(matrix[10], matrix[13]) ) ) ) );
det2 = cuCmul(matrix[1],
cuCadd( cuCsub( cuCmul( matrix[4],
cuCsub( cuCmul(matrix[10], matrix[15]),
cuCmul(matrix[11], matrix[14]) ))
, cuCmul(matrix[6],
cuCsub( cuCmul(matrix[8], matrix[15]),
cuCmul(matrix[11], matrix[12]) ) ) )
, cuCmul(matrix[7],
cuCsub( cuCmul(matrix[8], matrix[14]),
cuCmul(matrix[10], matrix[12]) ) ) ) );
det3 = cuCmul(matrix[2],
cuCadd( cuCsub( cuCmul( matrix[4],
cuCsub( cuCmul(matrix[9], matrix[15]),
cuCmul(matrix[11], matrix[13]) ))
, cuCmul(matrix[5],
cuCsub( cuCmul(matrix[8], matrix[15]),
cuCmul(matrix[11], matrix[12]) ) ) )
, cuCmul(matrix[7],
cuCsub( cuCmul(matrix[8], matrix[13]),
cuCmul(matrix[9], matrix[12]) ) ) ) );
det4 = cuCmul(matrix[3],
cuCadd( cuCsub( cuCmul( matrix[4],
cuCsub( cuCmul(matrix[9], matrix[14]),
cuCmul(matrix[10], matrix[13]) ))
, cuCmul(matrix[5],
cuCsub( cuCmul(matrix[8], matrix[14]),
cuCmul(matrix[10], matrix[12]) ) ) )
, cuCmul(matrix[6],
cuCsub( cuCmul(matrix[8], matrix[13]),
cuCmul(matrix[9], matrix[12]) ) ) ) );
det = cuCsub( cuCadd(det1, det3), cuCadd(det2, det4) );
return det;
}
__global__ void det_5x5(cuDoubleComplex *matrix, cuDoubleComplex *res) {
cuDoubleComplex det = make_cuDoubleComplex(0.0, 0.0);
printf("\t sum %.16E,%.16E \n",
cuCreal(matrix[24]),
cuCimag(matrix[24]));
// Calculate the determinant using cofactor expansions
for (int col = 0; col < N; col++) {
cuDoubleComplex submatrix[N - 1][N - 1];
int subrow = 0, subcol = 0;
// Create the submatrix by excluding the current row and column
for (int row = 1; row < N; row++) {
for (int c = 0; c < N; c++) {
if (c == col) {
continue;
}
submatrix[subrow][subcol] = matrix[row * N + c];
subcol++;
}
subrow++;
subcol = 0;
}
// Calculate the cofactor
cuDoubleComplex det_sign = make_cuDoubleComplex((col % 2 == 0 ? 1 : -1), 0.0);
cuDoubleComplex cofactor = cuCmul( cuCmul(det_sign, matrix[col]), det_4x4(&submatrix[0][0]));
// Accumulate the cofactors to calculate the determinant
det = cuCadd(det, cofactor);
}
res[0] = det;
return;
}
int main() {
double complex matrix[N * N] = {
2 + 2 * I, 3 + 0 * I, 1 + 0 * I, 5 + 0 * I, 7 + 0 * I,
4 + 0 * I, 5 + 0 * I, 3 + 0 * I, 2 + 0 * I, 8 + 0 * I,
1 + 0 * I, 3 + 0 * I, 2 + 0 * I, 4 + 0 * I, 3 + 0 * I,
3 + 0 * I, 4 + 0 * I, 3 + 0 * I, 1 + 0 * I, 2 + 0 * I,
5 + 0 * I, 6 + 0 * I, 4 + 0 * I, 3 + 0 * I, 2 + 2 * I
};
// double complex *matrix;
double complex *det;
det = (double complex*)malloc(sizeof(double complex));
memset(det, 0.0, sizeof(double complex));
cuDoubleComplex *d_matrix, *d_res;
cudaMalloc(&d_matrix, sizeof(cuDoubleComplex) * 25 );
cudaMalloc(&d_res, sizeof(cuDoubleComplex) );
cudaMemcpy(d_matrix, matrix, sizeof(cuDoubleComplex) * 25,
cudaMemcpyHostToDevice );
det_5x5 <<< 1, 1>>>(d_matrix, d_res);
cudaMemcpy(det, d_res, sizeof(cuDoubleComplex), cudaMemcpyDeviceToHost );
printf("The determinant is: %.16E + %.16E i\n", creal(det[0]), cimag(det[0]));
return 0;
}
```

However, I am wondering if this implementation is even efficient. I am seeking more efficient methods to calculate the determinants of small complex matrices (`5x5`

, `4x4`

, and `3x3`

) within a device array. Any suggestion or answers would be greatly appreciated!