无法理解以下代码。 以下代码如何创建正定矩阵

Question

//Cannot understand use of this function

    for(int i=0; i<n; i++) {
        for(int j=0; j<n; j++) {
            double sum = 0;
            for(int k=0; k<n; k++) {
                //Why is i*n+k used here?
                sum += A[i*n+k]*A[j*n+k];
            }
            C[i*n+j] = sum;

int main() {
    

    double *m4 = (double*)malloc(sizeof(double)*n*n);
//Why was gemm_ATA function used here?
    gemm_ATA(m3, m4, n); //make a positive-definite matrix
    printf("\n");
    //show_matrix(m4,n);

    
    
}

I am making a project for parallelizing Cholesky method and found a useful code. In the given, project this function is used and I have no idea why is it used. Also, can someone help me understand the code and its function used in the code given in the link:- http://coliru.stacked-crooked.com/a/6f5750c20d456da9

Answer 1

The function gemm_ATA takes an input matrix A and calculates C = A^T * A , which is positive semi-definite by definition (note the semi-definiteness, depending on the properties of the input matrix).

Mathematically, calculating this matrix would be:

c_i,j = sum_k  a_k,i * a_k,j

c_i,j is the entry of C in the i -th row and j -th column. The expressions i*n+k and j*n+k transform these 2D indices (row and column) to a 1D index of the underlying array.

Answer 2

gemm_ATA calculate AA^T and stores it in C . A^T is A but flipped over its diagonal. So A*A^T multiply each row of matrix A (call A[-,i] ) with each column of the matrix A (call A^T[j,-] ) which is also the row of A ( A[-,j] ).
Also, if you flatten an n*n 2D matrix to 1D matrix, you can access the first element of each i-th row by i*n+0 . So if you want k-th column of ith row you have it with A[i*n+k] .
Note that since you pass C by reference to your function, after calling the function, m4 is your positive definite matrix created from m3 .