Consider the following code, found on wikipedia, that implements the Crout decomposition algorithm:    void crout(double const **A, double **L, double **U, int n) {    int i, j, k;    double sum = 0;    for (i = 0; i < n; i++) {        U[i][i] = 1;    }    for (j = 0; j < n; j++) {        for (i = j; i < n; i++) {            sum = 0;            for (k = 0; k < j; k++) {                sum = sum + L[i][k] * U[k][j];              }            L[i][j] = A[i][j] - sum;        }        for (i = j; i < n; i++) {            sum = 0;            for(k = 0; k < j; k++) {                sum = sum + L[j][k] * U[k][i];            }            if (L[j][j] == 0) {                printf(det(L) close to 0!\n Can't divide by 0...\n);                exit(EXIT_FAILURE);            }            U[j][i] = (A[j][i] - sum) / L[j][j];        }    }}Which part is responsible for pivoting?The thing is, I've read that partial pivoting is still a rather non-trivial operation (on the order $O(n^2)$), but the above algorithm seems similar in terms of complexity to the naive LU decomposition without any pivoting.To check, I've run the algorithm on my 28x28 test matrix[ 0 0.291601633 0 -0.062262937 0 0 0 0 0 -0.22949092 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ][ 0.291601633 0 -0.062262937 0 0 0 0 0 -0.22949092 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ][ 0 -0.062262937 0 0.633860174 0 -0.203470344 0 -0.185648901 0 -0.182968732 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ][ -0.062262937 0 0.633860174 0 -0.203470344 0 -0.185648901 0 -0.182968732 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ][ 0 0 0 -0.203470344 0 0.386208493 0 -0.18368887 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ][ 0 0 -0.203470344 0 0.386208493 0 -0.18368887 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ][ 0 0 0 -0.185648901 0 -0.18368887 0 1.163032501 0 -0.044175498 0 0 0 -0.20912 0 0 0 -0.55618 0 0 0 0 0 0 0 0 0 0 ][ 0 0 -0.185648901 0 -0.18368887 0 1.163032501 0 -0.044175498 0 0 0 -0.20912 0 0 0 -0.55618 0 0 0 0 0 0 0 0 0 0 0 ][ 0 -0.22949092 0 -0.182968732 0 0 0 -0.044175498 0 0.702145321 0 -0.25202 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ][ -0.22949092 0 -0.182968732 0 0 0 -0.044175498 0 0.702145321 0 -0.25202 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ][ 0 0 0 0 0 0 0 0 0 -0.25202 0 0.883882188 0 0 0 0 0 0 0 0 0 -0.220414179 0 -0.28380561 0 -0.14610303 0 0 ][ 0 0 0 0 0 0 0 0 -0.25202 0 0.883882188 0 0 0 0 0 0 0 0 0 -0.220414179 0 -0.28380561 0 -0.14610303 0 0 0 ][ 0 0 0 0 0 0 0 -0.20912 0 0 0 0 0 0.49528 0 -0.17615 0 -0.11001 0 0 0 0 0 0 0 0 0 0 ][ 0 0 0 0 0 0 -0.20912 0 0 0 0 0 0.49528 0 -0.17615 0 -0.11001 0 0 0 0 0 0 0 0 0 0 0 ][ 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.17615 0 0.17615 0 0 0 0 0 0 0 0 0 0 0 0 ][ 0 0 0 0 0 0 0 0 0 0 0 0 -0.17615 0 0.17615 0 0 0 0 0 0 0 0 0 0 0 0 0 ][ 0 0 0 0 0 0 0 -0.55618 0 0 0 0 0 -0.11001 0 0 0 1.033363204 0 -0.090289125 0 0 0 0 0 0 0 -0.298767964 ][ 0 0 0 0 0 0 -0.55618 0 0 0 0 0 -0.11001 0 0 0 1.033363204 0 -0.090289125 0 0 0 0 0 0 0 -0.298767964 0 ][ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.090289125 0 0.299090395 0 -0.208861407 0 0 0 0 0 0 ][ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.090289125 0 0.299090395 0 -0.208861407 0 0 0 0 0 0 0 ][ 0 0 0 0 0 0 0 0 0 0 0 -0.220414179 0 0 0 0 0 0 0 -0.208861407 0 0.429181968 0 0 0 0 0 0 ][ 0 0 0 0 0 0 0 0 0 0 -0.220414179 0 0 0 0 0 0 0 -0.208861407 0 0.429181968 0 0 0 0 0 0 0 ][ 0 0 0 0 0 0 0 0 0 0 0 -0.28380561 0 0 0 0 0 0 0 0 0 0 0 0.570852385 0 -0.29792224 0 0 ][ 0 0 0 0 0 0 0 0 0 0 -0.28380561 0 0 0 0 0 0 0 0 0 0 0 0.570852385 0 -0.29792224 0 0 0 ][ 0 0 0 0 0 0 0 0 0 0 0 -0.14610303 0 0 0 0 0 0 0 0 0 0 0 -0.29792224 0 0.818338896 0 -0.387730558 ][ 0 0 0 0 0 0 0 0 0 0 -0.14610303 0 0 0 0 0 0 0 0 0 0 0 -0.29792224 0 0.818338896 0 -0.387730558 0 ][ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.298767964 0 0 0 0 0 0 0 -0.387730558 0 0.68647389 ][ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.298767964 0 0 0 0 0 0 0 -0.387730558 0 0.68647389 0 ]which is invertible (you can check here), when it returned a number NaNs and Infinities.
Where is pivoting done in the Crout decomposition algorithm?
matrices