 Bareiss algorithm

The Bareiss algorithm is fraction-free Gaussian elimination for exact determinants. For an integer matrix, put $A^{(0)}=A$ and $p_0=1$. At step $k$, choose a nonzero pivot element $p_k=a_{kk}^{(k-1)}$ and update the trailing entries bySet the eliminated entries below the pivot to zero. The determinant identities behind the method make division by $p_{k-1}$ exact, so intermediate entries remain integers. Without row exchanges, the last pivot $a_{nn}^{(n-1)}$ is $\det A$; row exchanges change its sign in the usual way. A zero pivot needs an exchange before continuing.

For $A=\left(\begin{array}{ccc}2& 1& 0\\ 1& 2& 1\\ 0& 1& 2\end{array}\right)$, the first trailing block becomes $\left(\begin{array}{cc}3& 2\\ 2& 4\end{array}\right)$. The last update is $(3\cdot 4-2\cdot 2)/2=4$, giving the determinant without rational intermediate entries. Exact division is different from having no division at all.^[ref]

The Bareiss algorithm is an efficient method used in numerical linear algebra for the computation of the determinant of a matrix. Developed by Hans Bareiss in the 1960s, this algorithm is particularly notable for its use of rational arithmetic, which helps in reducing numerical errors associated with floating-point computations.