Examining Equation 27 and Equation 28, you will observe that LU decomposition will fail when value the $a_{kk}^{(k)}$ (called the pivot element) is zero. In many applications, the possibility of a zero pivot is quite real and constitutes a serious impediment to the use of Gaussian elimination. This problem is compounded by the fact that Gaussian elimination is numerically unstable even if there are no zero pivot elements.

Numerical instability occurs when errors introduced by the finite precision representation of real numbers are of sufficient magnitude to swamp the true solution to a problem. In other words, a numerically unstable problem has a theoretical solution that may be unobtainable in finite precision arithmetic.

The other LU decomposition schemes examined in this section exhibit similar characteristics,
e.g. instability is introduced by the division by u_{jj} in Equation
30
and l_{ii} in Equation
34.