# Bareiss algorithm vs. LU-decomposition

I at the moment try to fully understand the Bareiss algorithm for calculating determinants. One question that came to my mind is the following: Why is LU-decomposition much more often used than the Bareiss algorithm? I mean, they both have a complexity of $$\mathcal{O}{(n^3)}$$, so what's the problem with Bareiss?

• Note that Bareiss is for calculating the determinant of an integer matrix by using only integer arithmetic. – Alone Programmer Feb 26 '20 at 17:24
• So LU-decomposition is used to be able to also calculate determinants of complex matrices? – user34175 Feb 26 '20 at 17:52
• I believe at least scipy.linalg.lu could handle complex matrices as well. – Alone Programmer Feb 26 '20 at 18:03
• Between integers and complex numbers there are real numbers, which are by far the most common case in practice. – Federico Poloni Feb 27 '20 at 7:56
• I know, but I don't really get the comment of @AloneProgrammer, because bareiss works fine with real numbers. – user34175 Feb 27 '20 at 7:58

Moreover, overflow and underflow are often practical issues that discourage the use of determinants (exercise: use your favorite programming language to compute $$\det(0.1 I_{350\times 350})$$). See also this answer of mine on [math.se], which has similar arguments.