On the Worst-case Complexity of Integer Gaussian Elimination

Gaussian elimination is the basis for classical algorithms for computing canonical forms of integer matrices. Experimental results have shown that integer Gaussian elimination may lead to rapid growth of intermediate entries. On the other hand various polynomial time algorithms do exist for such computations, but these algorithms are relatively complicated to describe and understand. Gaussian elimination provides the simplest descriptions of algorithms for this purpose. These algorithms have a nice polynomial number of steps, but the steps deal with long operands. Here we show that there is an exponential length lower bound on the operands for a well-defined variant of Gaussian elimination when applied to Smith and Hermite normal form calculation. We present explicit matrices for which this variant produces exponential length entries. Thus, Gaussian elimination has worst-case exponential space and time complexity for such applications. The analysis provides guidance as to how integer...

• Type: Conference
• Year: 1997
• Where: ISSAC
• Authors: Xin Gui Fang, George Havas
• Keywords: Algorithms, Gaussian Elimination, Integer Gaussian Elimination, ISSAC 1997, Mathematics