Comparison of time complexity growth for different methods/algorithms for rectangular determinant calculations

Abstract

This article examines the asymptotic time complexity development for several approaches to compute the rectangular determinants. In this analysis are taken into consideration the Cullis/Radic method which has asymptotic time complexity of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O\left(C\binom {n}{m}\cdot m^{3}\right)$</tex> , Laplace method with the asymptotic time complexity calculated as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(m!\cdot(n-m))$</tex> , Chios-like method that has asymptotic time complexity as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$o(m^{2}\cdot n\cdot(n-m))$</tex> and Dodgson's condensation methods which have asymptotic time complexity of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(2^{2m}\cdot(n-m)^{2})$</tex> . According to the calculations of time complexity growth for different order of matrices, the Chios-like method is the most useful method for computing rectangular matrices' determinant.