New parallel algorithms for finding determinants of N×N matrices

Abstract

Determinants has been used intensively in a variety of applications through history. It also influenced many fields of mathematics like linear algebra. Finding the determinants of a squared matrix can be done using a variety of methods, including well-known methods of Leibniz formula and Laplace expansion which calculate the determinant of any N×N matrix in O(n!). However, decomposition methods, such as: LU decomposition, Cholesky decomposition and QR decomposition, have replaced the native methods with a significantly reduced complexity of O(n^3). In this paper, we introduce two parallel algorithms for Laplace expansion and LU decomposition. Then, we analyze them and compare them with their perspective sequential algorithms in terms of run time, speed-up and efficiency, where new algorithms provided better results. At maximum, in Laplace expansion, it became 129% faster, whereas in LU Decomposition, it became 44% faster.