### A New LU Decomposition on Hybrid GPU-Accelerated Multicore Systems

#### Abstract

In this paper, we postulate a new decomposition theorem of a matrix A into two matrices, namely, a lower triangular matrix M, in which all entries are determinants, and an upper triangular matrix U whose entries are also in determinant form. From a well-known theorem on the pivot elements of the Doolittle-Gauss elimination process, we deduce a corollary to obtain a diagonal matrix D. With it, we scale the elementary lower triangular matrix of the Doolittle-Gauss elimination process and deduce a new elementary lower triangular matrix. Applying this linear transformation to A by means of both minimum and complete pivoting strategies, we obtain the determinant of A as if it had been calculated by means of a Laplace expansion. If we apply this new linear transformation and the above pivot strategy to an augmented matrix (A|b), we obtain a Cramer’s solution of the linear system of equations.

#### Keywords

New LU theorem, Cramer rule, Gauss elimination, Laplace expansion, determinants, GPU, multicore systems.