A New LU Decomposition on Hybrid GPU-Accelerated Multicore Systems
Héctor Eduardo González, Juan Robledo Carmona · Redalyc (Universidad Autónoma del Estado de México) · 2013
"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. These algorithms present an O n 3 para obtener una Matriz Diagonal D. Usando esta matriz, escalamos la Matriz Elemental Triangular Inferior obtenida durante el proceso de eliminación de Doolittle-Gauss y deducimos una Nueva Matriz Elemental Triangular Inferior. Aplicando esta transformación lineal a la matriz A, por medio de una estrategia de pivoteo total, se obtiene el determinante de A como si hubiera sido calculado a través de la Expansión de Laplace. Si aplicamos esta nueva transformación lineal y la estrategia de pivoteo anteriormente mencionada a la matriz aumentada (A|b) obtenemos la solución de la Regla de Cramer aplicada a un Sistema de Ecuaciones Lineales. Estos algoritmos presentan una complejidad computacional O n3 cuando A,b R se calcula en Sistemas Multi-Core Acelerados con GPU. n computational complexity when A,b Rn on hybrid"