A New LU Decomposition on Hybrid GPU-Accelerated Multicore Systems
Una nueva descomposición LU calculada en sistemas multi-core acelerados con GPU
Héctor Eduardo González and Juan Carmona
Information Technology Department, Instituto Nacional de Investigaciones Nucleares (ININ), AP 18-1027, 11801, D.F., Mexico. eduardo.gonzalez@inin.gob.mx, juan.carmona@inin.gob.mx
]]>Article received on 15/02/2013;
accepted on 10/08/2013.
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. These algorithms present an O(n3) computational complexity when (A,b)⊂Rn on hybrid GPU-accelerated multicore systems.
Keywords: New LU theorem, Cramer rule, Gauss elimination, Laplace expansion, determinants, GPU, multicore systems.
Resumen
En este trabajo se postula un Nuevo Teorema de Descomposición de una Matriz A en dos matrices: una Matriz triangular inferior M, cuyas entradas son todas expresadas en forma de determinantes, y una matriz triangular superior U cuyas entradas están también expresadas en forma de determinantes. A partir de un muy conocido Teorema sobre los elementos pivotales del proceso de eliminación de Doolittle-Gauss, deducimos un corolario
]]> 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)⊂Rn se calcula en Sistemas Multi-Core Acelerados con GPU.Palabras clave: Nuevo teorema LU, regla de Cramer, eliminación de Gauss, expansión de Laplace, determinantes, GPU, sistemas Multi-Core.
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