Um método para o cálculo da inversa de matrizes em blocos com uso limitado de memória

Abstract

The inversion of extremely high order matrices has been a challenging task because of the limited processing and memory capacity of conventional computers. In a scenario in which the data does not fit in memory, it is worth to consider exchanging more processing time for less memory usage in order to enable the computation of the inverse, which otherwise would be prohibitive. Therefore, this work introduces a novel algorithm to compute the inverse of block partitioned matrices with a reduced memory footprint. The algorithm works recursively to invert one block of a k×k block matrix M, with k ≥ 2, based on the successive splitting of M into lower order matrices. This algorithm, called Block Recursive Inverse (BRI), computes one block of the inverse at a time to limit memory usage during the entire processing. Considering that the low memory consumption, provided by the BRI, is counterbalanced by longer processing time, this work also discusses a parallel implementation of the algorithm in OpenMP to reduce the running time and to extend its applicability. Additionally, an improvement in the sequential algorithm is proposed. As a practical application, the proposed algorithm was applied in the cross-validation process for Least Squares Support Vector Machines (LS-SVM). This computational procedure uses the inverse matrix calculation to find the expected labels of the test samples in the cross-validation. Experimental results with BRI show that, despite increasing computational complexity, matrices that otherwise would exceed the memory-usage limit can be inverted using this technique