Partitioning Method: Computations and Applications

摘要

The original partitioning method has been introduced by Greville in [10]. Later in literature, several different proofs for the Greville's method were presented. Many extensions of the partitioning method have been developed. Three different classes of these extensions can be observed. In the first aspect, there are extensions of the partitioning method to various classes of generalized inverses. On the other hand, various extensions applicable to rational and polynomial matrices have been established. The first result in this approach is the extension on the Greville's algorithm to the set of one-variable polynomial and/or rational matrices, introduced in [37]. The extension of results from [37] to the set of the two-variable rational and polynomial matrices is introduced in [33]. Wang's partitioning method from [55], aimed in the computation of the weighted Moore-Penrose inverse, is extended to the set of one-variable rational and polynomial matrices in the paper [42]. Also an efficient algorithm for computing the weighted Moore-Penrose, appropriate for sparse polynomial matrices where only a few polynomial coefficients are nonzero, is established in [31]. In the paper [43] the Greville's recursive principle is generalized to {1}, {l,3},{l,4}-inverses and the Moore-Penrose inverse and extended to the set of the one-variable rational and polynomial matrices. The third type of generalization is presented in [50], where the recursive algorithm for finding the Moore-Penrose inverse was provided, with the assumption that a block of columns is added in each recursive step. Due to its computational dominance, this method became one of the most efficient algorithms for computing the Moore-Penrose inverse. A lot of applications of the partitioning method and its modifications are known in the literature. An application of the partitioning method in the image deblurring is also developed in this survey.