About the generalized LM-inverse and the weighted Moore-Penrose inverse

TL;DR

This paper introduces a new recursive algorithm for computing the generalized LM-inverse of matrices, demonstrating its equivalence to existing methods and evaluating its efficiency through computational experiments.

Contribution

The paper develops a sequential algorithm for the generalized LM-inverse and proves its equivalence to the weighted Moore-Penrose inverse algorithm, with implementation and performance analysis.

Findings

The new algorithm is equivalent to the Wang and Chen method.

Implementation in MATHEMATICA shows efficient computation.

Performance varies with matrix type and size.

Abstract

The recursive method for computing the generalized LM-inverse of a constant rectangular matrix augmented by a column vector is proposed in Udwadia and Phohomsiri (2007) [16] and [17]. The corresponding algorithm for the sequential determination of the generalized LM-inverse is established in the present paper. We prove that the introduced algorithm for computing the generalized LM-inverse and the algorithm for the computation of the weighted Moore-Penrose inverse developed by Wang and Chen (1986) in [23] are equivalent algorithms. Both of the algorithms are implemented in the present paper using the package MATHEMATICA. Several rational test matrices and randomly generated constant matrices are tested and the CPU time is compared and discussed.