Successive Matrix Squaring Algorithm for Computing the Generalized InverseAT,S(2)

Abstract

We investigate successive matrix squaring (SMS) algorithms for computing the generalized inverseAT,S(2)of a given matrixA∈Cm×n.

Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses

“…A lot of iterative methods for computing outer inverses with the prescribed range and null space have been developed. An outline of these numerical methods can be found in [5][6][7][8][9][10][11][12][13].…”

Approximating the Inverse of a Square Matrix with Application in Computation of the Moore-Penrose Inverse

“…Two important matters must be mentioned at this moment to ease up the perception of why a higher order (efficient) method such as (9) with 7 matrix-matrix products to reach at least the convergence order 9 is practical. First, by following the comparative index of informational efficiency for inverse finders [25], defined by = / , wherein and stand for the convergence order and the number of matrixmatrix products, then the informational index for (9), that is, 9/7 ≈ 1.28, beats its other competitors, 2/2 = 1 of (1), 3/3 = 1 of (2)-(3), and 3/4 = 0.75 of (4). And second, the significance of the new scheme will be displayed in its implementation.…”

“…Though there are certain and efficient ways for finding 0 , in general such initial approximations take a high number of iterations (see e.g., Figure 3, the blue color) to arrive at the convergence phase. On the other hand, each cycle of the implementation of such Schulz-type methods includes one stopping criterion based on the use of a matrix norm, and this would impose further burden and load in general, for the low order methods in contrast to the high order methods, such as (9). Because the computation of a matrix norm (usually ‖ ⋅ ‖ 2 for dense complex matrices and ‖ ⋅ ‖ for large sparse matrices) takes time and therefore higher number of steps/iterations (which is the result of lower order methods), it will be costlier than the lower number of steps/iterations of high order methods.…”

“…Much of the application of these solvers (specially the Schulz method) for structured matrices was investigated in [7] while the authors in [8] showed that the matrix iteration of Schulz is numerically stable. Further discussion about such iterative schemes might be found at [9][10][11][12].…”

Iterative Method to Find Core-EP Inverse

“…In the recent years, many have attempted to improvise these iterative methods to find MoorePenrose inverse, Drazin inverse and outer inverses with specific column space and row space. A few examples in this regard are work by Stanimirović with his collaborators (see [12,13,16,17]) and others (see, [5,6,11,18,19]). In this article, inspired by Stanimirović's work, we consider core-EP inverse as a special case of an outer inverse to propose an iterative method and discuss its convergence.…”