The notion of a generalized (or pseudo) inverse of a matrix extends the idea of the inverse of an ordinary square (and nonsingular) matrix to any matrix. The conventional application areas include linear optimization as well as least squares estimation. Computing the generalized inverse is also important in the distributive associative memory (DAM) paradigm. The paper is concerned with the problem of recursive computation of the generalized inverse, including both recursive addition and recursive deletion.

In section 2 the authors give a formal definition of the generalized inverse. An outline of Rust’s method for nonrecursive computation of the generalized inverse follows. This method serves as the basis for the authors’ newly developed recursive algorithm.

Section 3 is devoted to the main results of the paper in terms of recursive algorithms for the computation of the generalized inverse. A basic alternative representation of the main computational block of Rust’s development is derived first. Section 3.1 treats recursive column addition, and Section 3.2 discusses recursive deletion (elimination of a column). The algorithms are essentially modifications of Rust’s results.

In section 4, the authors compare their algorithm to Greville’s algorithm for recursive addition and to Fletcher’s algorithm for recursive deletion (each algorithm is the inverse of the other). The comparisons are for storage requirements, computational complexity, and numerical stability. Greville’s algorithm is better than the proposed algorithm in terms of storage requirements, but can give erroneous results in determining the dependence or independence of vectors.

In section 5, the authors describe a major application example. They first describe the DAM model in terms of representation and then discuss fault-tolerant aspects of DAM as applied to a new model of database retrieval.

This research paper is clearly written. It is mainly suited to those interested in efficient recursive implementations of the generalized inverse.