A generalization of the ABS algorithms and its application to some special real and integer matrix factorizations

In 1984, Abaffy, Broyden, and Spediacto (ABS) introduced a class of the so-called ABS algorithms to solve systems of real linear equations. Later, the scaled ABS algorithm, the extended ABS algorithm, the block ABS algorithm, and the integer ABS algorithm were introduced leading to various well-known matrix factorizations. Here, we present a generalization of ABS algorithms containing all matrix factorizations such as triangular, WZ, and ZW. We present the octant interlocking factorization and show that the generalized ABS algorithm is more general to produce the octant interlocking factorization.