Correction of 'J. Laderman, V. Pan, X. H. Sha, On practical Algorithms for Accelerated Matrix Multiplication, Linear Algebra and its Applications. Vol. 162-164 (1992) pp. 557-588'

TL;DR

This paper corrects a previously published trilinear formula for matrix multiplication algorithms, improving their accuracy and practical applicability despite a slight increase in computational complexity.

Contribution

The authors identify and correct errors in the original formulas for triple disjoint matrix multiplication, providing explicit bilinear forms for better practical use.

Findings

Corrected the trilinear formula for matrix multiplication for dimensions ≥ 2

Provided explicit bilinear formulas for practical implementation

Slightly increased algorithm time complexity due to corrections

Abstract

In this article we corrected the trilinear formula for triple disjoint matrix multiplication given in the article 'J. Laderman, V. Pan, X. H. Sha, On practical Algorithms for Accelerated Matrix Multiplication, Linear Algebra and its Applications. Vol. 162-164 (1992) pp. 557-588', which is incorrect for matrix dimensions equal to two or greater. That formula is a base of two algorithms, for disjoint and single matrix multiplication. The necessary correction made the amount of non scalar products raise, slightly increasing the algorithm time complexity. We also corrected explicit formulas, in the bilinear form, for triple disjoint matrix multiplication. They are explicit, thus convenient for practical use of fast matrix multiplication algorithms in question.