Using fast matrix multiplication to find basic solutions

Using Fast Matrix Multiplication to Find Basic Solutions

We consider the problem of nding a basic solution to a system of linear constraints in standard form given a non basic solution to the system We show that the known arithmetic complexity bounds for this problem admit considerable improvement Our technique which is similar in spirit to that used by Vaidya to nd the best complexity bounds for linear programming is based on reducing much of the co...

Fast matrix multiplication using coherent configurations

We introduce a relaxation of the notion of tensor rank, called s-rank, and show that upper bounds on the s-rank of the matrix multiplication tensor imply upper bounds on the ordinary rank. In particular, if the “s-rank exponent of matrix multiplication” equals 2, then ω = 2. This connection between the s-rank exponent and the ordinary exponent enables us to significantly generalize the group-th...

Fast Matrix Multiplication

We give an overview of the history of fast algorithms for matrix multiplication. Along the way, we look at some other fundamental problems in algebraic complexity like polynomial evaluation. This exposition is self-contained. To make it accessible to a broad audience, we only assume a minimal mathematical background: basic linear algebra, familiarity with polynomials in several variables over r...

Fast matrix multiplication

Using fast matrix multiplication to solve structured linear systems

Structured linear algebra techniques enable one to deal at once with various types of matrices, with features such as Toeplitz-, Hankel-, Vandermondeor Cauchy-likeness. Following Kailath, Kung and Morf (1979), the usual way of measuring to what extent a matrix possesses one such structure is through its displacement rank, that is, the rank of its image through a suitable displacement operator. ...