Asymptotic Complexity of Matrix Multiplication
This chapter is devoted to the discussion of fast matrix multiplication algorithms. We define the exponent ω of matrix multiplication, a quantity measuring the asymptotic complexity of this problem. Strassen’s original algorithm gives ω ≤ 2.81 (see Chap. 1). The robustness of the exponent with respect to various cost functions, and its invariance with respect to field extensions is shown. (Further motivation for the investigation of ω is given in Chap. 16.) We introduce the concept of border rank of tensors due to Bini, Capovani, Lotti, and Romani , and deduce Schönhage’s asymptotic sum inequality , which has become one of the main tools for gaining upper estimates of the exponent. Then we present the laser method  and its generalization by Coppersmith and Winograd , and prove their estimate ω < 2.39. Finally, an extension of the asymptotic sum inequality to the multiplication of partially filled matrices is considered. As an application, we describe Coppersmith’s construction of astonishingly rapid algorithms for the multiplication of rectangular matrices.
KeywordsEntropy Filtration Hull Bini
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