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Matrix multiplication is one of the most fundamental algorithmic problems in numerical linear algebra, distributed computing, scientific computing, and high-performance computing. Parallelization of matrix multiplication has been extensively studied (e.g., [21, 12, 24, 2, 51, 39, 36, 23, 45, 61]). It has been addressed using many theoretical approaches, algorithmic tools, and software engineering methods in order to optimize performance and obtain faster and more efficient parallel algorithms and implementations. To design efficient parallel algorithms, it is necessary not only to load balance the computation, but also to minimize the time spent communicating between processors. The interprocessor communication costs are in many cases significantly higher than the computational costs. Moreover, hardware trends predict that more problems will become communication-bound in the future [38, 35]. Even matrix multiplication, which is widely considered to be computation-bound, becomes communication-bound when a given problem is run on sufficiently many processors.

We establish basic information about border rank algorithms for the matrix multiplication tensor and other tensors with symmetry. We prove that border rank algorithms for tensors with symmetry (such as matrix multiplication and the determinant polynomial) come in families that include representatives with normal forms. These normal forms will be useful both to develop new efficient algorithms and to prove lower complexity bounds. We derive a border rank version of the substitution method used in proving lower bounds for tensor rank. We use this border-substitution method and a normal form to improve the lower bound on the border rank of matrix multiplication by one, to 2n^2- n+1. We also point out difficulties that will be formidable obstacles to future progress on lower complexity bounds for tensors because of the "wild" structure of the Hilbert scheme of points.

We further develop the group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families of wreath product groups that achieve matrix multiplication exponent less than 3, the asymptotically fastest of which achieves exponent 2.41. We present two conjectures regarding specific improvements, one combinatorial and the other algebraic. Either one would imply that the exponent of matrix multiplication is 2.

In this paper we present a class of multiresolution algorithms for fast application of structured dense matrices to arbitrary vectors, which includes the fast wavelet transform of Beylkin, Coifman and Rokhlin and the multilevel matrix multiplication of Brandt and Lubrecht. In designing these algorithms we first apply data compression techniques to the matrix and then show how to compute the desired matrix-vector multiplication from the compressed form of the matrix. In describing this class we pay special attention to an algorithm which is based on discretization by cell-averages as it seems to be suitable for discretization of integral transforms with integrably singular kernels. multiresolution analysis; fast matrix vector multiplication.

Let {\alpha} be the maximal value such that the product of an n x n^{\alpha} matrix by an n^{\alpha} x n matrix can be computed with n^{2+o(1)} arithmetic operations. In this paper we show that \alpha>0.30298, which improves the previous record \alpha>0.29462 by Coppersmith (Journal of Complexity, 1997). More generally, we construct a new algorithm for multiplying an n x n^k matrix by an n^k x n matrix, for any value k\neq 1. The complexity of this algorithm is better than all known algorithms for rectangular matrix multiplication. In the case of square matrix multiplication (i.e., for k=1), we recover exactly the complexity of the algorithm by Coppersmith and Winograd (Journal of Symbolic Computation, 1990). These new upper bounds can be used to improve the time complexity of several known algorithms that rely on rectangular matrix multiplication. For example, we directly obtain a O(n^{2.5302})-time algorithm for the all-pairs shortest paths problem over directed graphs with small integer weights, improving over the O(n^{2.575})-time algorithm by Zwick (JACM 2002), and also improve the time complexity of sparse square matrix multiplication.

Strassen's and Winograd's algorithms for matrix multiplication are investigated and compared with the normal algorithm. Floating - point error bounds are obtained, and it is shown that scaling is essential for numerical accuracy using Winograd's method. In practical cases Winograd's method appears to be slightly faster than the other two methods, but the gain is, at most, about 20%. An attempt to generalize Strassen's method is described.

We make an in-depth study of the known border rank (i.e. approximate) algorithms for the matrix multiplication tensor encoding the multiplication of an n x 2 matrix by a 2 x 2 matrix.

1Seven Stairs

By Stuart Brent

An Autobiography of Stuart Brent, who in 1946 followed his bliss and opened an independent bookstore in Chicago, which became very popular and highly regarded. Filled with stories and anecdotes of rubbing elbows with celebrities, starting a weekly TV show in which he reviewed books, along with the challenges of finding balance with raising a family. A must read for lovers and supporters of those quirky independent bookstores. - Summary by Phyllis Vincelli