On fast multiplication of a matrix by its transpose

J. Dumas, Clément Pernet, A. Sedoglavic
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引用次数: 5

Abstract

We present a non-commutative algorithm for the multiplication of a 2 × 2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any field of prime characteristic. We use geometric considerations on the space of bilinear forms describing 2 × 2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions. Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a prime field. To conclude, we show how to use our result in L · D · LT factorization.
关于矩阵的转置的快速乘法
在C或任何素数特征域上使用5个块积(3个递归调用和2个一般积)进行2 × 2块矩阵的转置乘法,给出了一种非交换算法。我们在描述2 × 2矩阵乘积的双线性形式空间上使用几何考虑来获得该算法,并展示了如何减少所涉及的加法数量。由此产生的任意维数的算法是将矩阵的乘法通过其转置简化为一般矩阵乘积,通过一个常数因子改进先前已知的简化。最后,我们提出了具有低内存占用的调度,以支持在主要字段上的快速和内存高效的实际实现。最后,我们展示了如何在L·D·LT分解中使用我们的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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