使用斯特拉森算法的矩阵链乘法混合优化技术
Srinivasarao Thota1, Thulasi Bikku2, Rakshitha T3
1Department of Mathematics, Amrita School of Physical Sciences, Amrita Vishwa Vidyapeetham, Amaravati, Andhra Pradesh, 522503, India.
F1000Research
|April 17, 2025
概括
本研究介绍了一种混合矩阵链乘法 (MCM) 方法,将动态编程与斯特拉森算法相结合. 优化的MCM显著加快了大型矩阵计算的速度,减少了执行时间和内存使用量.
科学领域:
- 计算数学 计算数学 计算数学
- 计算机科学 计算机科学
- 科学计算科学计算
背景情况:
- 矩阵链乘法 (MCM) 在科学计算,图形和机器学习中至关重要.
- 传统的MCM使用了带有Memoization的动态编程 (DP),但对于大矩阵而言,它存在O(n^3) 复杂性.
- 对于大规模的计算,标准矩阵乘法是低效的.
研究的目的:
- 为矩阵链乘法开发一种混合优化技术.
- 通过将Strassen的算法集成到MCM中来加速矩阵乘法.
- 为了减少计算复杂性,并提高大矩阵的效率.
主要方法:
- 一种两阶段的方法: (i) 优化矩阵链顺序,使用上下DP和记忆,以及 (ii) 混合乘法策略.
- 斯特拉森算法的选择性应用 (O(n^2.81)) 对n ≥128.8的矩阵.
- 通过计算实验与传统的MCM和独立的Strassen算法进行比较.
主要成果:
- 与传统方法相比,混合MCM方法实现了显著的加快速度 (4x-8x).
- 对于大规模应用程序来说,已证明减少了内存消耗.
- 保持了数值准确性,同时提高了性能.
结论:
- 拟议的混合MCM方法有效地减少了执行时间和内存使用量.
- 斯特拉森算法的选择性集成提高了大型矩阵的MCM效率.
- 在矩阵运算中为并行计算和GPU加速开辟了道路.
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