三角矩阵符号函数计算的高性能算法
High-Performance Algorithms for Computing the Sign Function of Triangular Matrices
AI总结：
本文提出基于Parlett替代算法的新型递归和缓存高效算法，用于计算三角矩阵的符号函数，通过优化缓存命中率和利用高效矩阵乘法库提升性能。
摘要：
本文提出基于Parlett替代算法的新型递归和缓存高效算法，用于计算三角矩阵的符号函数，通过优化缓存命中率和利用高效矩阵乘法库提升性能。
英文摘要：
Algorithms and implementations for computing the sign function of a triangular matrix are fundamental building blocks in algorithms for computing the sign of arbitrary square real or complex matrices. We present novel recursive and cache efficient algorithms that are based on Higham's stabilized specialization of Parlett's substitution algorithm for computing the sign of a triangular matrix. We show that the new recursive algorithms are asymptotically optimal in terms of the number of cache misses that they generate. One of the novel algorithms that we present performs more arithmetic than the non-recursive version, but this allows it to benefit from calling highly-optimized matrix-multiplication routines; the other performs the same number of operations as the non-recursive version, but it uses custom computational kernels instead. We present implementations of both, as well as a cache-efficient implementation of a block version of Parlett's algorithm. Our experiments show that the blocked and recursive versions are much faster than the previous algorithms, and that the inertia strongly influences their relative performance, as predicted by our analysis.
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