Communication efficient application of sequences of planar rotations to a matrix

TL;DR

This paper introduces a highly efficient algorithm for applying sequences of planar rotations to matrices, optimizing for register reuse, cache efficiency, and providing thorough analysis to enhance performance in numerical linear algebra tasks.

Contribution

The paper presents a novel algorithm with optimized kernel, blocking scheme, and detailed memory analysis, outperforming existing methods in applying planar rotations.

Findings

Achieves near-peak flop rate on modern hardware

Outperforms state-of-the-art algorithms in efficiency

Provides theoretical insights into memory operations

Abstract

We present an efficient algorithm for the application of sequences of planar rotations to a matrix. Applying such sequences efficiently is important in many numerical linear algebra algorithms for eigenvalues. Our algorithm is novel in three main ways. First, we introduce a new kernel that is optimized for register reuse in a novel way. Second, we introduce a blocking and packing scheme that improves the cache efficiency of the algorithm. Finally, we thoroughly analyze the memory operations of the algorithm which leads to important theoretical insights and makes it easier to select good parameters. Numerical experiments show that our algorithm outperforms the state-of-the-art and achieves a flop rate close to the theoretical peak on modern hardware.