Hardware-Accelerated Algorithm for Complex Function Roots Density Graph Plotting

TL;DR

This paper introduces a hardware-accelerated FPGA algorithm for plotting complex function roots density graphs, significantly improving energy efficiency over CPU methods and optimizing QR decomposition for small complex Hessenberg matrices.

Contribution

The paper presents a novel FPGA-based architecture that accelerates complex roots density graph plotting by optimizing QR decomposition for small complex Hessenberg matrices.

Findings

Achieves up to 65x higher energy efficiency than CPU-based methods.

Processes 16.5 million QR decompositions per second for 6x6 matrices.

Outperforms prior solutions in energy efficiency despite lower raw performance.

Abstract

Solving and visualizing the potential roots of complex functions is essential in both theoretical and applied domains, yet often computationally intensive. We present a hardware-accelerated algorithm for complex function roots density graph plotting by approximating functions with polynomials and solving their roots using single-shift QR iteration. By leveraging the Hessenberg structure of companion matrices and optimizing QR decomposition with Givens rotations, we design a pipelined FPGA architecture capable of processing a large amount of polynomials with high throughput. Our implementation achieves up to 65x higher energy efficiency than CPU-based approaches, and while it trails modern GPUs in performance. Compared with state-of-the-art QR decomposition solutions, our design specificly optimize QR decomposition for complex-valued Hessenberg matrices up to size 6x6, exhibiting a moderate throughput of 16.5M QR decompositions per second, while prior works have predominantly focused on 4x4 general matrices.