Hardware-Accelerated Algorithm for Complex Function Roots Density Graph Plotting
Ruibai Tang, Chengbin Quan
TL;DR
The paper proposes a hardware-accelerated pipeline on FPGA to visualize the density of complex function roots by approximating the function with polynomials on the unit disk and computing polynomial roots via single-shift QR iterations on Hessenberg companion matrices using Givens rotations. It demonstrates a streamlined, implicit QR approach that updates matrices in place, reducing storage and increasing throughput, and couples computation with a real-time 1920×1080 visualization module. The FPGA design achieves substantial energy-efficiency gains over a CPU baseline (approximately 65×) and remains competitive with GPUs in throughput, though GPUs still outperform in raw speed. The work provides an open-source, scalable framework for fast root-density visualization and serves as a reference for hardware acceleration of numerical linear algebra in root-finding and visualization tasks.
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.