Automatic Differentiation (AD) is a technique that computes the derivatives of numerical programs by systematically applying the chain rule, playing a critical role in domains such as machine learning, simulation, and control systems. However, parallelizing differentiated programs remains a significant challenge due to the conflict between tapes (a data structure for intermediate variable storage) and summations: the differentiation process inherently introduces inter-thread summation patterns, which require prohibitively expensive atomic operations; and traditional tape designs tightly couple data retrieval with the program’s control flow, preventing code restructuring needed to eliminate these costly dependencies.

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 65× 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.