Clock Math — a System for Solving SLEs Exactly

Abstract

In this paper, we present a GPU-accelerated hybrid system that solves ill-conditioned systems of linear equations exactly. Exactly means without rounding errors due to using integer arithmetics. First, we scale floating-point numbers up to integers, then we solve dozens of SLEs within different modular arithmetics and then we assemble sub-solutions back using the Chinese remainder theorem. This approach effectively bypasses current CPU floating-point limitations. The system is capable of solving Hilbert’s matrix without losing a single bit of precision, and with a significant speedup compared to existing CPU solvers.

Keywords

• integer arithmetic
• modular arithmetic
• Hilbert’s matrix
• error-free
• GPGPU
• solver
• OpenCL
• optimizations.