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.
Highlights
Solving linear algebraic equations solution is quite a frequent task in numerical mathematics
The underlying idea sounds quite simple – bypass floating point rounding limitations by using integer arithmetics. It consists of three parts – converting floating point numbers into integers, solving multiple systems of linear equations within their modules, and converting sub-solutions back using the Chinese remainder theorem
We benchmarked primarily against linSolve-0.7.15 [7], a highly optimized solver with performance critical parts written in x86 assembler
Summary
Solving linear algebraic equations solution is quite a frequent task in numerical mathematics. One method that we use relies on modular arithmetics [2] in order to solve dense systems of linear equations precisely. The underlying idea sounds quite simple – bypass floating point rounding limitations by using integer arithmetics It consists of three parts – converting floating point numbers into integers, solving multiple systems of linear equations within their modules, and converting sub-solutions back using the Chinese remainder theorem. We present a GPU-accelerated solver of ill-conditioned systems of linear equations.
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