Orthogononalization on a general purpose graphics processing unit with double double and quad double arithmetic

TL;DR

This paper presents a GPU-based implementation of orthogonalization algorithms using extended precision arithmetic to efficiently solve large, overdetermined linear systems arising in polynomial system solving with Newton's method.

Contribution

The paper introduces a massively parallel implementation of the modified Gram-Schmidt orthogonalization method on a GPU using double double and quad double arithmetic for high-precision linear algebra.

Findings

Achieved significant speedups with GPU implementation despite high-precision overhead.

Demonstrated that parallel orthogonalization can compensate for multiprecision arithmetic costs.

Enabled efficient solution of large linear systems in polynomial system solving applications.

Abstract

Our problem is to accurately solve linear systems on a general purpose graphics processing unit with double double and quad double arithmetic. The linear systems originate from the application of Newton's method on polynomial systems. Newton's method is applied as a corrector in a path following method, so the linear systems are solved in sequence and not simultaneously. One solution path may require the solution of thousands of linear systems. In previous work we reported good speedups with our implementation to evaluate and differentiate polynomial systems on the NVIDIA Tesla C2050. Although the cost of evaluation and differentiation often dominates the cost of linear system solving in Newton's method, because of the limited bandwidth of the communication between CPU and GPU, we cannot afford to send the linear system to the CPU for solving during path tracking. Because of large degrees, the Jacobian matrix may contain extreme values, requiring extended precision, leading to a significant overhead. This overhead of multiprecision arithmetic is our main motivation to develop a massively parallel algorithm. To allow overdetermined linear systems we solve linear systems in the least squares sense, computing the QR decomposition of the matrix by the modified Gram-Schmidt algorithm. We describe our implementation of the modified Gram-Schmidt orthogonalization method for the NVIDIA Tesla C2050, using double double and quad double arithmetic. Our experimental results show that the achieved speedups are sufficiently high to compensate for the overhead of one extra level of precision.