# Numerically Stable Recurrence Relations for the Communication Hiding Pipelined Conjugate Gradient Method

## TL;DR

This paper introduces a numerically stable variant of the pipelined Conjugate Gradient method, enhancing accuracy and parallel performance for large-scale linear system solutions without increasing computational cost.

## Contribution

It presents a new two-term recurrence relation that improves numerical stability of the pipelined Conjugate Gradient method, enabling high accuracy regardless of pipeline length.

## Key findings

- Achieves high accuracy independently of pipeline length
- Demonstrates excellent parallel performance
- Resolves stability issues in pipelined Krylov methods

## Abstract

Pipelined Krylov subspace methods (also referred to as communication-hiding methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in parallel. For symmetric and positive definite system matrices the pipelined Conjugate Gradient method outperforms its classic Conjugate Gradient counterpart on large scale distributed memory hardware by overlapping global communication with essential computations like the matrix-vector product, thus hiding global communication. A well-known drawback of the pipelining technique is the (possibly significant) loss of numerical stability. In this work a numerically stable variant of the pipelined Conjugate Gradient algorithm is presented that avoids the propagation of local rounding errors in the finite precision recurrence relations that construct the Krylov subspace basis. The multi-term recurrence relation for the basis vector is replaced by two-term recurrences, improving stability without increasing the overall computational cost of the algorithm. The proposed modification ensures that the pipelined Conjugate Gradient method is able to attain a highly accurate solution independently of the pipeline length. Numerical experiments demonstrate a combination of excellent parallel performance and improved maximal attainable accuracy for the new pipelined Conjugate Gradient algorithm. This work thus resolves one of the major practical restrictions for the useability of pipelined Krylov subspace methods.