Interprecision transfers in iterative refinement

TL;DR

This paper explicitly analyzes interprecision transfers in iterative refinement algorithms, providing new insights into their behavior, convergence, and variants, especially regarding precision promotion and triangular solve approaches.

Contribution

It introduces an explicit algorithmic framework for interprecision transfers in iterative refinement, offering novel convergence analysis and insights into algorithmic variants.

Findings

Iterative refinement with promoted precision solves the original problem in a modified form.

Explicit interprecision transfer analysis clarifies convergence behavior.

New approaches for interprecision transfer in triangular solves are discussed.

Abstract

We make the interprecision transfers explicit in an algorithmic description of iterative refinement and obtain new insights into the algorithm. One example is the classic variant of iterative refinement where the matrix and the factorization are stored in a working precision and the residual is evaluated in a higher precision. In that case we make the observation that this algorithm will solve a promoted form of the original problem and thereby characterize the limiting behavior in a novel way and obtain a different version of the classic convergence analysis. We also discuss two approaches for interprecision transfer in the triangular solves.