Interprecision transfers in iterative refinement
C. T. Kelley
TL;DR
This work extends iterative refinement by explicitly modeling interprecision transfers, examining how storing the matrix and its factorization in a low precision while evaluating the residual in a higher precision yields a promoted problem with novel convergence insights. It introduces IR-V1, which makes all interprecision transfers explicit and analyzes cases such as when triangular solves operate in the residual precision (TS = TR > TW), showing equivalence to solving the promoted problem and providing guidance on stability and termination. The paper also assesses transfer costs, contrasts on-the-fly and in-place strategies, and presents an integral-equation example demonstrating practical timing tradeoffs. Overall, the framework clarifies how precision choices affect convergence, storage, and performance in mixed-precision iterative refinement, with implications for exascale and high-performance linear solvers.
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.