Iterative Refinement with Low-Precision Posit Arithmetic

This study examines the mixed-precision iterative refinement technique using posit numbers instead of standard IEEE floating-point. The process is applied to a general linear system \(Ax = b\) where A is a large sparse matrix. Multiple scaling strategies, including row and column equilibration, scale matrix entries into higher-density regions of machine numbers before performing the \(O(n^3)\) factorization operation. Low-precision LU factorization followed by forward/backward substitution yields an initial estimate. The residual \(r = b - Ax\) is computed to a higher precision with a deferred rounding mechanism, then used as the right-hand side in a new linear system \(Ac = r\) . The corrector c is calculated and used to refine the previous solution. Results show a 16-bit posit configuration coupled with equilibration yields accuracy comparable to IEEE half-precision (fp16), showing potential for balancing efficiency and accuracy.