Evaluating the Numerical Stability of Posit Arithmetic

The Posit number format has been proposed by John Gustafson as an alternative to the IEEE 754 standard floating-point format. Posits offer a unique form of tapered precision whereas IEEE floating-point numbers provide the same relative precision across most of their representational range. Posits are argued to have a variety of advantages including better numerical stability and simpler exception handling. The objective of this paper is to evaluate the numerical stability of Posits for solving linear systems where we evaluate Conjugate Gradient Method to demonstrate an iterative solver and Cholesky-Factorization to demonstrate a direct solver. We show that Posits do not consistently improve stability across a wide range of matrices, but we demonstrate that a simple rescaling of the underlying matrix improves convergence rates for Conjugate Gradient Method and reduces backward error for Cholesky Factorization. We also demonstrate that 16-bit Posit outperforms Float16 for mixed precision iterative refinement especially when used in conjunction with a recently proposed matrix re-scaling strategy proposed by Nicholas Higham.