Numerical linear algebra is fundamental to computer and computational science and is the central tool of many application fields. This is well illustrated by the popularity of numerical linear algebra software (LINPACK, LAPACK, and so on) and turnkey packages (MATLAB, Maple, and Mathematica). Nevertheless, there are issues in obtaining the maximum accuracy for an arbitrary matrix. The paper provides a substantial contribution to the bridge involving classical error analysis, software standards development for mixed-precision floating point computation, the design of algorithmic implementation for iterative refinement techniques, and methods for extensive testing over a wide range of generated matrices. It thus covers the spectrum from classical numerical linear algebra to the tools for extending these methods in an upcoming release of a new version of LAPACK.

In summary, the authors present an algorithm for iterative refinement in solving linear systems that uses extended precision to compute the residual. There are, however, two obstacles to using it. First is the need for software standards for mixed precision in libraries such as LAPACK. However, recent extensions of the basic linear algebra subprograms library address this issue. Second, there has not been an available error bound for the computed result. This paper focuses on this issue. Furthermore, the paper includes a matrix-generation approach, and generates two million matrices designed to test various numerical properties and the capability of the algorithms to handle these cases. The results are presented in an extensive collection of histograms, which are interpreted in detail.