This book chapter is published open access.
The largest dense linear systems that are being solved today are of order . Single-precision arithmetic, which has a unit roundoff , is widely used in scientific computing, and half-precision arithmetic, with , is increasingly being exploited as it becomes more readily available in hardware. Standard rounding error bounds for numerical linear algebra algorithms are proportional to , with growing at least linearly with . Therefore we are at the stage where these rounding error bounds are not able to guarantee any accuracy or stability in the computed results for some extreme-scale or low-accuracy computations. We explain how rounding error bounds with much smaller constants can be obtained. Blocked algorithms, which break the data into blocks of size , lead to a reduction in the error constants by a factor or more. Two architectural features also reduce the error constants: extended precision registers and fused multiply–add operations, either at the scalar level or in mixed precision block form. We also discuss a new probabilistic approach to rounding error analysis that provides error constants that are the square roots of those of the worst-case bounds. Combining these different considerations provides new understanding of the numerical stability of extreme scale and low precision computations in numerical linear algebra.