## Recent results in fine-grained rounding error analysis

#### Dr. Claude-Pierre Jeannerod (Inria researcher in the AriC team at LIP - ENS de Lyon. )

Since the 1950s and Wilkinson's development of backward error analysis,
accuracy and stability results have been most often obtained using the
so-called standard model of floating-point arithmetic, and conveniently
expressed with bounds involving quantities of the form $\gamma_h := hu/(1-hu)$, where $u$ is the unit roundoff and $h$ typically depends on
the size of the problem. Recently, it was realized that for various core
algorithms these traditional worst-case bounds can be replaced by
simpler and sharper ones, sometimes without any size restriction like
$hu < 1$ or with $h$ provably minimal. This talk will survey these
results as well as some of the underlying techniques, and focus in
particular on the following aspects:

1. The best-known error bounds and their conditions of applicability
for various algorithms in linear algebra and complex arithmetic.
2. The influence of some IEEE floating-point features (fused
multiply-add, rounding modes) on fine-grained error analysis.
3. The construction of certificates parametrized by the precision
and aimed at showing that the bounds obtained are sharp or
reasonably sharp.