We explore the link between data representation and soft errors in dot
products. We present an analytic model for the absolute error introduced should
a soft error corrupt a bit in an IEEE-754 floating-point number. We show how
this finding relates to the fundamental linear algebra concepts of
normalization and matrix equilibration. We present a case study illustrating
that the probability of experiencing a large error in a dot product is
minimized when both vectors are normalized. Furthermore, when data is
normalized we show that the absolute error is less than one or very large,
which allows us to detect large errors. We demonstrate how this finding can be
used by instrumenting the GMRES iterative solver. We count all possible errors
that can be introduced through faults in arithmetic in the computationally
intensive orthogonalization phase, and show that when scaling is used the
absolute error can be bounded above by one
