4 research outputs found
Reducing the Influence of Tiny Normwise Relative Errors on Performance Profiles
It is a widespread but little-noticed phenomenon that the
normwise relative error
of vectors and of floating point numbers,
where is an approximation to ,
can be many orders of magnitude smaller than the unit roundoff.
We analyze this phenomenon and show that in the -norm
it happens precisely when
has components of widely varying magnitude
and every
component of of largest magnitude agrees with the corresponding
component of .
Performance profiles are a popular way to compare competing algorithms
according to particular measures of performance.
We show that performance profiles based on
normwise relative errors
can give a misleading impression
due to the influence of zero or tiny errors.
We propose a transformation
that reduces the influence of these extreme errors in a controlled manner,
while preserving the monotonicity of the underlying data and leaving
the performance profile unchanged at its left end-point.
Numerical examples with both artificial and genuine data illustrate
the benefits of the transformation
Reducing the influence of tiny normwise relative errors on performance profiles
It is a widespread but little-noticed phenomenon that the normwise relative error βx β yβ/βx β of vectors x and y of floating point numbers of the same precision, where y is an approximation to x, can be many orders of magnitude smaller than the unit roundoff. We analyze this phenomenon and show that in the β-norm it happens precisely when x has components of widely varying magnitude and every component of x of largest magnitude agrees with the corresponding component of y. Performance profiles are a popular way to compare competing algorithms according to particular measures of performance. We show that performance profiles based on normwise relative errors can give a misleading impression due to the influence of zero or tiny normwise relative errors. We propose a transformation that reduces the influence of these extreme errors in a controlled manner, while preserving the monotonicity of the underlying data and leaving the performance profile unchanged at its left end-point. Numerical examples with both artificial and genuine data illustrate the benefits of the transformation