How to Describe Relative Approximation Error? A New Justification for Gustafson’s Logarithmic Expression

How can we describe relative approximation error? When the value b approximate a value a, the usual description of this error is the ratio $$|b-a|/|a|$$ . The problem with this approach is that, contrary to our intuition, we get different numbers gauging how well a approximates b and how well b approximates a. To avoid this problem, John Gustafson proposed to use the logarithmic measure $$|\ln (b/a)|$$ . In this paper, we show that this is, in effect, the only regular scale-invariant way to describe the relative approximation error.