extrapolating from the arithmetic and geometric means
[under construction: needs arithmetic checked]
The usual norms/means are the root-sum/mean-power:
Suppose we try the fractional powers,
for example the half-norm, of two values, may be the
root-sum-square-root: square(sqrt(|a|)+sqrt(|b|)) ...
but this is just |a|+|b|+sqrt(|a*b|),
the sum of the 'Manhattan' norm -
double the arithmetic mean - and the geometric mean.
- the Euclidean norm is the two-norm, or
root-sum-square: sqrt(square(a)+square(b)), and
the Euclidean mean is the root-mean-square:
the mean replacing the sum:
sqrt(sum(square(ai))/N) : i=1,...,N;
- the 'Manhattan' norm is the one-norm, or
root-sum-absolute-values: (|a|+|b|), and
the 'Manhattan' mean is the absolute-average;
- the maximum value is the infinity-norm, or
or (a^N+b^N)^(1/N) n->oo -
its mean equals its norm.
© 1996 GrandAdmiralPetry@Lanthus.net