[Math] Applications of group theory in numerical analysis

Question

Are applications of group theory known to exist in numerical analysis?
One particular aspect I am curious about is whether matrix groups have been successfully used to derive algorithms.
Also, are there aspects of numerical analysis that would have been difficult to conceive or derive WITHOUT the help of group theory?

Answer 1

Alain Connes& K mentions some "Butcher group" ( http://arxiv.org/abs/hep-th/9904044 ):

... We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.

See also:

On the Hopf Algebraic Structure of Lie Group Integrators H. Z. Munthe-Kaas, W. M. Wright

Hopf algebras of formal diffeomorphisms and numerical integration on manifolds Alexander Lundervold, Hans Munthe-Kaas

Hopf algebras are in some sense "almost groups":) so hopefully this should qualify.

In general it seems to me that these series of works by Connes, Kreimer, Broadhurst, Moscovich & K on the Hopf algebras in various fields of math is quite fascinating...