Numerical cubature from Archimedes' hat-box theorem

Details Numerical cubature from Archimedes' hat-box theorem Numerical cubature from Archimedes' hat-box theorem

2004-05-19

arxiv.org/abs/math/0405366v2

Mathematics

Numerical Analysis

Dedicated to Krystyna Kuperberg on the occasion of her 60th birthday. This version has many minor changes

Scientific paper

Archimedes' hat-box theorem states that uniform measure on a sphere projects to uniform measure on an interval. This fact can be used to derive Simpson's rule. We present various constructions of, and lower bounds for, numerical cubature formulas using moment maps as a generalization of Archimedes' theorem. We realize some well-known cubature formulas on simplices as projections of spherical designs. We combine cubature formulas on simplices and tori to make new formulas on spheres. In particular $S^n$ admits a 7-cubature formula (sometimes a 7-design) with $O(n^4)$ points. We establish a local lower bound on the density of a PI cubature formula on a simplex using the moment map. Along the way we establish other quadrature and cubature results of independent interest. For each $t$, we construct a lattice trigonometric $(2t+1)$-cubature formula in $n$ dimensions with $O(n^t)$ points. We derive a variant of the M\"oller lower bound using vector bundles. And we show that Gaussian quadrature is very sharply locally optimal among positive quadrature formulas.

Affiliated with

Also associated with

No associations

Rating

Numerical cubature from Archimedes' hat-box theorem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Numerical cubature from Archimedes' hat-box theorem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Numerical cubature from Archimedes' hat-box theorem will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-254906