Gauss Integration Error

With the “best” selection of points, the value of the first integral, evaluated at N points is

Then the value of the second integral is

Define

So that

The error term is

The errors for each dimension are additive, while the number of function evaluations are multiplicative.

For an n dimensional integral with an equal number of evaluations in each dimension

The error in this integral is

The smallest number of points in each dimension is ~ 2. For n = 6, this is 64 points with an error of

. This does not seem small at all.

Doubling this number to 4 takes 64 times as long using 4096 points for an error of

.

Doubling this number to 8 takes 64 times as long using 262144 points for an error of

N=10⁶, then Well within reach.

12 dimensions – 4 electrons

N=2¹² = 4096

N=4¹² = 262144 - fast

N=5¹²=244,140,625 – noticeable – maybe a days run.

Note that there is no need to generate extensive sets of Gauss Quadrature points.